Influence of Vacancy on Structural Stability, Mechanical Properties and Electronic Structures of a Ti₅Sn₃ Compound from First-Principles Calculations

Abstract

Titanium alloy is widely used in biomedical materials. Ti-Sn alloy is a new type β titanium alloy with no toxicity. In this paper, the mechanical and electronic properties of Ti₅Sn₃ with vacancy defects have been studied by using first-principles method. The vacancy formation energy, vacancy formation enthalpy, elastic constant, elastic modulus, hardness and electronic structure of perfect Ti₅Sn₃ and Ti₅Sn₃ with different vacancies were also calculated and discussed. The results show that Ti₅Sn₃ is more likely to form vacancies at V_Ti2. In addition, the bulk deformation resistance of Ti₅Sn₃ is weakened by the vacancy, and the shear resistance, stiffness and hardness of Ti₅Sn₃ are increased by the Ti vacancy, but the brittleness of Ti₅Sn₃ is increased. On the contrary, the presence of Sn vacancy decreases the shear resistance, stiffness and hardness of Ti₅Sn₃, and increases the toughness of Ti₅Sn₃. By analyzing the change of electronic structure, it is found that removing the Ti atom at the V_Ti2 position can improve the interaction between atoms, while Sn vacancy can weaken the interaction.

1. Introduction

Titanium and titanium alloys have been widely used in the field of medical orthopedic materials due to their excellent properties such as high strength, low density, corrosion resistance, lack of toxicity and good biocompatibility [1,2,3]. The alloy used for these applications is based on the (α + β) type titanium alloy Ti-6Al-4V (TC4). However, the elastic modulus of TC4 does not match with human bone [4,5,6], and it contains elements harmful to the human body [7,8]. In recent years, with the development of materials science and biology, there is a requirement for higher performance of medical titanium and titanium alloy. Therefore, β titanium alloys with high strength, low elastic modulus, lack of toxicity and good biocompatibility have attracted extensive attention from researchers. With a better understanding of the microstructure and properties of β-type titanium alloys, the application of novel β-type titanium alloys without toxicity in orthopedic materials has become possible [9].

A large number of studies have shown that the addition of a certain amount of Sn in titanium alloys can help to improve the mechanical properties of titanium alloys [3]: (i) Sn can form solid solution strengthening in Ti, which increases the lattice parameters of Ti and increases the distance between alloy atoms [10,11]; (ii) Sn is a non-toxic element with good biocompatibility; (iii) Sn can reduce the initial martensite temperature, inhibit the formation of ω phase, and improve the elastic properties of the alloy [10,12]. Therefore, adding Sn is a good way of enhancing performance of titanium based medical orthopedic materials. In addition, Ti-Sn intermetallic compounds play an important role in improving the properties of alloys [13,14,15,16,17,18], and understanding the strengthening mechanism of intermetallic compounds is of positive significance to the study of Ti-Sn orthopedic materials. Nowadays, Ti-Sn intermetallic compounds have received more and more attention, and many studies have been conducted on the microstructure, elastic properties and phase stability of Ti₅Sn₃. For example, Tedenac et al. [19] have calculated the crystal structure and phase stability of Ti₅Sn₃ in Ti-Ni-Sn alloys by first-principles calculations. Chen et al. [20] have studied the structural stability, ideal strength and other properties of Ti₅Sn₃. Mechanical properties of solid materials are usually determined by the combination of different crystal structures and various defects (including vacancy, impurity, dislocation, etc.). Vacancy defects are inherent defects in solids and play an important role in determining the mechanical properties of solids. However, there are few studies on the physical properties of Ti₅Sn₃ and its vacancies. Therefore, the vacancy formation energy, elastic modulus, and electronic structure of Ti₅Sn₃ and Ti₅Sn₃ with different vacancies were studied by using first-principles calculations. This provides theoretical support for the design of Ti-Sn alloy.

2. Calculation Method and Model

First-principles calculations are implemented in the CASTEP module in Materials Studio based on density functional theory [21]. The correlation exchange energy between electrons is processed by the PBE function in the generalized gradient approximation (GGA) [22]. The interactions between the electrons and ion cores are employed by using an Ultrasoft pseudopotential for Ti and Sn atoms. [23]. After the convergence test, Ti₅Sn₃ truncation energy is set to 400 eV, the value of k-point in the Brillouin region is determined to be 4 × 4 × 4 by the Monkhorst-Pack method, as shown in

Table 1. Variation of total energies with different truncation energies at the same K point of Ti5Sn3.

Energy Cutoff (eV)	Total Energies (eV)
300	−16,608.69
350	−16,609.957
400	−16,610.513
450	−16,610.536
500	−16,610.538

and

Table 2. The K points, lattice parameters and total energies for pure Ti₅Sn₃.

K Point	a (Å)	c (Å)	E (eV)
2 × 2 × 4	8.074	5.468	−16,610.103
3 × 3 × 4	8.077	5.456	−16,610.238
4 × 4 × 4	8.081	5.449	−16,610.376
4 × 4 × 6	8.083	5.441	−16,610.404
5 × 5 × 7	8.083	5.439	−16,610.383
6 × 6 × 8	8.083	5.438	−16,610.385

. Geometric optimization of the established model was carried out through BFGS [24], and the most stable structure in the region was obtained. The convergence parameters of the crystal calculation process are as follows: the convergence precision of the total energy of the system is 1 × 10⁻⁵ eV/atom, the maximum Hellmann–Feynman force is 0.03 eV/Å, the maximum stress is less than 0.05 GPa, and the maximum ionic displacement is 1 × 10⁻³ Å. In order to confirm image–image interactions of the vacancies, the 1 × 1 × 1 and 1 × 1 × 2 supercell models of Ti₅Sn₃ with different vacancies were constructed in this work. The test results indicated that the effect of the periodic boundary condition is quite small, which is attributed to the distances between adjacent vacancies being at least 8 Å for the 1 × 1 × 1 model of Ti₅Sn₃ with vacancies.

The space group of Ti₅Sn₃ is P63/mcm, and the lattice constants are a = b = 8.049 Å, c = 5.454 Å, α = β = 90°, γ = 120° [25]. There are three kinds of atomic equivalent sites in Ti₅Sn₃ crystals, whose lattice coordinates are Ti 6g (0.230, 0, 0.250), Ti 4d (0.333, 0.667, 0) and Sn 6g (0.600, 0, 0.25), respectively. After geometric optimization, the calculated lattice parameters of Ti₅Sn₃ are a = b = 8.0805 Å and c = 5.449 Å, which are consistent with the experimental values and other theoretical results [19], which are also consistent with a = b = 8.0716 Å and c = 5.4823 Å. The original cell of Ti₅Sn₃ contains ten Ti atoms and six Sn atoms. The Ti₅Sn₃ compounds with various vacancies are obtained by removing single atoms from perfect Ti₅Sn₃ cells, then optimizing geometry again. In this way, the ground state structures of Ti₅Sn₃ with vacancies were established. Finally, the stability, electronic structure and elastic properties of the four optimized geometric structures were calculated and analyzed.

3. Results and Discussion

3.1. Structural Stability

In order to study the influence of vacancy defects on the structure and physical properties of Ti₅Sn₃, three crystal models with different vacancy defects were constructed (as shown in Figure 1), namely V_Ti1, V_Ti2 and V_Sn3. The vacancy formation energy (${\mathrm{E}}_{\mathrm{form}}^{\mathrm{x}}$) determines the structural stability of vacancy defects. In order to investigate the stability of different vacancies of Ti₅Sn₃, three vacancy formation energies are calculated, and the calculation formula is as follows [26]:

$${\mathrm{E}}_{\mathrm{form}}^{\mathrm{x}}={\mathrm{E}}_{\mathrm{hole}}^{\mathrm{x}}-{\mathrm{E}}_{\mathrm{host}}^{\mathrm{x}}+{\mathsf{\mu }}_x$$

(1)

In the formula, ${\mathrm{E}}_{\mathrm{hole}}^{\mathrm{x}}$ and ${\mathrm{E}}_{\mathrm{host}}^{\mathrm{x}}$ represent the total energy of Ti₅Sn₃ with vacancy defects and Ti₅Sn₃ without defects, respectively; μ_x (x = Ti1, Ti2, Sn) represents the chemical potential of the corresponding removed atoms. The calculation of chemical potential of Ti and Sn atoms should satisfy the following equation:

$$5{\mathsf{\mu }}_{\mathrm{Ti}}+3{\mathsf{\mu }}_{\mathrm{Sn}}={\mathsf{\mu }}_{\mathrm{Ti}5\mathrm{Sn}3}^{\mathrm{bulk}}$$

(2)

$${\mathsf{\mu }}_{\mathrm{Ti}}\le {\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{bulk}}$$

(3)

$${\mathsf{\mu }}_{\mathrm{Sn}}\le {\mathsf{\mu }}_{\mathrm{Sn}}^{\mathrm{bulk}}$$

(4)

$$\mathsf{\Delta }{\mathrm{H}}_{\mathrm{f}}\left({\mathrm{Ti}}_5{\mathrm{Sn}}_3\right)\le {\mathsf{\mu }}_{\mathrm{Ti}}-{\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{bulk}}\le 0$$

(5)

where ΔH_f (Ti₅Sn₃) is the formation energy of a perfect Ti₅Sn₃ compound. ${\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{bulk}}$ and ${\mathsf{\mu }}_{\mathrm{Sn}}^{\mathrm{bulk}}$ are the chemical potentials of Ti and Sn in the bulk perfect Ti₅Sn₃ compound.

Obviously, the chemical potential of Ti atoms depend on the environment in which the vacancy is created. That is, the limiting cases of Sn-rich and Ti-rich. In the Ti-rich environment, μ_Ti approaches the chemical potential of bulk Ti, μ_Ti = ${\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{bulk}}$= −1603.122 eV. The chemical potential of Sn can be determined from the thermochemical equilibrium condition in Equation (2), μ_Sn = (${\mathsf{\mu }}_{\mathrm{Ti}5\mathrm{Sn}3}^{\mathrm{bulk}}$ − 5${\mathsf{\mu }}_{\mathrm{Ti}}^{\mathrm{bulk}}$)/3 = −96.529 eV. In the Sn-rich environment, μ_Sn is close to the chemical potential of bulk Sn, μ_Sn = ${\mathsf{\mu }}_{\mathrm{Sn}}^{\mathrm{bulk}}$= −95.483 eV, μ_Ti = $({\mathsf{\mu }}_{\mathrm{Ti}5\mathrm{Sn}3}^{\mathrm{bulk}}$ − 3${\mathsf{\mu }}_{\mathrm{Sn}}^{\mathrm{bulk}}$) = −1603.750 eV. Therefore, the formation energy variation curve of Ti₅Sn₃ with different Ti and Sn vacancies in different environments can be obtained, as shown in Figure 2.

The lower the enthalpy of formation of a solid substance, the more stable its thermodynamic stability is. In order to study the ideal thermodynamic stability of Ti₅Sn₃ and Ti₅Sn₃ with different vacancies, the enthalpy of formation was calculated according to the given equation:

$$\mathsf{\Delta }\mathrm{H}\left({\mathrm{Ti}}_5{\mathrm{Sn}}_3\right)=\frac{1}{\mathrm{x}+\mathrm{y}}\left({\mathrm{E}}_{\mathrm{total}}\left({\mathrm{Ti}}_5{\mathrm{Sn}}_3\right)-{\mathrm{xE}}_{\mathrm{Ti}}-{\mathrm{yE}}_{\mathrm{Sn}}\right)$$

(6)

Table 3. Lattice parameters, vacancy formation energy and enthalpy of formation of perfect Ti₅Sn₃ and different vacant Ti₅Sn₃.

Compound	a (Å)	b (Å)	c (Å)	Ef (eV) Ti-Poor Ti-Rich		ΔH (kJ/mol)
Ti5Sn3	8.0805	8.0805	5.4490			−37.536
Other a	8.0716	8.0716	5.4823
Other b	8.0650	8.0650	5.4350			−34.870
VTi1	8.3301	7.8319	5.4883	1.094	1.722	−29.856
VTi2	7.9708	7.9704	5.4395	0.948	1.576	−31.008
VSn3	7.9322	7.9860	5.3740	2.311	3.357	−18.528

^a Ref. [20]. ^b Ref. [27].

below lists the row forming energy and enthalpy of formation of vacant positions at three different positions. If the enthalpy of formation of a solid material is negative, it indicates that the structure is thermodynamically stable in the ground state. It can be seen from Table 3 that the enthalpies of formation of intact Ti₅Sn₃ and different vacancy Ti₅Sn₃ are all less than zero, indicating that the Ti₅Sn₃ structures with these three vacancy defects are thermodynamically stable in the ground state. Among the three different vacancy models, the enthalpy of formation of V_Ti2 is the smallest, indicating that the removal of Ti atoms at this position is conducive to the formation of a more stable dynamic structure. In addition, the vacancy formation energy and formation enthalpy of Ti atomic vacancy are both smaller than those of the Sn atomic vacancy, indicating that the structure of Ti₅Sn₃ with the Ti atomic vacancy in the ground state is more stable than that with the Sn vacancy. The vacancy formation energy is mainly determined by the chemical potential. The lower the vacancy formation energy, the better the dynamic stability. It can be seen from Table 3 and Figure 2 that the vacancy formation energy of V_Ti2 is the lowest, which means that the possibility of V_Ti2 vacancy formation is greater for Ti₅Sn₃.

It can be seen from Table 3 that, compared with complete Ti₅Sn₃, the lattice constants of Ti₅Sn₃ with different vacancies show different degrees of shrinkage or expansion. The lattice constant of Ti₅Sn₃ with V_Ti2 vacancy shrinks along the a, b and c axis lattice, which enhances the interaction between the atoms. The V_Sn3 vacancy is the same as the V_Ti2 vacancy, which causes lattice contraction, but there are differences between the two. It can be clearly seen from Table 3 that vacancy formation energies of Ti₅Sn₃ containing Ti vacancy defects are all lower than those containing Sn vacancy, that is, Ti atomic vacancy has a stronger stability than Sn atomic vacancy.

3.2. Mechanical Properties

The elastic property is an important aspect of the mechanical property and structural stability of materials. The elastic property of Ti₅Sn₃ is calculated by the stress–strain method. Ti₅Sn₃ has five different second-order elastic constants, and the mechanical property stability is determined by the Born criterion [28,29,30]:

C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁ − C₁₂ > 0;
4C₁₃ + C₃₃ + 2(C₁₁ + C₁₂) > 0, C₁₁ + C₃₃ > 2C₁₃, C₆₆ = (C₁₁ − C₁₂)/2

(7)

The elastic constants of complete Ti₅Sn₃ and Ti₅Sn₃ with different vacancies are listed in

Table 4. Elastic constant C_ij (GPa) of Ti₅Sn₃ with perfect and different vacancies.

Compound	C11	C12	C13	C33	C44	C66
Ti5Sn3	192.18	116.80	45.03	220.82	48.02	37.69
Other a	182.23	111.22	46.60	199.72	47.43	35.51
VTi1	194.78	69.61	51.56	190.25	63.25	62.58
VTi2	204.20	73.12	45.55	191.47	56.63	65.54
VSn3	165.25	96.14	59.38	182.70	24.57	34.56

^a Ref. [13].

. For the perfect Ti₅Sn₃, although there is a very small difference between the calculated elastic constants and the theoretical data for slight fluctuations, it is consistent with the theoretical results. It is not difficult to see that both intact and different vacancies of Ti₅Sn₃ show mechanical stability, because their elastic constants meet the Born stability criterion. However, the C₁₁ value of intact Ti₅Sn₃ is smaller than that of intact Ti₅Sn₃ with Ti vacancy, which indicates that the existence of Ti vacancy makes the Ti₅Sn₃ crystal have stronger deformation resistance along the a-axis. On the contrary, the vacancy of Sn weakens the deformation resistance of Ti₅Sn₃ along the a-axis. In addition, the C₃₃ of perfect Ti₅Sn₃ is greater than that of vacant Ti₅Sn₃. Obviously, the vacancy weakens the deformation resistance along the c-axis. In addition, the C₄₄ and C₆₆ of Ti₅Sn₃ with Ti vacancy are greater than that of perfect Ti₅Sn₃, which indicates that the existence of the Ti vacancy enhances the shear resistance of Ti₅Sn₃. On the contrary, the Sn vacancy weakens the shear resistance of Ti₅Sn₃.

According to Table 4, through the Voigt-Reuss-Hill (VRH) relation, the elastic constant C_ij is used to calculate the macroscopic elastic parameters, such as the volume elastic modulus (B), shear elastic modulus (G) and Poisson’s ratio (ν). The formula is as follows:

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

(8)

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

(9)

The deformation resistance and elastic strain of the solid are mainly expressed by Vickers hardness, which is one of the important parameters of mechanics. Therefore, the intrinsic hardness of perfect Ti₅Sn₃ and Ti₅Sn₃ with vacancy was studied. Their Vickers hardness can be expressed by the following formula [31,32]:

H_v = 2((G/B)²·G)^0.585 − 3

(10)

Various macroscopic elastic parameters calculated are listed in

Table 5. Volume modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (ν) and intrinsic hardness (H_v) of perfect and different vacancy Ti₅Sn₃.

Compound	B (GPa)	G (GPa)	E (GPa)	B/G	ν	Hv (GPa)
Ti5Sn3	112.85	50.39	131.58	2.24	0.31	10.91
Other a	108.11	50.06	124.60	2.26	0.31
VTi1	99.21	61.19	152.27	1.62	0.24	15.82
VTi2	103.08	63.28	157.59	1.63	0.25	16.14
VSn3	103.74	25.71	71.24	4.04	0.38	3.64

^a Ref. [13].

. It can be seen from Table 5 that the bulk modulus of Ti₅Sn₃ with vacancy is smaller than that of perfect Ti₅Sn₃. This indicates that the removal of Ti and Sn atoms can weaken the volumetric deformations resistance of the crystal. In addition, the Ti vacancy can enhance the shear deformation resistance and stiffness of Ti₅Sn₃, because the shear modulus and Young’s modulus of Ti₅Sn₃ with Ti vacancy are greater than those of intact Ti₅Sn₃. Consistent with the shear modulus and Young’s modulus, the Vickers hardness of Ti vacancy Ti₅Sn₃ is greater than that of perfect Ti₅Sn₃, indicating that the Ti vacancy leads to the increase in Vickers hardness of Ti₅Sn₃. The value of Poisson’s ratio is generally between 0~0.5. The greater the hardness of the material, the smaller the Poisson’s ratio; On the contrary, the softer the material, the greater the Poisson’s ratio, which is fully proved by the data of Poisson’s ratio and hardness in Table 5. According to Pugh’s empirical rule [33,34,35], if the value of B/G is less than 1.75, the material has ductility, otherwise it does not. Obviously, the Pugh ratio of complete Ti₅Sn₃ obtained from Table 5 is 2.24, which is ductile, and the B/G of Sn vacancy Ti₅Sn₃ is 4.04, indicating that the Sn vacancy can improve the ductility of the material. However, Ti vacancy Ti₅Sn₃ is less than 1.75, indicating that the Ti vacancy can reduce the ductility of Ti₅Sn₃. Compared with Ti vacancy, Sn vacancy has a greater influence on Ti₅Sn₃. It can be clearly seen from Table 5 that the removal of Sn atom greatly reduces the shear modulus, Young’s modulus and hardness of Ti₅Sn₃.

3.3. Electronic Structure

In order to further study the chemical bonding and mechanical properties of Ti₅Sn₃ with different vacancies, densities of states of perfect Ti₅Sn₃ and Ti₅Sn₃ with different vacancies were calculated in this paper, as shown in Figure 3 below. The dotted line positions indicate the Fermi energy levels. It can be seen that the density of states of perfect Ti₅Sn₃ and Ti₅Sn₃ with different vacancies are very similar. Obviously, in Figure 3a, from −10.6 eV to −6.2 eV, the 5s state of Sn atom plays a major role. From −5.9 eV to the Fermi level, the density of this region is mainly contributed by Sn–5p and Ti–3d, and Sn–5p and Ti–3p also have a small contribution. In addition, charge transfer from the Sn–4s state to the 3p state occurs in this part. To the right of the Fermi level, the density of this segment is mainly contributed by Ti–3d, and there is a large overlap in this range, which indicates that there is a strong orbital hybridization between Ti and Sn, and between Ti and Ti, forming the Ti–Sn and Ti–Ti bonds. In addition, it can be seen from Figure 3b–d that the vacancy causes the valley phase change near the Fermi level. Ti vacancy makes the valley phase transition shallow, and Sn vacancy makes the valley phase increase significantly. As shown in Figure 3c, for V_Ti2, the density of states forms a small peak at the Fermi level, which indicates that the removal of Ti atoms at the V_Ti2 position is conducive to the interaction between charges, which strengthens the Ti–Sn bond. For V_Sn3, the density of states forms a valley phase at the Fermi level, indicating that the interatomic hybridization between Ti and Sn is weakened and the Ti–Sn bond is weakened.

It is well known that vacancies alter chemical bonding states and mechanical properties. In order to better explore the chemical bonding characteristics and mechanical properties of Ti₅Sn₃ and different vacant Ti₅Sn₃, this paper calculated their charge density difference respectively, as shown in Figure 4. Blue represents the absence of electrons, red represents the enrichment of electrons, and white represents no change in electron density. It can be seen from Figure 4a that on the (001) plane, electron deletion mainly occurs around Sn atoms and electron enrichment mainly occurs around Ti atoms, indicating the formation of Ti-Sn bond. For perfect Ti₅Sn₃, the Ti1–Sn, Ti2–Sn and Ti2–Ti2 bonds on (001) plane is 2.766 Å, 2.840 Å and 2.725 Å, respectively (as shown in

Table 6. The bond lengths in the (0 0 1) surface of Ti₅Sn₃ with different vacancies.

Bonds	Ti5Sn3 (Å)	VTi1 (Å)	VTi2 (Å)	VSn3 (Å)
Ti1-Sn	2.766	2.836	2.758	2.873
Ti2-Sn	2.840	2.890	2.840	2.899
Ti2-Ti2	2.725	2.699	2.720	2.729

). For the vacancy of Sn, the removal of the Sn atom weakens the bonding strength and hybridization between the Sn atom and Ti atom, so the deformation resistance becomes weak, which is also the reason for the decrease in the elastic modulus. It can be seen from Table 6 that the Ti vacancy, especially the removal of Ti atoms at the V_Ti2 position, is conducive to the improvement of the strength of Ti–Sn bond and Ti–Ti, which indicates that the interaction between Ti atoms and between Ti atoms and Sn atoms is enhanced.

4. Conclusions

In this paper, the mechanical properties and electronic structures of perfect Ti₅Sn₃ and different vacant Ti₅Sn₃ are studied by using first-principles calculations. The comprehensive calculation results draw the following conclusions:

(1)

According to vacancy formation energy and vacancy formation enthalpy, Ti vacancy has better structural stability than Sn vacancy, and Ti₅Sn₃ is more inclined to form vacancy at V_Ti2.

(2)

According to the calculation results of elastic properties, the vacancy weakens the bulk deformation capacity of Ti₅Sn₃, and the Ti vacancy increases the shear resistance, stiffness and hardness of Ti₅Sn₃, while the vacancy of Sn is the opposite.

(3)

Ti vacancy can increase the brittleness of Ti₅Sn₃, while Sn vacancy can strengthen the toughness of Ti₅Sn₃.

(4)

The changes in electronic structure and chemical bonds indicate that the removal of the Ti atom at the V_Ti2 position can strengthen the interatomic interaction, while the Sn vacancy will weaken the interaction.