#
First-Principles Calculations of High-Pressure Physical Properties of Ti_{0.5}Ta_{0.5} Alloy

^{1}

^{2}

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## Abstract

**:**

_{0.5}Ta

_{0.5}alloy with systematic symmetry under high pressure is conducted via first-principles calculations, and relevant physical parameters are calculated. The results demonstrate that the calculated parameters, including lattice parameter, elastic constants, and elastic moduli, fit well with available theoretical and experimental data when the Ti

_{0.5}Ta

_{0.5}alloy is under $T=0$ and $P=0$, indicating that the theoretical analysis method can effectively predict the physical properties of the Ti

_{0.5}Ta

_{0.5}alloy. The microstructure and macroscopic physical properties of the alloy cannot be destroyed as the applied pressure ranges from 0 to 50GPa, but the phase transition of crystal structure may occur in the Ti

_{0.5}Ta

_{0.5}alloy if the applied pressure continues to increase according to the TDOS curves and charge density diagram. The value of Young’s and shear modulus is maximized at $P=25\text{}\mathrm{GPa}$. The anisotropy factors ${A}_{\left(100\right)\left[001\right]}$ and ${A}_{\left(110\right)\left[001\right]}$ are equal to 1, suggesting the Ti

_{0.5}Ta

_{0.5}alloy is an isotropic material at 28 GPa, and the metallic bond is strengthened under high pressure. The present results provide helpful insights into the physical properties of Ti

_{0.5}Ta

_{0.5}alloy.

## 1. Introduction

_{0.5}Ta

_{0.5}alloy were revealed through theory and experiment, as far as we know, but high-pressure physical properties in Ti

_{0.5}Ta

_{0.5}alloy have not been studied yet, such as the structural, mechanical, and electronic properties. Therefore, the aim of this paper is to apply the first-principles calculations to investigate the high-pressure physical properties in Ti

_{0.5}Ta

_{0.5}alloy in the frame of density functional theory (DFT), and related physical parameters are computed to reveal the relationships between these parameters and applied pressure acting on Ti

_{0.5}Ta

_{0.5}alloy. In this paper, the main framework is listed below: In Section 2, the theoretical methodology and design parameters are introduced in detail. In Section 3, the calculated parameters of Ti

_{0.5}Ta

_{0.5}alloy are presented and discussed. Finally, a brief summary is provided in Section 4.

## 2. Theoretical Methodology

^{2}3d

^{2}and 6s

^{2}5d

^{3}, respectively. To ensure the convergence accuracy of the electronic calculations of the Ti

_{0.5}Ta

_{0.5}alloy, the k-mesh in the Brillouin zone is set as 13 × 13 × 13, and the cutoff energy is optimized as 400 eV. The energy convergence criterion in the self-consistent calculation is optimized for 1.0 × 10

^{-6}eV/atom. For Ti

_{0.5}Ta

_{0.5}alloy, it has the space group of Im-3m, and Figure 1 displays the crystal structure. The algorithm of Broyden–Fletcher–Goldfarb–Shanno (BFGS) [23] is selected for optimizing the geometric configuration of the alloy in the pressure range of -10 to 50 GPa. Meanwhile, the Hellmann–Feyman force of each atom is accurate to 0.01 eV/Å.

## 3. Analysis and Discussions

#### 3.1. Structure and Stability

_{0.5}Ta

_{0.5}alloy according to the Birch–Murnaghan equation of state, as depicted in Figure 2, in which the volume range is set to 0.9–1.1 V

_{0}, and each unit cell is optimized sufficiently. From the variation curve, the total energy gets the minimum value (${E}_{t}=-19.401\text{}\mathrm{eV}$) when the volume is ${V}_{0}=34.660\text{}{\mathsf{\AA}}^{3}$, which shows that the crystal structure reaches the most stable state when these physical parameters are taken, and then the theoretical prediction of corresponding lattice constant is obtained by derivation, namely, ${a}_{0}=3.260\text{}\mathsf{\AA}$, where ${V}_{0}$ and ${a}_{0}$ represents the primitive cell volume and equilibrium lattice constant under $T=0$ and $P=0$, respectively. Meanwhile, Table 1 lists the comparative results of the calculated lattice constant ${a}_{0}$ versus other data for Ti

_{0.5}Ta

_{0.5}alloy, and it fits well with these available data, indicating the validity of the theoretical analysis method.

_{0.5}Ta

_{0.5}alloy under different applied pressures to get the corresponding volumes and equilibrium lattice constants, which are used to investigate the impacts of applied pressure on the unit cell volume and lattice constant. Subsequently, the dependencies of dimensionless ratios $V/{V}_{0}$ and $a/{a}_{0}$ on the applied pressure are obtained in the range of −10 to 50 GPa, where the negative pressure denotes the tension, as shown in Figure 3. From the variation curves, it can be found that the two ratios decrease when applied pressure increases and the descent speed for $V/{V}_{0}$ is significantly faster compared to $a/{a}_{0}$, suggesting that applying pressure can greatly reduce interatomic distance, and then strengthen electron interactions between adjacent atoms in the Ti

_{0.5}Ta

_{0.5}alloy.

_{0.5}Ta

_{0.5}alloy under the applied pressure ranging 0 to 50 GPa. Results display that ${C}_{11}$ and ${C}_{12}$ have the same trend that the values of theoretical predictions increase monotonously with the increase of applied pressure, and the elastic constant ${C}_{44}$ increases slowly and then decreases gradually, where the increase of elastic constants indicates that the deformation resistance of Ti

_{0.5}Ta

_{0.5}alloy becomes stronger. Meanwhile, the calculated values of ${C}_{ij}$ always satisfy the stability criterion of crystal structure in the range of applied pressure, suggesting that the high-pressure mechanical stability of Ti

_{0.5}Ta

_{0.5}alloy cannot be destroyed.

#### 3.2. Mechanical Properties

_{0.5}Ta

_{0.5}alloy under the applied pressure in the range of 0 to 50 GPa. Obviously, bulk modulus B increases gradually as applied pressure increases, which indicates that high pressure for Ti

_{0.5}Ta

_{0.5}alloy can enhance its resistance to volume change. E and G first increase and then decrease gradually, and both of them get the maximum values at P = 25 GPa, which indicates that increasing pressure can improve the resistances to elastic and shear deformations between 0 and 25 GPa, but decline their resistances beyond 25 GPa. Furthermore, we compare the theoretical predictions of elastic moduli with available theoretical and experimental data for Ti

_{0.5}Ta

_{0.5}alloy under $T=0$ and $P=0$, and Table 3 shows that the calculated data of this work are well consistent with the research results of others [7,12,25].

_{0.5}Ta

_{0.5}alloy has good ductility. Additionally, there is a critical pressure P = 25 GPa; namely, the ductility of Ti

_{0.5}Ta

_{0.5}alloy is almost invariable at the pressure between 0 and 25 GPa, whereas the ductility of the alloy significantly increases when the applied pressure further increases.

#### 3.3. Anisotropy

_{0.5}Ta

_{0.5}alloy under applied pressure. The values of two anisotropy factors are equal to 1 when the applied pressure is about 28 GPa, indicating that the Ti

_{0.5}Ta

_{0.5}alloy is an isotropic material at 28 GPa. In the light of the trend of variation curves, the elastic anisotropy of Ti

_{0.5}Ta

_{0.5}alloy decreases with increasing pressure between 0 and 28 GPa, but increases under applied pressure ranging from 28 to 50 GPa, thereby the tangential force decreases first and then increases in promoting the cross slip of screw dislocations.

_{0.5}Ta

_{0.5}alloy under the applied pressure in this paper. For Poisson’s ratio $\sigma $, the value changes from −1 to 0.5, and a large value indicates good plasticity. Reed et al. [36] put forward an important conclusion that $\sigma =0.25$ and $\sigma =0.5$ were the smallest and largest bounds for central-force solids, respectively. Additionally, Fu et al. [35] found that the magnitude of $\sigma $ was used for defining the sort of interatomic bonding. In this paper, Poisson’s ratios ${\sigma}_{\left[001\right]}$ and ${\sigma}_{\left[111\right]}$ in the [001] and [111] crystallographic directions are calculated according to the theoretical predictions of elastic constants ${C}_{ij}$, and the calculation formulas are ${\sigma}_{\left[001\right]}={C}_{12}/\left({C}_{11}+{C}_{12}\right)$ and ${\sigma}_{\left[111\right]}=\left({C}_{11}+2{C}_{12}-2{C}_{44}\right)/2\left({C}_{11}+2{C}_{12}+{C}_{44}\right)$ [37]. Figure 8 plots the changing curves of Poisson’s ratios with respect to applied pressure in Ti

_{0.5}Ta

_{0.5}alloy. The values of ${\sigma}_{\left[001\right]}=0.441$ and ${\sigma}_{\left[111\right]}=0.409$ reveal that the central forces in the [001] and [111] crystallographic directions make a greater contribution to the interatomic bonding for the Ti

_{0.5}Ta

_{0.5}alloy in essence. Clearly, Poisson’s ratio ${\sigma}_{\left[001\right]}$ monotonously decreases with the increase of applied pressure, while Poisson’s ratio ${\sigma}_{\left[111\right]}$ is reversed and gradually approaches the value of upper bound 0.5, which shows that the high pressure can strengthen the interatomic bonding, thereby improving the plasticity of Ti

_{0.5}Ta

_{0.5}alloy in the [111] crystallographic direction.

_{0.5}Ta

_{0.5}alloy, including ${G}_{\left(100\right)\left[010\right]}$, ${G}_{\left(110\right)\left[1\overline{1}0\right]}$, and ${E}_{<100>}$, to further study the related mechanical performances, where ${G}_{\left(100\right)\left[010\right]}$ represents the shear modulus in the $\left(100\right)\left[010\right]$ direction, and ${G}_{\left(110\right)\left[1\overline{1}0\right]}$ indicates the one in the $\left(110\right)\left[1\overline{1}0\right]$ direction, respectively, and ${E}_{<100>}$ denotes Young’s modulus along the $<100>$ direction. These parameters can also be obtained by elastic constants ${C}_{ij}$, and the calculation formulas are ${G}_{\left(100\right)\left[010\right]}={C}_{44}$, ${G}_{\left(110\right)\left[1\overline{1}0\right]}=\left({C}_{11}-{C}_{12}\right)/2$, and ${E}_{<100>}=\left({C}_{11}-{C}_{12}+{C}_{11}{C}_{12}-{C}_{12}^{2}\right)/\left({C}_{11}+{C}_{12}\right)$ [32,38], and Figure 9a draws the variation curves of material moduli under applied pressure ranging from 0 to 50 GPa. It shows that the total varying trends of ${G}_{\left(110\right)\left[1\overline{1}0\right]}$ and ${E}_{<100>}$ increase gradually as the applied pressure increases, although ${E}_{<100>}$ decreases slightly at 50 GPa, indicating that high pressure strengthens the resistances to shear and elastic deformations of Ti

_{0.5}Ta

_{0.5}alloy in the $\left(110\right)\left[1\overline{1}0\right]$ and $<100>$ crystallographic directions. Different from ${G}_{\left(110\right)\left[1\overline{1}0\right]}$ and ${E}_{<100>}$, shear modulus ${G}_{\left(100\right)\left[010\right]}$ initially increases and then decreases when applied pressure increases, and reaches the maximum value at 25 GPa, which shows that the Ti

_{0.5}Ta

_{0.5}alloy has the strongest shear resistance along the $\left(100\right)\left[010\right]$ direction under P = 25 GPa, while the shear deformation resistance of the alloy will decline if the pressure continues to increase.

_{0.5}Ta

_{0.5}alloy. It is noted from the curve that the value of ${C}_{12}-{C}_{44}$ is positive at any pressures, and quickly increases for the Ti

_{0.5}Ta

_{0.5}alloy between 0 and 50 GPa, suggesting that the metallic bond is the main bonding form of Ti

_{0.5}Ta

_{0.5}alloy, and the metallic bond can be strengthened in the case of high pressure.

#### 3.4. Electronic Properties

_{0.5}Ta

_{0.5}alloy under applied pressure in this paper. Figure 10 exhibits the changing curves of the partial density of states (PDOS) and the total density of states (TDOS) at P = 0 GPa, where the red dash line stands for the Fermi level (${E}_{\mathrm{F}}=0\text{}\mathrm{eV}$). For the TDOS curve, it can be found that the value is not zero at ${E}_{\mathrm{F}}=0\text{}\mathrm{eV}$, suggesting that the Ti

_{0.5}Ta

_{0.5}alloy exhibits the metallicity, which verifies the conclusion in Figure 9b. For PDOS curves, they show that the Ti-3d and Ta-5d states have the greatest effect on DOS at 0 eV, while the effects of the Ti-4s and Ta-6s states on DOS are almost negligible.

_{0.5}Ta

_{0.5}alloy, which affects the microstructure and macroscopic physical properties of Ti

_{0.5}Ta

_{0.5}alloy. At the same time, according to the results of Figure 4, it can be found that the value of elastic constant ${C}_{44}$ may become negative as the applied pressure increases further, which indicates that the Ti

_{0.5}Ta

_{0.5}alloy may produce the structural phase transition under higher pressure, and the analysis result is consistent with the previous conclusion in Figure 11.

_{0.5}Ta

_{0.5}alloy, as shown in Figure 12, where the isosurface levels are set to 0.0425 ${r}_{0}^{-3}$ (${r}_{0}$ is the Bohr radius). With increasing applied pressure, the local chemical bonding between Ti and Ta atoms gradually increases, implying that the high pressure can enhance the electron interaction between the two atoms, thereby increasing the local chemical bonding between them, and the change increases as applied pressure increases. However, the strong interactions between two atoms may destroy the stability of the crystal structure. Therefore, the structural phase transition may occur in the Ti

_{0.5}Ta

_{0.5}alloy under higher pressure beyond 50 GPa, which coincides with the result in Figure 11.

## 4. Conclusions

_{0.5}Ta

_{0.5}alloy, including structural, mechanical, and electronic properties. Results reveal that the calculated data for equilibrium lattice parameter, elastic constants, elastic moduli are consistent with the research results of others for Ti

_{0.5}Ta

_{0.5}alloy under $T=0$ and $P=0$. ${C}_{11}$ and ${C}_{12}$ increase monotonously as applied pressure increases, but ${C}_{44}$ increases slowly and then decreases gradually. The applied pressure cannot destroy the mechanical stability of Ti

_{0.5}Ta

_{0.5}alloy because the stability criterion is always satisfied in the range of 0–50GPa. Bulk modulus B gradually increases with increasing applied pressure, but E and G reach the maximum values at P = 25 GPa, suggesting that high pressure increases the resistance to volume deformation, and the resistances to elastic and shear deformation are maximized under P = 25 GPa in Ti

_{0.5}Ta

_{0.5}alloy. $B/G$ indicates that the ductility of the alloy significantly increases as applied pressure increases further when P > 25 GPa. The Ti

_{0.5}Ta

_{0.5}alloy is an isotropic material when the applied pressure is 28 GPa because the values of anisotropy factors ${A}_{\left(100\right)\left[001\right]}$ and ${A}_{\left(110\right)\left[001\right]}$ are equal to 1. The major atomic bonding in Ti

_{0.5}Ta

_{0.5}alloy is characterized by a metallic bond, and high pressure can enhance the metallic bond. The TDOS curves and the isosurface contours of charge density indicate that the structural phase transition may occur in the Ti

_{0.5}Ta

_{0.5}alloy if the applied pressure increases further. The present results are valuable for the application of Ti

_{0.5}Ta

_{0.5}alloy under high pressure in the future, such as the wide applications in aerospace, marine, chemical, and biological fields under high pressure.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Dependencies of dimensionless ratios $V/{V}_{0}$ and $a/{a}_{0}$ on applied pressure in Ti

_{0.5}Ta

_{0.5}alloy.

**Figure 4.**Dependencies of elastic constants ${C}_{ij}$ on applied pressure in Ti

_{0.5}Ta

_{0.5}alloy.

**Figure 9.**Dependencies of calculated moduli (

**a**) and Cauchy pressure (

**b**) on applied pressure in Ti

_{0.5}Ta

_{0.5}alloy.

**Figure 10.**Changing curves of the partial density of states (PDOS) and the total density of states (TDOS) for Ti

_{0.5}Ta

_{0.5}alloy at 0 GPa.

**Figure 12.**Charge density diagrams of isosurface contours versus various pressures in Ti

_{0.5}Ta

_{0.5}alloy.

**Table 1.**Comparisons of lattice constant with experimental and theoretical data for Ti

_{0.5}Ta

_{0.5}alloy.

Ti_{0.5}Ta_{0.5} Alloy | Present | Experimental Data | Theoretical Data |
---|---|---|---|

Lattice constant a_{0} (Å) | 3.260 | 3.295 [11], 3.286 [14] | 3.278 [24], 3.274 [25] |

**Table 2.**Comparisons of three elastic constants with available theoretical data in Ti

_{0.5}Ta

_{0.5}alloy.

Ti_{0.5}Ta_{0.5} Alloy | Present | Theoretical Data |
---|---|---|

${C}_{11}$ (GPa) | 180.34 | 163.40 [24], 181.80 [25] |

${C}_{12}$ (GPa) | 142.16 | 132.80 [24], 138.86 [25] |

${C}_{44}$ (GPa) | 30.14 | 39.00 [24], 45.18 [25] |

**Table 3.**Comparisons of the current elastic moduli with other data in Ti

_{0.5}Ta

_{0.5}alloy (unit: GPa).

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**MDPI and ACS Style**

Yu, F.; Liu, Y.
First-Principles Calculations of High-Pressure Physical Properties of Ti_{0.5}Ta_{0.5} Alloy. *Symmetry* **2020**, *12*, 796.
https://doi.org/10.3390/sym12050796

**AMA Style**

Yu F, Liu Y.
First-Principles Calculations of High-Pressure Physical Properties of Ti_{0.5}Ta_{0.5} Alloy. *Symmetry*. 2020; 12(5):796.
https://doi.org/10.3390/sym12050796

**Chicago/Turabian Style**

Yu, Fang, and Yu Liu.
2020. "First-Principles Calculations of High-Pressure Physical Properties of Ti_{0.5}Ta_{0.5} Alloy" *Symmetry* 12, no. 5: 796.
https://doi.org/10.3390/sym12050796