#
Mechanical and Thermal Properties of Low-Density Al_{20+x}Cr_{20-x}Mo_{20-y}Ti_{20}V_{20+y} Alloys

^{*}

## Abstract

**:**

_{20+x}Cr

_{20-x}Mo

_{20-y}Ti

_{20}V

_{20+y}((x, y) = (0, 0), (0, 10), and (10, 15)) were computationally studied to obtain a low density and a better mechanical property. The density functional theory (DFT) method was employed to compute the structural and mechanical properties of the alloys, based on a large unit cell model of randomly distributed elements. Debye–Grüneisen theory was used to study the thermal properties of Al

_{20+x}Cr

_{20-x}Mo

_{20-y}Ti

_{20}V

_{20+y}. The phase diagram calculation shows that all three RHEAs have a single body-centered cubic (BCC) structure at high temperatures ranging from 1000 K to 2000 K. The RHEA Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}has shown a low density of 5.16 g/cm

^{3}and a hardness of 5.56 GPa. The studied RHEAs could be potential candidates for high-temperature application materials where high hardness, ductility, and low density are required.

## 1. Introduction

^{3}) or are brittle at room temperature, which limits their applications. Recently, Xu et al. [6] designed three RHEAs Ti

_{50-x}Al

_{x}V

_{20}Nb

_{20}Mo

_{10}(x = 10, 15, 20) with low densities ranging from 6.01 g/cm

^{3}to 5.876 g/cm

^{3}. These alloys have high strength and good plasticity at high temperatures. So far, Vickers hardness of these alloys has not been reported. Khaled group [7] also designed a low-density nanocrystalline high-entropy alloy Al

_{20}Li

_{20}Mg

_{10}Sc

_{20}Ti

_{30}with a hardness of 5.9 GPa. However, this alloy has a low melting point which is not appropriate for high-temperature applications. Kang et al. [8] prepared an equimolar low-density RHEA AlCrMoTiV. They found a single body-centered cubic (BCC) phase in AlCrMoTiV using the X-ray diffraction (XRD) method. The measured Vickers hardness of AlCrMoTiV is 5.54 GPa. No computer simulation is found on this RHEA AlCrMoTiV. The density and other physical properties of this alloy can be further tuned by optimizing the component concentration. It is highly desirable to develop a better RHEA based on this AlCrMoTiV, with a lower density, a higher melting point, and enhanced hardness and ductility.

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}with the aim of decreasing the density for a more balanced mechanical property when compared to the previously discussed RHEAs. The molar percentages of Mo (10.28 g/cm

^{3}) and Cr (7.14 g/cm

^{3}) were reduced, as they both have high density. The molar percentages of the low-density elements V (6.11 g/cm

^{3}) and Al (2.7 g/cm

^{3}) were increased to ensure the low density of the alloy. As the melting point of Ti is lower than that of V, it was kept constant to keep the alloy highly solidus at high temperatures.

## 2. Computational Methods

## 3. Results and Discussion

#### 3.1. Structural Properties

_{mix}) [38], mixing enthalpy (ΔH

_{mix}) [39], atomic size differences (δ) [40], and valence electron concentration (VEC) [41,42]. For the current RHEAs, corresponding values of ΔS

_{mix}

_{,}ΔH

_{mix}, VEC, and δ were calculated using the following formula, and the results are listed in Table 1:

^{th}and j

^{th}elements, respectively, ${r}_{i}$ is the radius of the i

^{th}element, $\overline{r}$ is averaged atomic radius, and ${\left(VEC\right)}_{i}$ is the valence electron concentration of the i

^{th}element.

_{mix}should be −15 ≤ ΔH

_{mix}≤ 5 kJ/mol, while Guo et al. [41] reported that ΔH

_{mi}

_{x}should be −22 ≤ ΔH

_{mix}≤ 7 kJ/mol. These conditions were statistically calculated, and the data had some deviations in the previous findings [38,42,43,44]. Other criteria to form a stable solid solution are 12 ≤ ΔS

_{mix}≤ 17.5 and δ ≤ 6.6%. Another parameter to predict the crystal structure of HEAs is VEC. If VEC is ≥ 8, the alloy will form a face-centered cubic (FCC) crystal, whereas if VEC is < 6.87, it will form a BCC crystal. Our calculated data in Table 1 show that the present RHEAs could form solid solutions with stable BCC crystal structures.

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}, Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}, and Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}were 2190 K, 2137 K, and 2025 K, respectively. The XRD patterns reported by Kang et al. [8] show the existence of a single BCC crystal structure in Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}. This experimental finding agrees with our prediction for Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}. Considering the experimental agreement and validation of the TCHEA1 database prediction according to a previous publication [19,20,21,22], we expect that future experimental results may confirm our phase prediction findings for Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}and Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}.

#### 3.2. Mechanical Properties

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}, Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}, and Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}were 1447.42 Å

^{3}, 1425.36 Å

^{3}, and 1461.51 Å

^{3}, respectively. From these optimized volumes, lattice constants and densities were calculated. From the calculation, Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}was found to have a low density of 5.16 g/cm

^{3}.

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}is 5.48 GPa while the experimentally measured ${H}_{v}$ value of Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}is 5.55 GPa [8]. The excellent agreement of calculated hardness with the experimental result confirms the reliability of our predicted value of bulk modulus and shear modulus. The excellent agreement of calculated hardness with the experimental result also confirms the reliability of our predicted value of elastic properties. The randomized model of the supercell used in our calculation may be used in predicting the mechanical and thermal properties of other future RHEAs. Our prediction may stimulate further experimental investigations of Al

_{30}Cr

_{10}Mo

_{5}V

_{35}Ti

_{20}, as it has a low density, is very ductile, and has a high melting point.

#### 3.3. Thermal Properties

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}, Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}and Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}RHEAs is shown in Figure 3a. The calculated Debye temperature (θ

_{D}) of Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}, Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}, and Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}were 428 K, 443 K, and 451 K, respectively. The higher value of θ

_{D}in materials reflects the strength of the covalent bond component in materials [50]. This reveals that the covalent bond component inside Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}could be stronger than the other two RHEAs, as it has a higher value of θ

_{D}. We found that the α increases rapidly as temperature increases from 0 K to 200 K, and slowly becomes linear at temperatures above 600 K for the given RHEAs. The value of α for Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}is similar to that of Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}and Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}at temperatures below 200 K, but its α increases rapidly at temperatures above 298 K. This variation of the thermal coefficient of linear expansion indicates that the α of Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}is very sensitive at temperatures above 298 K. It may be due to the addition of low-density elements such as Al and V in the RHEAs. Figure 3b shows the specific heat capacity at constant volume, C

_{v}, as a function of temperature between Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}, Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}, and Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}. At low temperatures below 50 K, the C

_{v}of all three RHEAs has identical behavior. At higher temperatures above 50 K, the C

_{v}of Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}> C

_{v}of Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}> C

_{v}of Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}. This trend implies that the lowering of the high-density elements Mo and Cr increases the C

_{v}of Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}. Since the specific heat capacity of Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}is higher, it can absorb more heat than the other two RHEAs. We are not aware of the experimental report for the thermal properties of the present RHEAs, and future experiments will likely confirm our thermal properties results.

## 4. Conclusions

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}RHEA has a low density of 5.16 g/cm

^{3}, a hardness of 5.56 GPa, and maintains a stable BCC structure at high temperatures of 1000–2000 K. It also has a B/G ratio of 2.7, which indicates its ductile nature. Calculations of thermal properties show that the addition of elements Al and V increases the covalent bond component, thermal expansion coefficient, and the specific heat capacity of Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}when compared with the previous RHEA Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}. The present study reveals that based on structural and physical property calculations, Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35}could be a potential candidate for high-temperature applications. It is expected that further experimental exploration may validate our results on the structural, mechanical and thermal properties of these RHEAs.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of a random 100-atom unit cell model used for densitiy functional theory (DFT) calculations. (

**b**) Plot between total pair distribution function (PDF) of all pairs in the supercell and total PDF of pairs as a function of distance.

**Figure 2.**Calculated equilibrium phase diagrams of the RHEAs at high temperature: (

**a**) Al

_{20}Cr

_{20}Mo

_{20}Ti

_{20}V

_{20}; (

**b**) Al

_{20}Cr

_{20}Mo

_{10}Ti

_{20}V

_{30}; (

**c**) Al

_{30}Cr

_{10}Mo

_{5}Ti

_{20}V

_{35.}

**Figure 3.**(

**a**) Thermal coefficient of linear expansion as a function of temperature; (

**b**) Specific heat capacity at constant volume as a function of temperature.

**Table 1.**Calculated values of entropy of mixing (ΔS

_{mix})

_{,}mixing enthalpy (ΔH

_{mix}), valence electron concentration (VEC), and atomic size differences (δ) of the refractory high-entropy alloys (RHEAs).

Name of Alloys | ΔS_{mix} (J/K.mol) | ΔH_{mix} (kJ/mol) | δ (%) | VEC |
---|---|---|---|---|

Al_{20}Cr_{20}Mo_{20}Ti_{20}V_{20} | 13.38 | −12.16 | 4.83 | 4.8 |

Al_{20}Cr_{20}Mo_{10}Ti_{20}V_{30} | 12.94 | −13.04 | 4.92 | 4.7 |

Al_{30}Cr_{10}Mo_{5}Ti_{20}V_{35} | 11.89 | −16.98 | 4.46 | 4.3 |

**Table 2.**Calculated three independent elastic constants (GPa);${C}_{11}$ ;${C}_{12}$ ;${C}_{44}$, Cauchy pressure ${C}_{11-}{C}_{44}$ (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratio (ν), Pugh’s ratio (B/G), lattice constant a (Å), density ρ (gm/cm

^{3}) and hardness H

_{v}(GPa) at zero pressure and Kelvin, respectively.

Alloys | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{44}$ | ${\mathit{C}}_{11-}{\mathit{C}}_{44}$ | B | G | E | ν | B/G | ρ | a | H_{v} |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Al_{20}Cr_{20}Mo_{20}Ti_{20}V_{20} | 255 | 119 | 69 | 186 | 165 | 61 | 163 | 0.33 | 2.72 | 6.28 | 3.07 | 5.48 |

Al_{20}Cr_{20}Mo_{10}Ti_{20}V_{30} | 247 | 111 | 64 | 183 | 157 | 60 | 161 | 0.32 | 2.62 | 5.85 | 3.05 | 5.66 |

Al_{30}Cr_{10}Mo_{5}Ti_{20}V_{35} | 246 | 90 | 63 | 183 | 143 | 56 | 149 | 0.32 | 2.71 | 5.16 | 3.08 | 5.56 |

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**MDPI and ACS Style**

Bhandari, U.; Zhang, C.; Yang, S.
Mechanical and Thermal Properties of Low-Density Al_{20+x}Cr_{20-x}Mo_{20-y}Ti_{20}V_{20+y} Alloys. *Crystals* **2020**, *10*, 278.
https://doi.org/10.3390/cryst10040278

**AMA Style**

Bhandari U, Zhang C, Yang S.
Mechanical and Thermal Properties of Low-Density Al_{20+x}Cr_{20-x}Mo_{20-y}Ti_{20}V_{20+y} Alloys. *Crystals*. 2020; 10(4):278.
https://doi.org/10.3390/cryst10040278

**Chicago/Turabian Style**

Bhandari, Uttam, Congyan Zhang, and Shizhong Yang.
2020. "Mechanical and Thermal Properties of Low-Density Al_{20+x}Cr_{20-x}Mo_{20-y}Ti_{20}V_{20+y} Alloys" *Crystals* 10, no. 4: 278.
https://doi.org/10.3390/cryst10040278