Lattice Stability and Elastic Properties of Zr-Ti-X Alloys (X = Al, V) by the First Principles Study

Abstract

Based on a certain ratio of Zr and Ti atomic fractions according to Zr₄₇Ti₄₅Al₅V₃ (wt.%), the lattice constants, lattice stability, and elastic properties of Zr-Ti-X alloys (X = Al, V) in body-centered cubic (BCC) (β phase) and hexagonal close-packed (HCP) (α phase) crystal structures were studied using first-principles calculations. It is shown that Al acts as an α stabilizer for Zr-Ti-Al alloys and V can stabilize the β phase for Zr-Ti-V alloys. As the mass fraction of Al increases from 4 wt.% (Zr₅₅Ti₄₁Al₄) to 6.8 wt.% (Zr_53.2Ti₄₀Al_6.8), these alloys all have relatively good strength, hardness, and rigidity, however, their ductility deteriorated with the increasing of Al mass fraction. When the mass fraction of V in Zr-Ti-V alloys is 2.4 wt.%, Zr_55.6Ti₄₂V_2.4 (wt.%) achieved the best strength, hardness, and rigidity, when the mass fraction of V increases from 0 (Zr₅₇Ti₄₃) to 12 wt.% (Zr_50.2Ti_37.8V₁₂), their ductility improved. The changes of phase compositions and structure with Al content or V content distinctly affect mechanical properties of ternary Zr-Ti-X alloys (X = Al, V), the amount of Zr and Ti could be factors that impact the mechanical properties of the multiphase Zr₄₇Ti₄₅Al₅V₃ from the point of view of ternary Zr-Ti-Al and Zr-Ti-V compositions.

1. Introduction

In recent decades, with the increasing number of spacecraft launched by various countries, the space environment has become increasingly complex, and the development of high-performance structural materials suitable for the space environment has become an inevitable choice for the spacecraft in the future [1,2,3,4]. For the critical components and structural parts of spacecraft, high-strength steel with higher density is the main material commonly utilized in the past, but the application of traditional steel is greatly restricted because of the demand for lightweight spacecraft. Therefore, there is an urgent need to develop high-strength and lightweight materials suitable for spacecraft structural parts. Many studies have shown that titanium alloys are expected to replace high-strength steel as critical components and structural materials in spacecraft, all because of their low density, high specific strength, good corrosion resistance, and low temperature performance [5,6,7]. Among hundreds of titanium alloys, Ti₉₀Al₆V₄ (wt.%) is widely used owing to its excellent mechanical performance. However, the service life of Ti₉₀Al₆V₄ (wt.%) will be greatly reduced because of poor impact resistance in the face of space debris, narrow working parameter range, complex manufacturing processes, as well as large deformation, and high strain rate at high temperature, if it is applied to spacecraft structural parts [8,9,10]. As is known to all that adding different elements to the alloys can greatly affect the lattice stabilities and mechanical properties of the alloys [11,12,13,14], therefore, it is expected to enhance the applicability of Ti₉₀Al₆V₄ (wt.%) in space by adding alloying elements. In 2012, from the perspective of adding zirconium element to Ti₉₀Al₆V₄ (wt.%), Jing et al. developed a new type of alloy—Zr₄₇Ti₄₅Al₅V₃ (wt.%) [15]. The chemical properties of Zr and Ti are similar because these elements belong to the same primary group (IVB), have two kinds of crystal structure (α- and β-type), and can form an infinitely miscible solid solution [16]. Compared with Ti₉₀Al₆V₄ (wt.%), Zr₄₇Ti₄₅Al₅V₃ (wt.%) has great mechanical properties. The hardness of Zr₄₇Ti₄₅Al₅V₃ (wt.%) is above 400 HV, which is nearly 30% higher than Ti₉₀Al₆V₄ (wt.%). Its material strength after heat treatment is above 1400 MPa, and its density is only about 2/3 of high-strength steel [17], thus, Zr₄₇Ti₄₅Al₅V₃ (wt.%) is a new type of alloy with great potential.

Since the Zr₄₇Ti₄₅Al₅V₃ (wt.%) was proposed, there are research findings which reported the relation of macroscopic rheological characteristic with microstructures of Zr₄₇Ti₄₅Al₅V₃ (wt.%) subjected to hot tensile, compression, or torsion deformation. In 2012, Liang et al. [16] studied the structural and mechanical properties of hot-rolled Zr-Ti-Al-V alloys based on X-ray diffraction and metallographic analyses. Results showed that the microstructure of alloys went through a series of changes from the α (α’)-phase to part or all of the β-phase, thus affecting the mechanical properties as Zr increased. In 2013, Liang et al. [18] experimentally studied the effect of Al content on the microstructure and mechanical properties of hot rolled Zr₄₇Ti₄₅Al₅V₃ (wt.%), results showed that the Zr-Ti-Al-V alloys with Al contents between 3.3 wt.% and 5.6 wt.% have excellent mechanical properties. Zhang et al. [19] conducted impact compression experiments on Zr₄₇Ti₄₅Al₅V₃ (wt.%), and results showed that no obvious evidence of phase transition is found in the shock compression pressure range. In 2016, Tan et al. [20] studied the hot deformation behavior of Zr₄₇Ti₄₅Al₅V₃ (wt.%) with an initial lamellar α-phase structure through a compression test in the temperature range of 823 K to 1073 K, researches showed that the flow curves exhibited a continuous flow softening in the α + β phase field. Liang et al. [21] investigated the effects of V content (0–7 wt.%) on the structure and mechanical properties of Zr-Ti-Al-V alloys, and results showed that the phase composition changes as α’→α’+ (α’’ + β)→α’’ + β→β+ (α’’)→β with increased V content from 0 wt.% to 7 wt.%, and the elastic modulus and yield strength show an antiparabolic trend with increased V content. In 2018, Tan et al. [22] studied the effect of thermal deformation on the α→β phase transition of Zr₄₇Ti₄₅Al₅V₃ (wt.%), results showed that with decreasing strain rate and increasing deformation temperature, the volume fraction and size of globular α phase increased.

At present, most of the research [23,24,25] on experimental methods discussed the influence of alloy processing technology on structure and mechanical properties. For Zr-Ti-Al-V alloys, there is a lack of in-depth research about the effect of element composition and phase structure on their mechanical properties. However, mastering the relation between the element composition, phase structure, and mechanical properties is an effective way to optimize the alloy properties [26,27,28,29,30,31,32], and it is of great significance to study the lattice stabilities and elastic properties of Zr-Ti-Al-V alloy systems. Marker et al. [33] studied the elastic properties of the Zr-Ti-X (X = Nb, Sn, Ta) alloys in the BCC structure, and predicted the single crystal elastic stiffness coefficients and polycrystalline aggregate properties using DFT-based first-principles calculations. Liao et al. [34] established the single-crystal elastic constant model of the quaternary BCC Zr-Ti-Nb-V alloys across full composition space using first principles, based on the research of Marker et al. [33], the accuracy of the model was confirmed by comparing with previous calculations and experiments. Konopatsky et al. [35] carried out the multi-cycle mechanical testing on ternary Ti₆₈Zr₁₈Nb₁₄ (at.%), quaternary Ti₆₈Zr₁₈Nb₁₃Ta₁ (at.%), Ti₆₈Zr₁₈Nb₁₂Ta₂ (at.%), and Ti₆₈Zr₁₈Nb₁₁Ta₃ (at.%) alloys, during multi-cycle mechanical testing, the following regularity appeared for all the alloys tested: during 10–15 first loading-unloading cycles, the mechanisms of plastic deformation and stress-induced martensite formation and stabilization prevail.

Many researchers have proved that phase composition of the Zr-Ti-Al-V alloys significantly depends on the amount of Zr and Ti [16], and changes of phase compositions and structure with alloying element distinctly affected mechanical properties of Zr-Ti-Al-V alloys [21]. It is of great significance to study the alloying effects of Al and V to provide guidance for further optimizing the lattice stabilities and mechanical properties regulation of quaternary Zr₄₇Ti₄₅Al₅V₃ (wt.%), in other words, the research of alloying element content could provide direction and assistance in guiding the research of quaternary alloys from the point of view of ternary alloys.

In this paper, we performed first-principles studies of Zr-Ti-X (X = Al, V) alloys using the special quasi-random structure (SQS) models to explore the relationship between element composition, phase structure, lattice stabilities, and elastic properties based on previous studies [36]. This article starts from the basic composition of Zr₄₇Ti₄₅Al₅V₃ (wt.%), the lattice constants, total energies, and elastic constants of the Zr-Ti-X (X = Al, V) alloys are calculated. These calculations provide an effective guide for further optimizing the composition of the Zr-Ti-X (X = Al, V) alloys, and provide guidance and help in exploring the complex relationship between the ternary alloys and quaternary alloys.

2. Materials and Methods

Based on the ratio of the Zr atomic fraction and the Ti atomic fraction in the Zr₄₇Ti₄₅Al₅V₃ (wt.%), in order to investigate the effect of alloying elements Al and V on the crystal structures and mechanical properties of Zr₄₇Ti₄₅Al₅V₃ (wt.%) alloy, two types of ternary alloy systems were designed—Zr-Ti-Al alloy systems and Zr-Ti-V alloy systems. For these two types of alloy systems, the hcp and bcc structures were studied to model α and β phase because a mixture of two phases usually appears in these alloys, which gives a better understanding of the phase properties of two competitive phases. In this paper, the special quasi-random structure (SQS) method was used in the alloy systems with hexagonal close-packed structure (HCP, α phase) and body-centered cubic structure (BCC, β phase).

To simplify the calculation, the atomic ratio of Zr and Ti is approximately 0.35:0.50. Figure 1 shows the SQS model of the Zr₃₅Ti₅₀Al₁₅ (at.%) alloys in HCP structure and BCC structure used in modeling in this article; the SQS model of the Zr₃₅Ti₅₀V₁₅ (at.%) alloys in HCP structure and BCC structure is shown in Figure 2.

Table 1. Zr-Ti-Al alloy systems in HCP structure and BCC structure with different atomic fraction of Al.

Structures	System	Composition (at.%)
HCP	Zr-Ti-Al	Zr41Ti59
		Zr40Ti57Al3
		Zr39Ti55Al6
		Zr38Ti53Al9
		Zr36Ti52Al12
		Zr35Ti50Al15
BCC		Zr41Ti59
		Zr40Ti57Al3
		Zr39Ti55Al6
		Zr38Ti53Al9
		Zr36Ti52Al12
		Zr35Ti50Al15

shows the ${\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{Al}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}$ (at.%) alloys (y:z = 0.35:0.50, and the atomic fraction of Al goes from 0 to 15 at.% with a gradient of 3 at.%) in HCP and BCC structures.

Table 2. Zr-Ti-V alloy systems in HCP structure and BCC structure with different atomic fraction of V.

Structures	System	Composition (at.%)
HCP	Zr-Ti-V	Zr41Ti59
		Zr40Ti57V3
		Zr39Ti55V6
		Zr38Ti53V9
		Zr36Ti52V12
		Zr35Ti50V15
BCC		Zr41Ti59
		Zr40Ti57V3
		Zr39Ti55V6
		Zr38Ti53V9
		Zr36Ti52V12
		Zr35Ti50V15

shows the ${\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{V}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}$ alloys (at.%) (y:z = 0.35:0.50, and the atomic fraction of V goes from 0 to 15 at.% with a gradient of 3 at.%) in HCP and BCC structures. In this paper, the randomness of atomic configuration was reproduced while considering the local relaxation in the supercell through modeling.

In this article, the CASTEP (Cambridge Sequential Total Energy Package, Cambridge, UK) total energy calculation software package, based on the density functional theory (DFT) [37] pseudopotential plane wave method [38], was used to analyze the Zr-Ti-Al alloys and Zr-Ti-V alloys in HCP and BCC structures for first-principles calculations. Density functional theory (DFT) as an effective approximation method is used to solve the structure of multi-electronic systems [39,40,41,42,43,44,45,46], and it obtains the total energy of multi-electronic systems based on the energy minimization method. The pseudopotentials describe the valence configurations of the elements, which are as follows: Zr 4s²4p⁶4d²5s², Ti 3s²3p⁶3d²4s², Al 3s²3p¹and V 3s²3p⁶3d³4s². The electronic exchange-correlation energy was treated within the generalized gradient approximation (GGA) with the PBE revised for solids [47]. The convergence criteria of 5 × 10⁻⁷ eV per atom was applied to be self-consistent, all calculations were executed in reciprocal space to improve the speed and accuracy, in addition, the calculation and convergence were controlled by the cut-off point of the wave energy of the Ultrasoft pseudopotential plane. After convergence is tested, the Monkhorst-Pack grid of 2 × 2 × 3 is adopted for the brillouin zone integration, the plane wave kinetic energy cut-off was determined as 350 eV, and the unit cell parameters of Zr-Ti-Al alloys and Zr-Ti-V alloys are calculated by first principles. About the lattice stability, formation energy ${\mathrm{E}}_{\mathrm{f}}\left({\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{X}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}\right)$ was defined by:

$${\mathrm{E}}_{\mathrm{f}}\left({\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{X}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}\right)=\left\{\mathrm{E}\left({\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{X}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}\right)-\mathrm{yE}\left(\mathrm{Zr}\right)-\mathrm{zE}\left(\mathrm{Ti}\right)-\left[100-\left(\mathrm{y}+\mathrm{z}\right)\right]\mathrm{E}\left(\mathrm{X}\right)\right\}/100$$

(1)

Here, $\mathrm{E}\left({\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{X}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}\right)$ is the total energy per atom of the ${\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{X}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}$ (at.%) alloy with alloying element concentration. $\mathrm{E}\left(\mathrm{Zr}\right)$, $\mathrm{E}\left(\mathrm{Ti}\right),$ and $\mathrm{E}\left(\mathrm{X}\right)$ are the energies per atom of Zr, Ti, and X in their ground state structures.

In this paper, according to our modeling system based on static modeling, the total energy and forming energy of the system calculated in this paper are considered from the level of electron exchange. They are themselves relative values because most of the energy is subtracted based on mass–energy equivalence. The energy of the atom itself is approximated as the absolute value of the total non-electron energy, which may not be considered in this article. We performed a static total energy calculation, based on that, the lattice stabilities of Zr-Ti-X (X = Al, V) alloys are evaluated with analyzing the total energy dependence on elements species, alloying fraction, and phase structure using the first principles in our work.

We determined the independent elastic coefficients, C_ij, based on expanding the total energy, U, of the crystal as a function of lattice strain Є [48], which is expressed as

$$\mathrm{U}={\mathrm{U}}_0+\mathrm{V}{\displaystyle \sum }_{\mathrm{i}}{\mathsf{\sigma }}_{\mathrm{i}}{\mathsf{\epsilon }}_{\mathrm{i}}+\frac{1}{2}\mathrm{V}{\displaystyle \sum }_{\mathrm{i}}{\displaystyle \sum }_{\mathrm{j}}{\mathrm{C}}_{\mathrm{ij}}{\mathsf{\epsilon }}_{\mathrm{i}}{\mathsf{\epsilon }}_{\mathrm{j}}+\dots$$

(2)

where V is the volume and U₀ is the total energy of the undistorted lattice system. The linear term in the expansion vanished, and the cubic or higher powers of ε_i terms in the polynomial expansion are neglected for sufficiently small strains.

In this paper, the elastic constant, C_ij, that describes the response of the crystal to external forces, is generally used to determine the strength of a material [49]. Based on the research findings of Zhou et al. [50], we theoretically predicted the elastic properties of Zr-Ti-X (X = Al, V) alloys of the alloying elements.

A small strain (δ) is applied to the solid lattice of the ${\mathrm{Zr}}_{\mathrm{y}}{\mathrm{Ti}}_{\mathrm{z}}{\mathrm{V}}_{100-\left(\mathrm{y}+\mathrm{z}\right)}$(at.%) alloys to construct a quadratic function relationship between the total energy changes (∆U) and the strains (δ) according to the Hooks law, as follows:

$$\Delta \mathrm{U}=\frac{\mathrm{V}}{2}{\displaystyle \sum }_{\mathrm{I},\mathrm{j}=1}^6{\mathrm{C}}_{\mathrm{ij}}{\mathsf{\epsilon }}_{\mathrm{i}}{\mathsf{\epsilon }}_{\mathrm{j}}$$

(3)

where ∆U is the total energy changes, V is the original unit cell volume, C_ij are the elastic constants, ε_i and ε_j are the components of strain matrix.

In order to investigate these elastic constants, as shown in

Table 3. The relationship between the strains (δ) and the total energy changes (∆U) of BCC metal.

Strain	Change of Total Energy
Є = (0, 0, 0, δ, δ, δ)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\frac{3}{2}{\mathrm{C}}_{44}{\mathsf{\delta }}^2$
Є = (δ, δ, 0, 0, 0, 0)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\left({\mathrm{C}}_{11}+{\mathrm{C}}_{12}\right){\mathsf{\delta }}^2$
Є = (δ, δ, δ, 0, 0, 0)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\frac{3}{2}\left({\mathrm{C}}_{11}+2{\mathrm{C}}_{12}\right){\mathsf{\delta }}^2$

, three sets of specific strains (δ) are applied to BCC metal in different directions. We also list the different strain configurations to compute the five independent elastic constants of HCP metal in

Table 4. The relationship between the strains (δ) and the total energy changes (∆U) of HCP metal.

Strain	Change of Total Energy
Є = (δ, δ, 0, 0, 0, 0)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\left({\mathrm{C}}_{11}+{\mathrm{C}}_{12}\right){\mathsf{\delta }}^2$
Є = (0, 0, 0, 0, 0, δ)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\frac{1}{4}\left({\mathrm{C}}_{11}-{\mathrm{C}}_{12}\right){\mathsf{\delta }}^2$
Є = (0, 0, δ, 0, 0, 0)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\frac{1}{2}{\mathrm{C}}_{33}{\mathsf{\delta }}^2$
Є = (0, 0, 0, δ, δ, 0)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}={\mathrm{C}}_{44}{\mathsf{\delta }}^2$
Є = (δ, δ, δ, 0, 0, 0)	$\frac{\Delta \mathrm{U}}{\mathrm{V}}=\frac{3}{2}\left({\mathrm{C}}_{11}+2{\mathrm{C}}_{12}\right){\mathsf{\delta }}^2$

. The elastic constants can be obtained by fitting the relationship between the total energy changes (∆U) and the applied strains (δ).

For different unit cell structures, the stability criteria are also different according to Born-Huang’s mechanical stability theory. In this paper, the HCP Zr-Ti-Al alloys and HCP Zr-Ti-V alloys are calculated by first principles to obtain five independent elastic constants, C₁₁, C₁₂, C₁₃, C₃₃, and C₄₄, their stability is determined from the elastic constants by:

$${\mathrm{C}}_{11}>0$$

(4)

$${\mathrm{C}}_{44}>0$$

(5)

$${\mathrm{C}}_{11}>{\mathrm{C}}_{12}$$

(6)

$$\left({\mathrm{C}}_{11}+{\mathrm{C}}_{12}\right){\mathrm{C}}_{33}-2{\mathrm{C}}_{13}^2>0$$

(7)

The BCC Zr-Ti-Al alloy systems and BCC Zr-Ti-V alloy systems are calculated by first principles to obtain three independent elastic constants, C₁₁, C₁₂ and C₄₄, their stability is determined from the elastic constants by:

$${\mathrm{C}}_{11}+2{\mathrm{C}}_{12}>0$$

(8)

$${\mathrm{C}}_{11}>{\mathrm{C}}_{12}$$

(9)

$${\mathrm{C}}_{44}>0$$

(10)

The mechanical parameters of the Zr-Ti-Al alloys and Zr-Ti-V alloys can be calculated by using single crystal elastic constants according to the Voigt-Reuss-Hill scheme [51,52,53,54]. The bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν) of the Zr-Ti-Al alloys and Zr-Ti-V alloys are calculated as follows:

$${\mathrm{B}}_{\mathrm{V}}=\frac{1}{9}\left({\mathrm{C}}_{11}+{\mathrm{C}}_{22}+{\mathrm{C}}_{33}\right)+\frac{2}{9}\left({\mathrm{C}}_{12}+{\mathrm{C}}_{13}+{\mathrm{C}}_{23}\right)$$

(11)

$$\frac{1}{{\mathrm{B}}_{\mathrm{R}}}=\left({\mathrm{S}}_{11}+{\mathrm{S}}_{22}+{\mathrm{S}}_{33}\right)+2\left({\mathrm{S}}_{12}+{\mathrm{S}}_{13}+{\mathrm{S}}_{23}\right)$$

(12)

$${\mathrm{B}}_{\mathrm{H}}=\frac{{\mathrm{B}}_{\mathrm{R}}+{\mathrm{B}}_{\mathrm{V}}}{2}$$

(13)

$${\mathrm{G}}_{\mathrm{V}}=\frac{1}{15}\left({\mathrm{C}}_{11}+{\mathrm{C}}_{22}+{\mathrm{C}}_{33}\right)-\frac{1}{15}\left({\mathrm{C}}_{12}+{\mathrm{C}}_{13}+{\mathrm{C}}_{23}\right)-\frac{1}{5}\left({\mathrm{C}}_{44}+{\mathrm{C}}_{55}+{\mathrm{C}}_{66}\right)$$

(14)

$$\frac{15}{{\mathrm{G}}_{\mathrm{R}}}=4\left({\mathrm{S}}_{11}+{\mathrm{S}}_{22}+{\mathrm{S}}_{33}\right)-4\left({\mathrm{S}}_{12}+{\mathrm{S}}_{13}+{\mathrm{S}}_{23}\right)+3\left({\mathrm{S}}_{44}+{\mathrm{S}}_{55}+{\mathrm{S}}_{66}\right)$$

(15)

$${\mathrm{G}}_{\mathrm{H}}=\frac{{\mathrm{G}}_{\mathrm{V}}+{\mathrm{G}}_{\mathrm{R}}}{2}$$

(16)

$$\mathrm{E}=\frac{9{\mathrm{B}}_{\mathrm{H}}{\mathrm{G}}_{\mathrm{H}}}{3{\mathrm{B}}_{\mathrm{H}}+{\mathrm{G}}_{\mathrm{H}}}$$

(17)

$$\mathsf{\upsilon }=\frac{3{\mathrm{B}}_{\mathrm{H}}-2{\mathrm{G}}_{\mathrm{H}}}{2\left(3{\mathrm{B}}_{\mathrm{H}}+{\mathrm{G}}_{\mathrm{H}}\right)}$$

(18)

The ratio of bulk modulus to shear modulus (B_H/G_H) proposed by Pugh [55] has been widely used to evaluate the toughness and brittleness of materials. It should be noted that a lower value of B means the atomic bond is weaker. Consequently, materials with a low value of B exhibit relatively poor resistance to various forms of localized corrosion (such as intergranular corrosion, pitting corrosion and stress-corrosion cracking) [56,57,58]. Furthermore, Clerc et al. [59] have demonstrated that the hardness of the annealed metal is proportional to G. It is commonly known that materials with a high value of the B/G ratio (>1.75) often result in ductility, while materials with a low value of the B/G ratio (<1.75) often exhibit brittleness [56,58,60,61]. Vitos et al. investigated the elastic properties of the ${\mathrm{Fe}}_{100-\left(\mathrm{c}+\mathrm{n}\right)}{\mathrm{Cr}}_{\mathrm{c}}{\mathrm{Ni}}_{\mathrm{n}}$ alloys (13.5 at.% < c < 25.5 at.% and 8 at.% < n < 24 at.%) [56], in that work, shear modulus (G) and bulk modulus (B) fall into the range of 74–81 GPa and 161–178 GPa, respectively.

In addition, the brittleness and ductility of materials can also be judged according to the value of Poisson’s ratio ν and the Cauchy pressure, that is, the difference between C₁₂ and C₄₄ [62]. The larger the value of Poisson’s ratio ν, the better the ductility of the material, and if the calculated Cauchy pressure of materials is negative or their Poisson’s ratio ν is lower than 0.26, these materials can be considered brittle. Jiang et al. [63] investigated the mechanical properties and Debye temperature of W-TM (TM = Cr, Cu, Fe, Mn, Mo, and Ni, respectively) alloys based on the first principles method, and results showed that the ductile/brittle properties of W-TM alloys can be evaluated based on the mechanical characteristic of B/G ratio, Poisson’s ratio (ν), and Cauchy pressure (C’).

3. Results and Discussion

Single cell unit cell parameters of alloy systems are shown in Section 3.1. In Section 3.2, the lattice stability of Zr-Ti-X (X = Al, V) alloys are evaluated with analyzing the total energy dependence on elements species, alloying fraction, and phase structure. The elastic constants of BCC and HCP Zr-Ti-X (X = Al, V) alloys, are shown in Section 3.3. Finally, the bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν) of BCC and HCP Zr-Ti-X (X = Al, V) alloys are then examined.

3.1. Lattice Constants

The calculated lattice constants a, b, and c (Å) for HCP-structured and BCC-structured Zr-Ti-Al alloy systems with different Al atomic fractions are shown in

Table 5. Effect of Al atomic fraction on single crystal cell parameters of HCP structure and BCC structure Zr-Ti-Al alloy systems.

Structures	System	Composition (at.%)	Lattice Parameters (Å)
			a	b	c
HCP	Zr-Ti-Al	Zr41Ti59	3.039	3.039	4.924
		Zr40Ti57Al3	3.028	3.028	4.873
		Zr39Ti55Al6	3.012	3.012	4.825
		Zr38Ti53Al9	2.993	2.993	4.763
		Zr36Ti52Al12	2.982	2.982	4.689
		Zr35Ti50Al15	2.971	2.971	4.597
BCC		Zr41Ti59	3.382	3.382	3.382
		Zr40Ti57Al3	3.368	3.368	3.368
		Zr39Ti55Al6	3.354	3.354	3.354
		Zr38Ti53Al9	3.341	3.341	3.341
		Zr36Ti52Al12	3.333	3.333	3.333
		Zr35Ti50Al15	3.325	3.325	3.325

, the single crystal cell parameters for HCP-structured and BCC-structured Zr-Ti-V alloy systems with various V concentrations are listed in

Table 6. Effect of V atomic fraction on single crystal cell parameters of HCP structure and BCC structure Zr-Ti-V alloy systems.

Structures	System	Composition (at.%)	Lattice Parameters (Å)
			a	b	c
HCP	Zr-Ti-V	Zr41Ti59	3.039	3.039	4.924
		Zr40Ti57V3	3.029	3.029	4.878
		Zr39Ti55V6	3.014	3.014	4.831
		Zr38Ti53V9	2.998	2.998	4.772
		Zr36Ti52V12	2.989	2.989	4.694
		Zr35Ti50V15	2.978	2.978	4.622
BCC		Zr41Ti59	3.382	3.382	3.382
		Zr40Ti57V3	3.371	3.371	3.371
		Zr39Ti55V6	3.359	3.359	3.359
		Zr38Ti53V9	3.346	3.346	3.346
		Zr36Ti52V12	3.338	3.338	3.338
		Zr35Ti50V15	3.329	3.329	3.329

.

Considering the concentration dependence, the lattice parameters decrease with increasing Al and V content, and the lattice parameters of the Zr-Ti-Al alloys and Zr-Ti-V alloys are close to each other.

3.2. Lattice Stability

As shown in Figure 3, the lattice stability of Zr-Ti-X (X = Al, V) alloys are evaluated with analyzing the total energies dependence on elements species, alloying fraction and phase structure using the first principles. Lattice stability studies are carried out with the purpose of explaining the complex relationships between the mechanical properties and the alloying elements, deformation, and temperature.

As shown in Figure 3a, the Zr-Ti-Al alloys in BCC structure generally have higher total energy than HCP Zr-Ti-Al alloys, and the introduction of Al atoms does not change the fact that the HCP Zr-Ti alloys is more stable than BCC [36]. When the Al atomic fraction reaches about 12 at.%, that is, when the Al mass fraction reaches about 5 wt.%, the HCP Zr-Ti-Al alloys are the most stable because of the lowest total energy, while their stability will decrease if the Al atomic fraction continues to increase. For the Zr-Ti-Al alloys in BCC structure, the introduction of Al atoms will enhance the stability because of micro alloying. When the mass fraction of Al goes from 0 to 1 wt.%, the structural stability of BCC Zr-Ti alloys will be improved, and reach the maximum when the mass fraction of Al is about 1 wt.%, but their stability will gradually decrease after the mass fraction of Al exceeds 1 wt.%. Therefore, the Al element can be regarded as a stable element of α phase of Zr-Ti alloys, which verifies the experimental research of Cordeiro [64].

It can be seen in Figure 3b, the introduction of V atoms can greatly enhance the stability of BCC-structured Zr-Ti alloys. When the V atomic fraction reaches about 15 at.%, the total energies of the BCC Zr-Ti-V alloys are lower than that of the HCP Zr-Ti-V alloys. For the HCP-structured Zr-Ti alloys, because of the micro alloying effect brought by trace V elements, their stability will be enhanced. But when the V atomic fraction exceeds 3 at.%, the continued addition of V atoms has little effect on the structural stability of HCP Zr-Ti-V alloys. In addition, the increase of V atomic fraction will greatly improve the structural stability of BCC Zr-Ti-V alloys. Thus, the V element can be regarded as a stable element of β phase in the Zr-Ti alloys, which also verifies the experimental research of Cordeiro [64].

3.3. Elastic Properties

In this study, the elastic constants of Zr-Ti-Al alloy systems (

Table 7. Independent elastic constants of HCP structure and BCC structure Zr-Ti-Al alloy systems.

Structures	System	Composition (at.%)	Elastic Constants/GPa
			C11	C12	C13	C33	C44	C66
HCP	Zr-Ti-Al	Zr41Ti59	178.1	45.2	70.1	184.2	27.1	65.8
		Zr40Ti57Al3	183.4	46.1	72.3	189.5	27.6	66.3
		Zr39Ti55Al6	189.2	46.8	75.4	192.3	28.2	66.8
		Zr38Ti53Al9	191.4	47.5	77.9	198.2	28.5	67.2
		Zr36Ti52Al12	194.5	47.9	79.3	203.3	28.9	67.5
		Zr35Ti50Al15	201.1	48.4	82.8	205.9	32.5	68.1
BCC		Zr41Ti59	118.3	94.7	\	\	33.6	\
		Zr40Ti57Al3	119.5	95.4	\	\	35	\
		Zr39Ti55Al6	120.2	95.8	\	\	36.3	\
		Zr38Ti53Al9	121.8	96.2	\	\	38.5	\
		Zr36Ti52Al12	122.2	97.9	\	\	39.1	\
		Zr35Ti50Al15	123.6	99.3	\	\	40.5	\

) in HCP structure and BCC structure and Zr-Ti-V alloy systems (

Table 8. Independent elastic constants of HCP structure and BCC structure Zr-Ti-V alloy systems.

Structures	System	Composition (at.%)	Elastic Constants/GPa
			C11	C12	C13	C33	C44	C66
HCP	Zr-Ti-V	Zr41Ti59	178.1	45.2	70.1	184.2	27.1	65.8
		Zr40Ti57V3	183.4	46.1	72.3	189.5	27.6	66.3
		Zr39Ti55V6	189.2	46.8	75.4	192.3	28.2	66.8
		Zr38Ti53V9	191.4	47.5	77.9	198.2	28.5	67.2
		Zr36Ti52V12	194.5	47.9	79.3	203.3	28.9	67.5
		Zr35Ti50V15	201.1	48.4	82.8	205.9	32.5	68.1
BCC		Zr41Ti59	118.3	94.7	\	\	33.6	\
		Zr40Ti57V3	119.5	95.4	\	\	35	\
		Zr39Ti55V6	120.2	95.8	\	\	36.3	\
		Zr38Ti53V9	121.8	96.2	\	\	38.5	\
		Zr36Ti52V12	122.2	97.9	\	\	39.1	\
		Zr35Ti50V15	123.6	99.3	\	\	40.5	\

) in HCP structure and BCC structure were calculated by first principles. Figure 4 shows the effect of Al atomic fraction on the independent elastic constants of HCP-structured and BCC-structured Zr-Ti-Al alloy systems, Figure 5 shows the effect of V atomic fraction on the independent elastic constants of HCP-structured and BCC-structured Zr-Ti-V alloy systems.

As can be seen from Figure 4 and Figure 5, in the calculated elastic constants of HCP Zr-Ti-Al alloys and HCP Zr-Ti-V alloys, C₁₁ and C₃₃ are much larger than other elastic constants, which shows that they are highly incompressible under uniaxial stress along the a and c axes. Moreover, as a hexagonal structure unit cell, the value of C₃₃ is greater than the value of C₁₁, indicating that the c-axis of these materials is more difficult to compress than a-axis. In addition, it can be seen in Figure 4 and Figure 5, the C₄₄ of the HCP Zr-Ti-Al alloys is only about 28 GPa, and the C₄₄ of the HCP Zr-Ti-V alloys is only about 25 GPa. These results indicate that HCP Zr-Ti-Al alloys and HCP Zr-Ti-V alloys have very weak resistance to shear deformation in the (100) plane.

In addition, for all Zr-Ti-Al alloys and Zr-Ti-V alloys constructed in this study, the Cauchy pressure calculated under the set environmental conditions is positive, which are generally higher than those of the Zr-Ti alloys [36]. These results indicate that these two alloy systems have higher ductility than Zr-Ti alloy systems, therefore, the introduction of alloying elements Al and V can improve the ductility of Zr-Ti alloy systems, and verified the experimental research results by Liang et al. [18,21].

Based on HCP structure and BCC structure, the elastic properties of the Zr-Ti-Al alloys of different Al atomic fractions and the Zr-Ti-V alloys of different V atomic fractions are calculated by first principles in this paper, the calculation results using Equations (11)–(18) are shown in

Table 9. Elastic properties of HCP structure and BCC structure Zr-Ti-Al alloy systems.

Structures	System	Composition (at.%)	Elastic Properties
			BH(GPa)	GH(GPa)	BH/GH	E(GPa)	ν
HCP	Zr-Ti-Al	Zr41Ti59	89.5	39.2	2.28	102.6	0.309
		Zr40Ti57Al3	91.9	40.2	2.29	105.3	0.309
		Zr39Ti55Al6	94.3	41.3	2.28	108.1	0.309
		Zr38Ti53Al9	95.5	41.7	2.29	109.2	0.309
		Zr36Ti52Al12	96.8	42.4	2.28	111	0.309
		Zr35Ti50Al15	99.3	46.1	2.15	120	0.299
BCC		Zr41Ti59	102.6	22.1	4.64	61.9	0.4
		Zr40Ti57Al3	103.4	22.8	4.53	63.7	0.397
		Zr39Ti55Al6	103.9	23.5	4.42	65.6	0.395
		Zr38Ti53Al9	104.7	24.8	4.22	69	0.39
		Zr36Ti52Al12	106.9	24.5	4.36	68.3	0.394
		Zr35Ti50Al15	107.4	25.1	4.27	69.9	0.392

and

Table 10. Elastic properties of HCP structure and BCC structure Zr-Ti-V alloy systems.

Structures	System	Composition (at.%)	Elastic Properties
			BH(GPa)	GH(GPa)	BH/GH	E(GPa)	ν
HCP	Zr-Ti-V	Zr41Ti59	89.5	39.2	2.28	102.6	0.309
		Zr40Ti57V3	91.1	35.1	2.60	93.3	0.292
		Zr39Ti55V6	91.7	38.6	2.38	101.6	0.315
		Zr38Ti53V9	93.4	38.1	2.45	100.6	0.320
		Zr36Ti52V12	93.7	37.8	2.47	100.0	0.322
		Zr35Ti50V15	94.3	36.9	2.56	98.0	0.327
BCC		Zr41Ti59	102.6	22.1	4.64	61.9	0.400
		Zr40Ti57V3	104.1	20.0	5.21	56.6	0.410
		Zr39Ti55V6	102.3	16.2	6.31	46.2	0.425
		Zr38Ti53V9	102.7	17.4	7.13	41.3	0.433
		Zr36Ti52V12	102.6	15.0	6.84	42.9	0.431
		Zr35Ti50V15	103.1	13.3	7.75	38.3	0.438

, as well as Figure 6 and Figure 7.

For Zr-Ti-Al alloys, it can be seen from Figure 6a that the bulk modulus of the HCP structure is less than that of BCC, indicating that α phase of Zr-Ti-Al alloys is lower than that of β phase. However, the HCP Zr-Ti-Al alloys have higher shear modulus and Young’s modulus than BCC, which indicates that the hardness and stiffness of α phase are higher than that of β phase. In addition, the bulk modulus, shear modulus, and Young’s modulus of HCP-structured Zr-Ti-Al alloys are positively correlated with Al atomic fraction. When the mass fraction of Al increases between 0 and 6.8 wt.%, the strength, hardness, and stiffness of α phase in Zr-Ti-Al alloys will be improved with the increasing of Al. For BCC Zr-Ti-Al alloys, their bulk modulus, shear modulus, and Young’s modulus are basically unchanged when the Al mass fraction exceeds 4 wt.%, and then these values continue to increase. In general, when the mass fraction of Al in the Zr-Ti-Al alloys is from 4 wt.% to 6.8 wt.%, the Zr-Ti-Al alloys have the highest strength, hardness, and rigidity.

As shown in Figure 6b, for the Zr-Ti-V alloys, the bulk modulus of the HCP structure are lower than that of BCC, which shows that the strength of α phase of Zr-Ti-V alloys is lower than that of β phase. In addition, the bulk modulus of HCP-structured Zr-Ti-V alloys is positively correlated with the atomic fraction of V, but the shear modulus and Young’s modulus are negatively correlated. When the mass fraction of V increases from 0 to 12 wt.%, the strength of α phase in the Zr-Ti-V alloys will be improved, while the hardness and rigidity of them will be weakened.

With the increasing of atomic fraction of V, the bulk modulus of BCC Zr-Ti-V alloys increases first and then remains unchanged. When the mass fraction of V increases from 0 to 2.4 wt.%, the strength of β phase in the Zr-Ti-V alloys will be enhanced, and when the mass fraction of V exceeds 2.4 wt.%, the strength of β phase in the Zr-Ti-V alloys is basically unchanged with the increasing mass fraction of V. In addition, the shear modulus and Young’s modulus of the BCC Zr-Ti-V alloys are negatively correlated with the element V, which indicates that the hardness and stiffness of β phase in Zr-Ti-V alloys will be reduced with the increasing of V element. In general, it can be seen that with the increasing of V mass fraction, the strength of Zr-Ti-V alloys improves, while the hardness and stiffness are opposite. When the mass fraction of V is around 2.4 wt.%, the Zr-Ti-V alloys achieve the best strength, hardness, and stiffness.

As shown in Figure 7, for Zr-Ti-Al alloys and Zr-Ti-V alloys, the B_H/G_H value of the BCC structure is greater than that of HCP. These two types of alloy systems have similar characteristics in Poisson’s ratio, which indicates that the ductility of α phase is lower than that of β phase in Zr-Ti-Al alloys and Zr-Ti-V alloys. In addition, for HCP Zr-Ti-Al alloys, when the Al mass fraction is between 0 and 5.2 wt.%, the increasing of Al atomic fraction has almost no effect on the B_H/G_H value of the HCP Zr-Ti-Al alloys. However, the B_H/G_H value of the HCP structure Zr-Ti-Al alloys tends to decrease after the Al mass fraction exceeds 5.2 wt.%, and the Poisson’s ratio of the HCP Zr-Ti-Al alloys also follows a similar rule. For BCC Zr-Ti-Al alloys, the increase of Al atomic fraction will reduce the B_H/G_H value and Poisson’s ratio of these alloy systems. It can be concluded that with the increasing of the Al mass fraction, the ductility of the β phase of the Zr-Ti-Al alloys will be weakened. On the contrary, for the Zr-Ti-V alloy systems, the ductility of these alloys will be improved as the V atomic fraction increases by observing the B_H/G_H value and Poisson’s ratio in Figure 7, which indicates that the ductility of Zr-Ti-V alloys will be enhanced when the mass fraction of V increases from 0 to 12 wt.%.

In this paper, we studied the influence of Zr and Ti on the phase composition of ternary Zr-Ti-X alloys (X = Al, V) by the content of different Al and V. In addition, based on the research that the changes of phase compositions and structure with alloying element content distinctly affect the mechanical properties of quaternary Zr-Ti-Al-V alloys, first-principles calculations could provide an effective guide for further exploring the effects of the amount of Zr and Ti on the phase composition and structure of quaternary Zr-Ti-Al-V alloys, thereby providing direction and help in discussing the relationship between Zr-Ti-X (X = Al, V) compositions and the mechanical properties of the multiphase Zr₄₇Ti₄₅Al₅V₃ alloy.

4. Conclusions

To improve our fundamental understanding of the roles of alloying elements in Zr-Ti-X alloys (X = Al, V), first-principles calculations were performed with the SQS method to consider the disordered atomic configurations in solid solution alloys. In this paper, the relationship between the microstructure and properties of the Zr-Ti-X alloys (X = Al, V) is studied on the basis of Zr₄₇Ti₄₅Al₅V₃ (wt.%) alloy, based on a certain ratio of Zr and Ti atomic fractions, the effects of different alloying elements (Al, V) mass fractions and different phase structures (HCP and BCC structure, α phase and β phase) on the lattice stability and elastic properties of Zr-Ti-X alloys (X = Al, V) are calculated by the first principles. The main conclusions are as follows.

• Al acts as an α stabilizer for Zr-Ti-Al alloys and V can stabilize the β phase for Zr-Ti-V alloys.
• For Zr-Ti-Al alloy systems, the strength of α phase is lower than that of β phase, but the hardness and stiffness of the α phase are higher than that of β phase. For Zr-Ti-V alloy systems, the hardness and stiffness of α phase are higher than that of β phase, while the ductility of β phase is generally higher than that of α phase.
• When the mass fraction of Al in Zr-Ti-Al alloys increases from 4 wt.% to 6.8 wt.%, Zr-Ti-Al alloys have relatively good strength, hardness, and rigidity, however, the ductility of Zr-Ti-Al alloys becomes worse as the mass fraction of Al increases. When the mass fraction of V in Zr-Ti-V alloys reaches 2.4 wt.%, Zr-Ti-V alloy achieves the best strength, hardness, and rigidity, and the ductility of Zr-Ti-V alloys improved as the mass fraction of V increases from 0 to 12 wt.%.
• Mechanical properties of quaternary Zr₄₇Ti₄₅Al₅V₃ alloy significantly depend on the amount of Zr and Ti; in addition, the changes of phase compositions and structure with Al content or V content distinctly affect mechanical properties of ternary Zr-Ti-X alloys (X = Al, V), thereby impacting the mechanical properties of the multiphase Zr₄₇Ti₄₅Al₅V₃ alloy from the point of view of the amount of Zr and Ti.