First-Principles Study of the Effects of Ti Content on Mechanical Properties and Microscopic Mechanism in Cu₂AlMn_1–xTi_x Alloys

Abstract

It has been found that the addition of Ti can improve the strength of Cu-Al-Mn alloys and adjust their mechanical properties. However, the internal mechanism has not been fully understood. In order to clarify the influence of Ti content on the mechanical properties and microscopic mechanism of Cu-Al-Mn alloys, the mechanical, structural, and electronic properties of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.50, 0.75, 1) alloys were studied by first-principles calculations. Results show that the substituted Ti prefers to occupy the Mn site directly due to the lower formation energy. With the increase of Ti substitution content, the L2₁ phase stability of the alloy improves. Moreover, the elastic modulus of the alloy increases and the anisotropy factor decreases. Further analysis shows that the proportion of antibonding states under the Fermi energy of the alloy decreases and the covalent bond is enhanced after Ti substitutes Mn, which is the main mechanism for the enhancement of stability and mechanical properties. Mulliken charges change little after Ti replaces Mn, indicating that Ti has little effect on the ionic bond strength.

1. Introduction

Cu-Al-Mn alloys are widely used to produce vibration-damping components because they exhibit significant superelastic properties at room temperature and are also considered one of the most promising superelastic materials for large-scale applications due to their low manufacturing costs [1,2,3,4,5,6,7]. However, Cu-Al-Mn alloys have the disadvantages of low strength and high anisotropy factor, which greatly limits their extended application [7]. Experiments have shown that the superelasticity hysteresis curve of Cu-Al-Mn alloys varies significantly with crystal orientation under uniaxial stretching [8,9,10]. The superelastic strain of [10] crystal orientation is obviously larger than that of other crystal orientations. It is widely believed that Cu-Al-Mn alloys have obvious mechanical anisotropy, which leads to extremely strong strain incongruence during deformation and stress concentration at grain boundaries. Columnar crystal alloys with specific crystal orientations instead of common polycrystalline alloys have been proposed as an effective strategy to optimize the superelastic properties of Cu-Al-Mn alloys [11,12].

Since it is challenging to explain the internal mechanism of alloys’ mechanical property change in experiments, first-principles simulation has become more popular as a useful tool [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The differential charge has shown that the significant anisotropy in the mechanical properties of Cu-Al alloys is due to the apparent directivity of the strong bonding between the nearest-neighbor Cu atoms [16]. The addition of Mn has little effect on the mechanical anisotropy of Cu-Al-based alloys because the elastic constants are not sensitive to the changes of Mn content [17]. In terms of phase stability of alloys, some studies on Hessler alloys have shown that the superelastic effect decreases with the increase of the stability of the L2₁ phase during tetragonal distortion [18,19]. With respect to the electronic properties of alloys, interatomic bonding was usually used to explain the variation in mechanical properties [20,21]. For example, in Cu-Ti alloys, the atomic bonds in Cu-Ti intermetallics are mixed bonds and the covalent bond nature determines the phase stability and mechanical properties [20]. In Cu-Zn-Sb alloys, the strength of the Cu-Sb bond is stronger than that of other bonds [21].

In addition, doping certain elements (such as Ti, Ni, Zn, and Ga) to Cu-Al-Mn alloys can improve their mechanical properties [29,30,31,32,33,34,35]. In particular, doping Ti into Cu-Al-Mn-based alloys can improve their strength because the Ti-rich phase formed can effectively inhibit grain growth [34]. In Cu-Al-Mn-Ti alloys, the dispersive L2₁-Cu₂TiAl precipitates strengthen the stabilization of stress-induced 2H(γ′₁) martensite from the L2₁-Cu₂AlMn parent, which results in superelasticity at deformation and shape memory effect after unloading by heating [35]. It was also found that the shape memory effects of the alloys decrease with the increase of the addition of Ti when the content of Ti is over 4.6wt%. However, the internal mechanism of the effect of Ti on the mechanical properties of Cu-Al-Mn alloys has not been fully understood. In this paper, Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, 1) alloys were selected as the research object, based on the phenomenon that Ti replaces Mn to form a new alloy phase with Cu and Al. To study the effects of Ti content on the mechanical properties and microscopic mechanism of Cu₂AlMn_1–xTi_x alloys, the structural, mechanical, and electronic properties were examined using the first-principles method.

2. Materials and Methods

The austenite of Cu-Al-Mn alloy is a highly ordered L2₁ structure that contains four types of lattice points and its chemical formula is Cu₂AlMn [22]. As shown in Figure 1, Type I and Type II are occupied by Mn and Al atoms respectively, and Type III and Type IV are occupied by Cu atoms. Each of the four types of lattice sites individually forms a face-centered cubic (FCC) lattice.

All simulations were implemented using the Vienna Ab-initio Simulation Package (VASP) [36,37] based on density functional theory. Generalized gradient approximation (GGA) [38] and Perdew-Burke-Enzerh exchange-correlation (PBE) functional [39] were used to describe the exchange-correlation potential. The projector augmented-wave (PAW) pseudopotential approach was used to describe atomic potentials [37,40]. Typical electronic configurations of Cu, Al, Mn, and Ti are 3p⁶3d¹⁰4s¹, 3s²3p¹, 3p⁶3d⁵4s², and 3p⁶3d²4s², respectively. A kinetic energy cutoff of 700 eV, an energy convergence criterion of 10⁻⁷ eV during the electronic self-consistency loop, and a force convergence criterion of 0.005 eV/Å during the structural relaxation were adopted. The first Brillouin zone was sampled using 13 × 13 × 13 k-points in the Monkhorst-Pack scheme [41]. The lattice constant of the unit cell after relaxation is 5.933 Å, which agrees well with the experimentally measured values of 5.965 Å [42], and 5.957 Å [43].

3. Results and Discussion

3.1. Structural Properties and Phase Stability

In order to study the effects of Ti content on mechanical properties and microscopic mechanism in Cu₂AlMn_1–xTi_x alloys, it is necessary to determine the atom occupation mode. All possible configurations for Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1) alloys were constructed. Figure 2 shows the configurations when x = 0.25. When Ti replaces Mn, five different structural models may exist. One of them is constructed by direct substitution, i.e., Ti is located directly at the Mn site (Figure 2b, termed as Ti_Mn). The other four structural models belong to the indirect substitution, including: (1) Ti is located at the Al site, and the Mn site is occupied by Al (Figure 2c, termed as Ti_AlAl_Mn); (2) Ti is located at the Cu site and the Mn site is occupied by Cu (Figure 2d, termed as Ti_CuCu_Mn); (3) Ti is located at the Al site, the Al is located at Cu site and Mn site is occupied by Cu (Figure 2e, termed as Ti_AlAl_CuCu_Mn); (4) Ti is located at Cu site, Cu is located at Al site and Mn site is occupied by Al (Figure 2f, termed as Ti_CuCu_AlAl_Mn). In addition, when x > 0.25, mixed substitution was also considered. Atomic positions in various structural models of Cu₂AlMn_1–xTi_x (x = 0.25, 0.5, 0.75, and 1) alloys were shown in

Table 1. The formation energy of Cu₂AlMn_1–xTi_x alloys (x = 0, 0.25, 0.50, 0.75, and 1).

x	Occupation Manner	Atomic Site Occupation			Ef (eV/atom)
		Ⅰ	Ⅱ	Ⅲ + Ⅳ
0	No substitution	Mn	Al	Cu2	–0.128
0.25	Direct substitution	Mn0.75Ti0.25	Al	Cu2	–0.157
	Indirect substitution	Mn0.75Al0.25	Al0.75Ti0.25	Cu2	–0.087
		Mn0.75Cu0.25	Al	Cu1.75Ti0.25	–0.114
		Mn0.75Cu0.25	Al0.75Ti0.25	Cu1.75Al0.25	–0.082
		Mn0.75Al0.25	Al0.75Cu0.25	Cu1.75Ti0.25	–0.096
0.5	Direct substitution	Mn0.50Ti0.50	Al	Cu2	–0.197
	Indirect substitution	Mn0.50Al0.50	Al0.50Ti0.50	Cu2	–0.122
		Mn0.50Cu0.50	Al	Cu1.50Ti0.50	–0.068
		Mn0.50Cu0.50	Al0.50Ti0.50	Cu1.50Al0.50	–0.109
		Mn0.50Al0.50	Al0.50Cu0.50	Cu1.50Ti0.50	–0.079
	Mixed substitution	Mn0.50Ti0.25Al0.25	Al0.75Ti0.25	Cu2	–0.102
		Mn0.50Ti0.25Cu0.25	Al	Cu1.75Ti0.25	–0.141
0.75	Direct substitution	Mn0.25Ti0.75	Al	Cu2	–0.247
	Indirect substitution	Mn0.25Al0.75	Al0.25Ti0.75	Cu2	–0.180
		Mn0.25Cu0.75	Al	Cu1.25Ti0.75	–0.153
		Mn0.25Cu0.75	Al0.25Ti0.75	Cu1.25Al0.75	–0.129
		Mn0.25Al0.75	Al0.25Cu0.75	Cu1.25Ti0.75	–0.137
	Mixed substitution	Mn0.25Ti0.25Al0.50	Al0.50Ti0.50	Cu2	–0.142
		Mn0.25Ti0.25Cu0.50	Al	Cu1.50Ti0.50	–0.096
		Mn0.25Ti0.50Al0.25	Al0.75Ti0.25	Cu2	–0.167
		Mn0.25Ti0.50Cu0.25	Al	Cu1.75Ti0.25	–0.171
1	Direct substitution	Ti	Al	Cu2	–0.306

. It is worth noting that all structural models are cubic except for x = 0.5 (tetragonal).

The formation energy (E_f) of alloys, which can be used to analyze the thermodynamic stability, was calculated by Equation (1).

$$E_{\mathrm{f}}=\frac{E_{\mathrm{total}}-N_{\mathrm{Cu}}{\mu }_{\mathrm{Cu}}-N_{\mathrm{Al}}{\mu }_{\mathrm{Al}}-N_{\mathrm{Mn}}{\mu }_{\mathrm{Mn}}-N_{\mathrm{Ti}}{\mu }_{\mathrm{Ti}}}{N_{\mathrm{total}}}$$

(1)

where E_total represents the total ground-state energy of the unit cell, N_X (X = Cu, Al, Mn, Ti) represents the number of corresponding elements X in the cell, and μ_X (X = Cu, Al, Mn, Ti) represents the average atomic energy of the element X. The calculated formation energy was listed in Table 1. It is known that lower formation energy means that the thermodynamic stability of the compound is higher. As shown in Table 1, the configuration constructed by direct substitution always has the lowest formation energy. Therefore, the configurations constructed by direct substitution were utilized for further calculation.

To clarify the effect of Ti on phase stability, tetragonal distortion under the Bain path was applied to L2₁ lattices of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1) alloys. Figure 3 displays the energy changes (E–E₀) of the Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1) alloys under different tetragonal distortions (Δa/a). The L2₁ phase (Δa/a = 0) always has the lowest total energy for different substitution content of Ti, indicating that the alloys have no transition tendency to martensite. With the increase of Ti substitution content, it can be found that the energy curves near the origin become steeper, which indicates that the L2₁ phase stability of Cu₂AlMn_1–xTi_x alloys increases. This is consistent with the phenomenon that the superelastic effect properties of Cu-Al-Mn-Ti alloys start to deteriorate when the addition of Ti exceeds 4.6wt % [35].

3.2. Mechanical Properties

The elastic constants (C_ij) are the comprehensive representation of the elastic state and the mechanical properties of crystalline materials [28]. There are three independent nonzero elastic constants (C₁₁, C₁₂, C₄₄) for cubic crystals and six independent nonzero elastic constants (C₁₁, C₃₃, C₁₂, C₁₃, C₄₄, C₆₆) for tetragonal crystals, which were both determined by performing six finite distortions of the lattice and deriving the elastic constants from the strain-stress relationship [44].

Table 2. Lattice volumes and elastic constants of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1).

Alloys	Source	V0 (Å3)	C11/C33 (GPa)	C12/C13 (GPa)	C44/C66 (GPa)
Cu2AlMn	This work.	208.83	136.94	120.40	106.15
	Experimental [47].	212.08	128.1	101.5	104.4
	Vanderbilt [17].	210.63	138.8	111.3	102.0
	FPLAPW [48].	——	137	115	112
	FPLAPW [49].	208.95	143.7	116.1	117.6
Cu2AlMn0.75Ti0.25	This work.	212.78	144.38	120.15	101.27
Cu2AlMn0.50Ti0.50	This work.	213.98	151.67/162.11	126.22/116.31	96.32/103.50
Cu2AlMn0.25Ti0.75	This work.	218.02	164.65	114.06	94.14
Cu2AlTi	This work.	219.76	178.95	115.02	101.32

shows the lattice volumes and elastic constants of Cu₂AlMn_1–xTi_x. It is found that the calculated results in this work are in good agreement with the previous experimental and theoretical data. Based on Born stability criteria [45], for cubic crystals the elastic constants need to satisfy:

$$C_{44}>0,C_{11}>\left|C_{12}\right|,C_{11}+2C_{12}>0,$$

(2)

and for tetragonal crystals the elastic constants need to satisfy [46]:

$$\begin{array}{l}{C}_{11}>\left|C_{12}\right|,C_{44}>0,C_{66}>0,\\ (C_{11}+C_{12})C_{33}>2C_{13}^2.\end{array}$$

(3)

All five structural models in this work satisfied the mechanical stability.

As shown in Table 2, the lattice volume increases after Ti substitute Mn, resulting in a decrease in the density. C₁₁ and C₃₃ are significantly higher than other elastic constants, indicating that the resistance to normal stresses of the alloys is higher. With the increase of Ti substitution content, the increase of C₁₁ and C₃₃ is relatively obvious, while the decrease of C₁₂, C₁₃, C₄₄, and C₆₆ is relatively small. This is different from the results of the effects of Mn content on the mechanical properties of Cu-Al alloys. In Cu-Al-Mn alloys, the decrease in C₁₂ value is most obvious with the change in Mn content [17]. Adding Mn can improve the stability of Cu-Al alloys, but has no significant effect on axial deformation resistance. The substitution of Mn for Ti not only improves the stability of the alloy but also improves the axial deformation resistance of the alloy.

To further investigate the elastic anisotropy of Cu₂AlMn_1–xTi_x, a three-dimensional surface representation of the elastic anisotropy was employed to show the variation of the elastic modulus with crystal orientation. It is well known that the anisotropy of mechanical properties of materials mainly arises from the atomic regularity of the spatial arrangement. Young’s modulus of arbitrary crystal orientation in cubic crystals can be expressed as [50]:

$$E_n=\frac{1}{(S_{11}-(2S_{11}-2S_{12}-S_{44})(l_1^2{l}_2^2+l_1^2{l}_3^2+l_2^2{l}_3^2))}$$

(4)

Young’s modulus of arbitrary crystal orientation in tetragonal crystals can be expressed as [50]:

$$E_n=\frac{1}{(S_{11}(l_1^4+l_2^4)+2S_{12}{l}_1^2{l}_2^2+2S_{13}(l_1^2{l}_3^2+l_2^2{l}_3^2)+S_{33}{l}_3^4+S_{44}(l_1^2{l}_3^2+l_2^2{l}_3^2)+S_{66}{l}_1^2{l}_2^2)}$$

(5)

where l_i is the cosine of the angle between the chosen direction n and the i-th crystal basis vector. Equations (4) and (5) displays the relationship between Young’s modulus (E_n) in any crystal orientation and the constants of the flexibility matrix (S_ij, as shown in

Table 3. Flexibility matrix constants of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1).

Alloys	S11/S33 (GPa−1)	S12/S13 (GPa−1)	S44/S66 (GPa−1)
Cu2AlMn	0.0401	−0.0187	0.0094
Cu2AlMn0.75Ti0.25	0.0284	−0.0129	0.0099
Cu2AlMn0.50Ti0.50	0.0242/0.0154	−0.0065/−0.0151	0.0103/0.0095
Cu2AlMn0.25Ti0.75	0.0140	−0.0057	0.0106
Cu2AlTi	0.0112	−0.0044	0.0099

).

Figure 4 shows the three-dimensional surface of Young’s modulus anisotropy for different x values based on Equations (4) and (5). The crystal direction and the corresponding elastic modulus were represented by the coordinates and lengths of points on the surface, respectively. The closer the shape of the three-dimensional surface is to a sphere, the lower the anisotropy of Young’s modulus is. As shown in Figure 4, with the increase of Ti substitution content, the three-dimensional surface becomes closer and closer to a sphere, which indicates that Ti substituting Mn reduces the magnitude of mechanical anisotropy of Cu₂AlMn alloy.

Then, in order to quantitatively understand the mechanical properties and the magnitude of anisotropy of polycrystals, the V-R-H model approximation was used to calculate the bulk modulus, shear modulus, and Young’s modulus of the alloys. The V-R-H model gives the following Equations (6)–(9) [51], and the anisotropy factors of alloys were calculated by Equation (10) [16,52]. The results were shown in

Table 4. The bulk modulus (B_H), shear modulus (G_H), Young’s modulus (E), Poisson’s ratio, anisotropy factor, and G/B of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, 1).

Alloys	B (GPa)	G (GPa)	E (GPa)	υ	AE	AG	G/B
Cu2AlMn	125.66	42.98	115.45	0.35	0.52	0.56	0.34
Cu2AlMn0.75Ti0.25	128.22	45.64	122.39	0.34	0.39	0.44	0.36
Cu2AlMn0.50Ti0.50	133.66	48.84	130.55	0.34	0.31	0.34	0.37
Cu2AlMn0.25Ti0.75	130.92	55.90	146.63	0.31	0.17	0.19	0.43
Cu2AlTi	136.33	63.96	166.14	0.30	0.13	0.15	0.47

.

Voigt and Reuss models of cubic crystals:

$$\begin{array}{l}{B}_{\mathrm{V}}=B_{\mathrm{R}}=(C_{11}+2C_{12})/3\\ G_{\mathrm{V}}=(C_{11}-C_{12}+3C_{44})/5\\ G_{\mathrm{R}}=5(C_{11}-C_{12})C_{44}/[4C_{44}+3(C_{11}-C_{12})]\end{array}$$

(6)

Voigt and Reuss models of tetragonal crystals:

$$\begin{array}{l}M=C_{11}+C_{12}+2C_{33}-4C_{13}\\ C^2=(C_{11}+C_{12})C_{33}-2C_{13}^2\\ B_{\mathrm{V}}=[2(C_{11}+C_{12})+C_{33}+4C_{13}]/9\\ G_{\mathrm{V}}=(M+3C_{11}-3C_{12}+12C_{44}+6C_{66})/30\\ B_{\mathrm{R}}=C^2/M\\ G_{\mathrm{R}}=15{\{(18B_{\mathrm{V}}/C^2)+[6/(C_{11}-C_{12})]+6/C_{44}+3/C_{66}\}}^{-1}\end{array}$$

(7)

Hill model averages Voigt and Reuss models:

$$B_{\mathrm{H}}=(B_{\mathrm{V}}+B_{\mathrm{R}})/2,G_{\mathrm{H}}=(G_{\mathrm{V}}+G_{\mathrm{R}})/2$$

(8)

Young’s modulus and Poisson’s ratio are calculated by the following equation:

$$E=9B_{\mathrm{H}}{G}_{\mathrm{H}}/(3B_{\mathrm{H}}+G_{\mathrm{H}}),\nu =(3B_{\mathrm{H}}-2G_{\mathrm{H}})/(6B_{\mathrm{H}}+2G_{\mathrm{H}})$$

(9)

The anisotropy factors of Young’s modulus and shear modulus were calculated using the following equation:

$$A_E=(E_{\mathrm{V}}-E_{\mathrm{R}})/(E_{\mathrm{V}}+E_{\mathrm{R}}),A_G=(G_{\mathrm{V}}-G_{\mathrm{R}})/(G_{\mathrm{V}}+G_{\mathrm{R}})$$

(10)

The results in Table 4 show that the substitution of Mn by Ti can increase the bulk modulus, shear modulus, and Young’s modulus of polycrystals. When Mn is completely substituted by Ti, the shear modulus and Young’s modulus of the alloys reach the highest values. This indicates that Ti substituting Mn can improve the deformation resistance of the alloy. The calculation of the mechanical anisotropy factor of the polycrystal shows that the substitution of Mn by Ti reduces the anisotropy of the alloy, which is consistent with the previous three-dimensional surface. The decrease of mechanical anisotropy can increase the strain coordination between grains with different crystal orientations during deformation and thus reduce the stress concentration at grain boundaries. In addition, the values of G/B were calculated to represent the ductility of the alloy, where the material is ductile when G/B is less than 0.5. As shown in Table 4, all alloys are ductile materials and the value of G/B of the alloys increases with the increase of x, indicating that adding Ti can make the alloy brittler.

In general, the addition of Ti can improve the deformation resistance and reduce the mechanical anisotropy of Cu₂AlMn alloys.

3.3. Density of States and Bonding Characteristics

In this section, the bonding characteristics were analyzed to examine the underlying mechanism for the change in mechanical properties.

To clarify the variation in covalent bonding, the total densities of states (TDOS) near Fermi Energy of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1) alloys were calculated, as shown in Figure 5. At Fermi energy, there is a sufficient density of states for all systems with different x values, which indicates that compounds are metallic in nature. In spin-down TDOS, the distribution of peaks is essentially unchanged with the increase of x. However, in spin-up TDOS, the peak between −2.50 and −1.50 eV gradually disappears with the increase of x, while a new peak is generated between 0.50 and 1.12 eV. To further understand the bonding properties, the crystal orbital Hamiltonian population (COHP) analysis method [53,54,55] was employed to determine the bonding characteristics. The −COHP values for the interactions of all the first and second nearest neighbor atomic pairs in a unit cell were calculated, and the average −COHP values were used to represent the covalent bonding characteristic of the entire structure (Figure 6). As shown in Figure 6, the spin-up −COHP peaks between −2.50 and −1.50 eV are antibonding states, which decrease continuously as the proportion of Ti increases. However, there is no significant change in the spin-down −COHP curve in Figure 6. This indicates that the decrease of spin-up TDOS peaks between −2.50 and −1.50 eV reduces the proportion of antibonding states below Fermi energy, which makes the alloy more stable and difficult to undergo a phase transition. This may be the reason for the addition of excess Ti weakening the superelastic strain of Cu-Al-Mn-Ti alloys. Specifically, the negative Integrated COHP Values at the Fermi Level were calculated to evaluate the strength of different bonds (Listed in

Table 5. Negative integrated COHP values at the Fermi Level for Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1).

Alloys	Bond
	Mn-Al	Mn-Cu	Al-Cu	Cu-Cu	Ti-Al	Ti-Cu
Cu2AlMn	0.649	0.351	0.918	0.051
Cu2AlMn0.75Ti0.25	0.614	0.35	0.837	0.047	0.924	0.526
Cu2AlMn0.50Ti0.50	0.607	0.349	0.804	0.044	0.964	0.528
Cu2AlMn0.25Ti0.75	0.577	0.357	0.831	0.036	0.955	0.529
Cu2AlTi			0.821	0.032	0.954	0.535

). As shown in Table 5, the covalent bond strength between the same atomic pairs in Cu₂AlMn_1–xTi_x alloys changes little under different Ti substitution content. The strength of Ti-Al and Ti-Cu bonds is stronger than that of Mn-Al and Mn-Cu bonds, which causes the increase of Young’s modulus. Compared to other bonds, the strength of the Cu-Cu bond is very weak and contributes little to the covalent bonding of alloys.

Then, the variation of covalent bonding strength during tetragonal distortion was examined. Figure 7 displays the electron localization function (ELF) maps of Cu₂AlMn and Cu₂AlTi for tetragonal distortion at different levels. Larger ELF values imply stronger covalent bonds, and these regions are marked in red in Figure 7. As shown in Figure 7, the ELF values between Mn and Al as well as those between Ti and Al decrease rapidly with the increase of tetragonal distortion, which indicates that the covalent bond strength of Al-Mn and Al-Ti is very sensitive to the tetragonal distortion. In order to quantitatively analyze the change of covalent bonding, the −COHP values of Cu-Mn, Cu-Al, Cu-Ti, Al-Mn, Al-Ti, and Cu-Cu bonds were integrated at Fermi energy, as shown in Figure 8. The strength of Al-Mn and Al-Ti covalent bonds declines rapidly in the early stage of the tetragonal distortion process, which is consistent with the previous analysis. However, the strength of Cu-Mn, Cu-Al, and Cu-Ti covalent bonds changes relatively slowly and the strength of Cu-Cu covalent bonds is always very weak. The result indicates that the strength of Al-Mn and Al-Ti covalent bonds is unstable and greatly affected by tetragonal distortion.

In addition, the ionic bonding properties of Cu₂AlMn_1–xTi_x alloys were also investigated. Mulliken population analysis [56] results were used to analyze electronic transfer information between different atoms, as shown in

Table 6. Atomic Mulliken charge of Cu₂AlMn_1–xTi_x (x = 0, 0.25, 0.5, 0.75, and 1).

Alloys	Elements	s	p	d	Total	Charge (e)
Cu2AlMn	Cu	0.96	6.00	9.68	16.64	+0.36
	Al	1.14	3.20		4.34	−1.34
	Mn	0.76	6.00	5.62	12.38	+0.62
Cu2AlMn0.75Ti0.25	Cu	0.97	6.00	9.67	16.64	+0.36
	Al-I (0.25)	1.14	3.27		4.41	−1.41
	Al-II (0.75)	1.14	3.17		4.31	−1.31
	Mn	0.76	6.00	5.61	12.37	+0.63
	Ti	0.71	6.00	2.71	9.42	+0.58
Cu2AlMn0.5Ti0.5	Cu	0.97	6.00	9.67	16.64	+0.36
	Al-I (0.50)	1.14	3.23		4.37	−1.37
	Al-II (0.50)	1.14	3.15		4.29	−1.29
	Mn	0.76	6.00	5.59	12.35	+0.65
	Ti	0.72	6.00	2.70	9.42	+0.58
Cu2AlMn0.25Ti0.75	Cu	0.97	6.00	9.66	16.63	+0.37
	Al-I (0.75)	1.14	3.19		4.33	−1.33
	Al-II (0.25)	1.14	3.16		4.30	−1.30
	Mn	0.74	6.00	5.58	12.32	+0.68
	Ti	0.71	6.00	2.73	9.44	+0.56
Cu2AlTi	Cu	0.97	6.00	9.65	16.62	+0.38
	Al	1.14	3.19		4.33	−1.33
	Ti	0.72	6.00	2.71	9.43	+0.57

. In Cu₂AlMn_1–xTi_x alloys, Cu, Mn, and Ti are the elements that lose electrons while Al is the element that gains electrons, which indicates the ionic nature of alloys. As the Ti substitution content increases, there is no significant change in the number of electrons from Cu and Mn to Al. The ionic bonding strength between Cu, Mn, and Al is insensitive to the Ti element.

4. Conclusions

In this paper, the effect of Ti replacing Mn on the mechanical properties of Cu₂AlMn_1−xTi_x(x = 0, 0.25, 0.5, 0.75, and 1) alloys were studied by the first-principles method, and the internal mechanism was analyzed from the perspective of structural and electronic properties. The following conclusions could be summarized:

(1)

In terms of mechanical properties, all of the Cu₂AlMn_1–xTi_x alloys presented in this work are mechanically stable. With the increase of x, Young’s modulus, shear modulus, and volume modulus increase, and the mechanical anisotropy factor decreases, which indicates that the addition of Ti improves the mechanical properties of the alloy.

(2)

In terms of structural properties and phase stability, Ti atoms prefer to occupy the position of Mn atoms directly in Cu₂AlMn_1–xTi_x alloys, and the L2₁ phase is the most stable phase in the process of tetragonal distortion. With the increase of Ti substitution content, the phase stability of the alloy improves.

(3)

In terms of microscopic mechanism, the proportion of antibonding states under the Fermi energy of Cu₂AlMn_1–xTi_x alloys decreases with the increase of Ti substitution content, which is the main reason for the increase in stability and mechanical properties. The strength of Ti-Al and Ti-Cu bonds is stronger than that of Mn-Al and Mn-Cu bonds. In particular, the covalent bond strength of Al-Mn and Al-Ti is relatively sensitive to tetragonal distortion. In addition, the addition of Ti has little effect on the ionic bond strength of Cu₂AlMn alloy.