The Alloying Strategy to Tailor the Mechanical Properties of θ-Al₁₃Fe₄ Phase in Al-Mg-Fe Alloy by First-Principles Calculations

Abstract

As an important strengthening phase in Al-Mg-Fe alloy, the elastic and ductile–brittle characteristics of Al₁₃Fe₄ intermetallics hold prime significance in ascertaining the mechanical properties and potential application of Al-Mg-Fe alloys. In this study, multialloying of Co, Cu, Cr, Mn, and Ni has been adopted for tuning the mechanical characteristics of the Al₁₃Fe₄ phase; their effects on mechanical features and electronic structure of the Al₁₃Fe₄ phase have been scrutinized systematically by first-principles calculations employing the density functional theory. The replacement of Fe with M (M = Co, Cu, Cr, Mn, and Ni) is energetically advantageous at 0 K, as evidenced by the negative cohesive energy and mixing enthalpy of all Al₁₃(Fe,M)₄ phases. Cu and Ni, on the contrary, have a detrimental impact on Al₁₃Fe_4′s modulus and hardness due to the evolution of chemical bonding strength. Co, Cr, and Mn are thus, interesting candidate elements. In the light of B/G and Poisson’s ratio (σ) criteria, Al₁₃Fe₄, Al₁₃(Fe,Cu)₄, and Al₁₃(Fe,Ni)₄ have superior ductility; however, Al₁₃(Fe,Co), Al₁₃(Fe,Mn), and Al₁₃(Fe,Cr)₄ tend to be brittle materials. Calculation-based findings show that Co, Cr, and Mn are appropriate alloying elements for enhancing fracture toughness, whereas Mn reduces Al₁₃Fe_4′s elastic anisotropy. The electronic structure assessment found that the mechanical properties of the intermetallics are predominantly influenced by the Al-M bonds when the alloying element M replaced Fe.

1. Introduction

During the solidification of aluminium, iron usually precipitates as detrimental Fe-containing intermetallics which can be attributed to the low solubility of Fe in solid α-Al [1,2,3]. The undesired Fe-containing intermetallics that might surface in the course of solidification of Al-Mg alloys are θ-Al₁₃Fe₄, Al_m(FeMn), Al₃(FeMn), Al₆(FeMn), β-Al₅FeSi (β-Fe) and α-Al₁₅ (FeMn)₃Si₂ (α-Fe), [4]. Needle-like or plate-like Fe-containing intermetallics, which are brittle, often serve as stress concentration points, leading to crack initiation, propagation, and fracture of alloys [5,6,7,8,9]. To mitigate the deleterious effects of Fe in Al oalloys, several approaches have been utilized, including element neutralization, Fe-containing intermetallics separation, and improved melt cooling rate [10,11,12]. Studies have shown that increasing the solidification rate and the inclusion of Mn, V and Co can prevent plate-like intermetallics from forming and boost alloy yield strength [13,14]. Additionally, it was discovered that Fe, when it exists in the equilibrium form of the θ-Al₁₃Fe₄ phase, can be used a3s a h4elpful entity to enhance the mechanical properties of alloys comprising aluminium, magnesium, and manganese [15]. This discovery paves the way for developing a new strategy for enhancing the mechanical properties of secondary alloys made of aluminium, magnesium, and manganese via the generation of an Al-AlFe eutectic microstructure [15]. When Fe content is less than 2.0 wt%, Zhu et al. suggested that the creation of eutectic θ-Al₁₃Fe₄ can increase the ultimate tensile strength of Al-Mg-Mn-Fe. The ultimate tensile strength (UTS) and yield strength (YS) of recycled Al-Mg-Mn-Fe alloys underwent an increase from 146 MPa to 289 MPa and 122 MPa to 244 Mpa, respectively, according to Zhao et al.’s research. This is because of the blockage of slip lines by the Fe-rich phases [15]. Al-Mg-Fe alloys’ ability to further improve their mechanical characteristics is constrained by the inherent brittleness of the -Al₁₃Fe₄ phase and this phase’s mechanical and elastic anisotropy has a significant impact on the applications for these alloys [16].

In Fe-containing intermetallics, alloying elements such as Mn, Cr, Si, Ni, and Cu may be introduced [17], which do not only modify the crystal structure but also alter the lattice parameters, thereby changing the overall mechanical characteristics [18]. This offers a new set of guidelines for the tuning and optimization of the Al₁₃Fe₄ phase’s mechanical characteristics and elastic anisotropy. Numerous researchers have undertaken investigations on the composition design and characteristics optimization of intermetallics that contain iron (Fe) [19,20,21]. According to Chen et al. estimations of the elastic properties of FeAl intermetallics with V, Cr, or Ni doping, these alloying elements were reported to increase the ductility of FeAl alloys, which is presumably because of the reduction in directional chemical bonding in FeAl intermetallics, following V, Cr, and Ni doping. Density functional theory (DFT) simulations were used by Fang et al. to explore Si solution in θ-Al₁₃Fe₄ in a systematic manner. The results showed that Si doping in θ-Al₁₃Fe₄ will result in greater Si-Fe chemical bonding, thereby leading to a change in the phase’s mechanical properties.

However, it is still unclear as to how exactly the alloying metals such as Co, Cu, Cr, Mn, and Ni affect the mechanical characteristics and elastic anisotropy of θ-Al₁₃Fe₄. At present, there is a lack of experimental results on the elastic properties of θ-Al₁₃Fe₄, and the study of θ-Al₁₃Fe₄ phase does not consider the effect of substitution of these alloying elements on its mechanical properties. For the design and optimization of the mechanical properties of the Al₁₃Fe₄ phase, the underlying mechanism must be clearly understood. A trustworthy and effective method for forecasting the stability, electrical structure, and mechanical characteristics of diverse intermetallic and alloy systems is a first-principles computation based on density functional theory (DFT) [22].

In the current work, first-principles calculations employing DFT were employed to examine the impact of Co, Cu, Cr, Mn, and Ni doping upon the stability, mechanical properties, and electronic structures of θ-Al₁₃Fe₄ phases. This work investigates the mixing enthalpy, Mulliken’s bond population, the spin-polarized density of states (spin-DOS), and mechanical characteristics for understanding the alteration in the mechanical and physical properties of the Co, Cu, Cr, Mn, and Ni doping in θ-Al₁₃Fe₄.

2. Experiments Methods and Details of Calculations

2.1. Experimental Methods

Pure Al (99.99%), pure Mg (99.99%), and master alloys of Al-20Fe are used to create the experimental alloys (all mentioned as mass fractions hereafter, unless specified otherwise). By using a direct-reading spectrometer (Spectrometer LAB, Ametek, kleve, Germany), the experimental chemical compositions of the alloys (as they were made) were determined and are displayed in

Table 1. The chemical compositions of alloys investigated in the experiment (wt.%).

Mg	Fe	Ti	Cu	Si	Mn	Al
4.56	1.81	<0.05	<0.03	<0.02	<0.03	Bal

. In an electromagnetic induction furnace, the experimental alloys were first melted for 30 min at 750 °C. After that, they were transported to a steel mould, where they are allowed to cool gradually at a rate of 5 °C/min. The alloys are then subjected to composition and microstructure analysis.

Samples are mechanically ground first for microstructural analysis and then polished with OP-U colloidal silica suspension having a grain size of 50 nm. With the aid of a scanning electron microscope (SEM, phenom XI, Phenom-Scientific, The Netherlands), the microstructures of the alloys are examined, and an energy-dispersive spectrometer (EDS) coupled with an SEM is used to ascertain the composition of primary θ-Al₁₃Fe₄. The transmission electron microscopy (TEM) method is used to characterize the high-resolution lattice and atomic pictures of the θ-Al₁₃Fe₄ phase (TEM, JEM-2100, JEOL, Tokyo, Japan). The ion thinning device (Gatan 691) is used to treat the samples for TEM to ensure that the thickness is less than 10 nm.

2.2. Calculation Details

First-principles calculations are used to determine the mixing enthalpy, cohesive energy, the spin polarised density of states, mechanical properties, and Mulliken’s bond population of the Al₁₃(Fe,M)₄ (M = Co, Cu, Cr, Mn, and Ni) phases to better understand how stability, electronic structure, and mechanical characteristics of -Al₁₃Fe₄ vary with different alloying elements doping. The Cambridge Serial Total Energy Package (CASTEP) code [23] implements first-principles calculations based on the density functional theory (DFT), with a 3 × 3 × 2 Monkhorst–Pack k-point grid and plane-wave cutoff energy of 500 eV, following a convergence test. The interactions of valence electrons and ionic cores are shown by norm-conserving pseudopotentials (NCPPs). For Al, Fe, Mn, Cr, Co, Ni, and Cu, the valence electrons taken into consideration include 3s²3p¹, 3d⁷4s¹, 3d⁵4s², 3d⁵4s¹, 3d⁵4s¹, 3d⁸4s², and 3d¹⁰4s¹ [24]. The crystal structure is optimized using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method until the force per atom is lower than 0.02 eV/Å and the total energy change converges to 1 × 10⁻⁵ eV. In this study, exchange-correlation energy calculations are performed using the generalized gradient approximation (GGA) within the Perdew, Burke, and Ernzerhof solid approach (PBEsol) scheme [25], which is improved for the surfaces of densely packed solids and the solids themselves. The atomic coordinates and lattice parameters are relaxed along with all of the structures. By deformation of the unit cell, the elastic constant matrix is calculated by making use of the established stress–strain ratios based on generalized Hooke’s law [26]. The Voigt–Reuss–Hill approximation yields the elastic modulus [27].

Thermodynamic calculations are performed using thermo-calc software to explore the precipitate’s solidification sequences in the Al-Mg-Fe alloys based on the TCAL6 database.

3. Results and Discussions

3.1. Determination of Composition and Crystal Structure of Al₁₃Fe₄ Phase

Figure 1a shows the calculated solidification sequence and phase fraction of the Al-4.5Mg-1.8Fe alloy in this study. In light of the calculation results, it is evident that θ-Al₁₃Fe₄ first precipitates as a primary phase at 664 °C, then eutectic α-Al + θ-Al₁₃Fe₄ and Al₃Mg₂ precipitate at 634 °C and 451 °C, respectively. The microstructure of Al-Mg-Fe alloy has been shown in Figure 1b. It indicates that the primary θ-Al₁₃Fe₄ phase manifests an acicular morphology. The mass percentages of Al and Fe in θ-Al₁₃Fe₄ are 64.77% and 35.23%, respectively (Figure 1c). The existence of the θ-Al₁₃Fe₄ phase is further confirmed by EBSD mapping in Figure 1d, which indicates that the microstructure contains θ-Al₁₃Fe₄ (red) and α-Al (green) regions. To identify the crystal structures of the θ-Al₁₃Fe₄ accurately, the θ-Al₁₃Fe₄ phase in Al-4.56Mg-1.81Fe alloy is examined using TEM. The TEM bright field image of the θ-Al₁₃Fe₄ phase has been presented in Figure 1e. The contrast of θ-Al is bright, and it displays a dark strip. Figure 1f displays the high-resolution lattice and atomic picture of θ-Al₁₃Fe₄, and Figure 1g inserts the corresponding selected area electron diffraction (SAED) pattern (f). The monoclinic crystal structure of primary -Al₁₃Fe₄ with the C2/m space group was discovered using TEM analysis from at least three different zone directions. The lattice parameter of the θ-Al₁₃Fe₄ phase is ascertained as a = 15.43 ± 0.01 Å, b = 8.03 ± 0.01 Å, c = 12.44 ± 0.01 Å, presenting a slight deviation from the documented value as a = 15.49 ± 0.01 Å, b = 8.07 ± 0.01 Å, c = 12.47 ± 0.01 Å in the literature [26,27] and a = 14.54 ± 0.01 Å, b = 7.95 ± 0.01 Å, c = 12.42 ± 0.01 Å in the literature [3,16,26]. There is a disparity between the calculation conditions and the experimental conditions of thermoccalc. However, in this paper, we try to ensure that the solidification speed of the alloy is slow as far as possible, and that the experimental conditions are as close to the steady state as possible when preparing the alloy. Although the equilibrium conditions are still not reached, the calculated phase should be consistent with the experimental phase, so it makes sense for DFT calculation. The results of the composition and crystal analysis furnish the necessary modeling data for the first-principles calculations that are to follow. Aluminium and iron each have their own unique set of atom coordinates, which may be found in [26]. The unit cell of θ-Al₁₃Fe₄ has a total of 102 atoms, 24 of which are iron, whereas 78 are aluminium [27]. The crystal structure of a 25% atom substitution at Fe sites by M (where M can be any of the elements Co, Cu, Cr, Mn, or Ni) has been established and has been presented θ in Figure 2. The coordinates of the substituted atoms of Fe, as well as the calculated crystal lattice constants and unit cell volume of θ-Al₁₃Fe₄, can be found in

Table 2. Structural information, lattice parameters (Å), cell volume (ų), and formation enthalpy (E_f) (at 0 K) for M occupying Fe sites in the θ-Al₁₃Fe₄ phase.

Al13(Fe,M)4	Atom Coordinates of M	Lattice Constants (Å)
		a	b	c	V (Å3)
Al13Fe4	M1 (0.098, 0.5, 0.38) M2 (0.414, 0.5, 0.62) M3 (0.59, 0.5, 0.99) M4 (0.43, 0.5, 0.98)	15.43	8.03	12.44	1542.5	α = 89.9° β = 107.7° γ = 90.1°
Al13(Fe,Co)4		15.41	8.06	12.43	1543.7
Al13(Fe,Cu)4		15.42	8.15	12.46	1566
Al13(Fe,Cr)4		15.64	8.07	12.57	1586.6
Al13(Fe,Mn)4		15.5	8.04	12.5	1558.2
Al13(Fe,Ni)4		15.42	8.08	12.42	1547.6

.

3.2. Stability and Electronic Structure of Al₁₃(Fe,M)₄

The stability of the Al₁₃(Fe,M)₄ phase can be estimated via the calculation of the cohesive energy and mixing enthalpy. This energy parameter can be defined in the following equations:

$$\Delta H_r\left(A{l}_{13}{(Fe,M)}_4\right)=\frac{E_{tot}\left(A{l}_{13}{(Fe,M)}_4\right)-x{E}_{bin}\left(Al\right)-y{E}_{bin}\left(M\right)-z{E}_{bin}\left(Fe\right)}{x+y+z}$$

(1)

$$E_{coh}\left(A{l}_{13}{(Fe,M)}_4\right)=\frac{E_{tot}\left(A{l}_{13}{(Fe,M)}_4\right)-x{E}_{iso}\left(Al\right)-y{E}_{iso}\left(M\right)-z{E}_{iso}\left(Fe\right)}{x+y+z}$$

(2)

where $\Delta H_r\left(A{l}_{13}{(Fe,M)}_4\right)$ and $E_{coh}\left(A{l}_{13}{(Fe,M)}_4\right)$ are the mixing enthalpy and cohesive energy of Al₁₃(Fe,M)₄ phases. E_tot(Al₁₃(Fe,M)₄) is the total energy of Al₁₃(Fe,M)₄ per unit cell, and E_bin(Al,Fe,M) is the chemical potential of pure Al, Fe, and M atom. E_iso(Al,Fe,M) is the total energy of a single Al, Fe, and M atom. x, y, and z mean the number of Al, M, and Fe atoms. Al₁₃(Fe,M)_4′s cohesive energy and mixing enthalpy findings from calculations are displayed in

Table 3. The calculated mixing enthalpy and cohesive energy (eV/atom) of Al₁₃(Fe,M)₄ phases with Co, Cu, Cr, Mn, and Ni doping.

Species	Cohesive Energy (eV/atom)	Mixing Enthalpy (eV/atom)
Al13Fe4	−4.69	−0.28
Al13(Fe,Co)4	−5.05	−0.34
Al13(Fe,Cu)4	−5.04	−0.39
Al13(Fe,Cr)4	−4.92	−0.15
Al13(Fe,Mn)4	−4.97	−0.34
Al13(Fe,Ni)4	−4.98	−0.3

. The alloying elements are easier to substitute in the Al₁₃Fe₄ phase when the mixing enthalpy is less than zero. The mixing enthalpy for Al₁₃Fe₄, Al₁₃(Fe,Co)₄, Al₁₃(Fe,Cu)₄, Al₁₃(Fe,Cr)₄, Al₁₃(Fe,Mn)₄, and Al₁₃(Fe,Ni)₄ are −0.28, −0.34, −0.39, −0.15, −0.34, and −0.3 eV/atom as shown by the calculated results in Table 3. It shows that all alloy elements can easy to dissolve in Al₁₃(Fe,M)₄ phases. The calculated cohesive energy for Al₁₃Fe₄, Al₁₃(Fe,Co)₄, Al₁₃(Fe,Cu)₄, Al₁₃(Fe,Cr)₄, Al₁₃(Fe,Mn)₄, and Al₁₃(Fe,Ni)₄ is −4.96, −5.05, −5.04, −4.92, −4.97, and −4.98 eV/atom, which indicates an improved interatomic bonding force following Co, Cu, Mn, and Ni doping and corroborate well with mixing enthalpy calculation results.

First-principles computation has the benefit of illuminating the physical mechanism underlying the macroscopic physical characteristics of the electronic scale. The partial density of states (PDOS) and total density of states (TDOS) are explored by the computed electronic structures in an attempt to comprehend and elaborate on the physical process of the multialloying impact on the mechanical characteristics of Al₁₃(Fe,M)₄ phases, as illustrated in Figure 3. It implies that all Al₁₃(Fe,M)₄ phases exhibit metallic character as a result of positive TDOS values at the Fermi level. Fe-3d bands dominate the TDOS in the −3.7 eV to 1.7 eV energy range for all Al₁₃(Fe,M)₄ phases. In addition, Al-3p and Fe-3d bands overlap in the −3.7 to 1.7 eV energy range, indicating that these two states have hybridized to produce two bond stats of bonds at various energies. After doping of Co, Cr, and Mn the hybridizations of Al-3p and Co-3d in −2.5 eV to 0 eV energy range, Al-3p and Cr-3d in −2 eV to 0 eV energy range, Al-3p and Mn-3d in −2 eV to 0 eV energy range, contribute to the strong chemical bonding of Al-Co, Al-Cr, and Al-Mn. Al-p bands do not correspond with the Cu and Ni-d bands, which is a sign of a lesser bonding strength than a covalent link, confirming the atomic character of the Cu and Ni doping. Al₁₃(Fe,M)₄ phases with 20% Co, Cu, Cr, Mn, and Ni doping are not magnetic, according to the approximately symmetric spin DOS in Figure 3.

The spatial localization of the electron probability density distribution, as represented in the domains assigned to localized bond and lone pair electrons, is graphically depicted by the electron density difference [28,29], the electron density difference function is the difference between the molecular electronic density and the superimposed densities of the constituent non-interacting atoms. The maps of Al₁₃(Fe,M)₄ phases on the (001) plane with Co, Cu, Cr, Mn, and Ni doping are shown in Figure 4. The fact that more electrons are localized in a circular pattern around the Co, Cu, Cr, Mn, and Ni atoms than around the Al atoms indicates that these metals have covalent interactions with Co, Cu, Cr, Mn, and Ni.

3.3. Mechanical Properties and Anisotropy

The elastic and ductile–brittle characteristics of θ-Al₁₃Fe₄ phases are crucial for the overall performance and applications of Al-Mg-Fe alloys since they are a necessary strengthening phase in Al-Mg-Fe alloys. Their elastic constants have been computed and displayed in

Table 4. Estimated elastic constants (in GPa) of Al₁₃(Fe,M)₄ phases with different elements doping.

Species	Cij
	C11	C22	C33	C44	C55	C66	C12	C13	C23	C15	C25	C35	C46
Al13Fe4	225.7	219.4	235.2	61.52	72.6	70.6	83.2	68.3	39.8	−1.74	−2.9	5.8	−2.1
Al13(Fe,Co)4	222.1	217.4	236.1	64.6	75.1	68.2	82.1	70.0	44.0	−0.51	−3.18	3.6	−4.3
Al13(Fe,Cu)4	205.9	174.2	224.8	50.4	68.8	59.51	95.4	66.3	42.6	−0.42	−1.71	2.2	−1.3
Al13(Fe,Cr)4	218.7	213.2	235.6	57.9	50.2	52.5	81.0	67.9	40.9	−1.7	−2.4	3.5	4.7
Al13(Fe,Mn)4	236.6	214.7	240.5	65.4	78.6	76.0	82.2	70.6	46.1	−3.7	−1.5	−4.1	−2.0
Al13(Fe,Ni)4	221.3	203.9	239.6	18.4	74.0	40.9	88.8	67.9	45.6	−1.75	−2.53	3.1	5.2

to examine the impact of Co, Cu, Cr, Mn, and Ni doping on the mechanical characteristics evolution of Al₁₃(Fe,M)₄ phases. Born-lattice Huang’s dynamical theory allows us to express the mechanical stability of Al₁₃(Fe,M)₄ criteria as:

C₁₁ > 0; C₂₂ > 0; C₃₃ > 0; C₄₄ > 0; C₅₅ > 0; C₆₆ > 0;
C₁₁ + C₂₂ + C₃₃ + 2 (C₁₂ + C₁₃ + C₂₃) > 0;
C₁₁ + 2C₁₂ > 0; (C₄₄C₆₆ − C₄₆²) > 0;
C₂₂ (C₃₃C₅₅ − C₃₅²) + 2C₂₃C₂₅C₃₅ − C₂₃²C₃₅ − C₂₅²C₃₃ > 0;
2 [C₁₅C₂₅ (C₃₃C₁₂ − C₁₃C₂₃) + C₁₅C₃₅ (C₂₂C₁₃ − C₁₂C₂₃) + C₂₅C₃₅ (C₁₁C₂₃ − C₁₂C₁₃)] − [C₁₅² (C₁₁C₂₂ − C₁₂²) + C₅₅ (C₁₁C₂₂C₃₃ − C₂₃²C₁₁ − C₁₃²C₂₂ − C₁₂²C₃₃ + C₁₂C₁₃C₂₃)] > 0

(3)

According to Table 4, the calculated elastic constants of these Al₁₃(Fe,M)₄ phases meet the aforementioned criteria, indicating that they are mechanically stable at 0 K when doped with Mn, Cr, Co, Ni, and Cu.

The following equations for Al₁₃(Fe,M)₄ phases compute the shear and bulk modulus based on the elastic constant:

$$B_V=\left(\frac{1}{9}\right)\left[2\left(C_{11}+C_{12}+C_{23}\right)+C_{13}+22+C_{33}\right]$$

(4)

$$G_V=\left(\frac{1}{15}\right)\left(C_{11}+C_{12}+C_{33}-C_{12}-C_{13}-C_{23}\right)+\left(\frac{1}{5}\right)\left(C_{44}+C_{55}+C_{66}\right)$$

(5)

$$B_R=1/\left[\left(S_{11}+S_{22}+S_{33}\right)+2\left(S_{12}+S_{13}+S_{23}\right)\right]$$

(6)

$$G_R=1/\left[4\left(S_{11}+S_{22}+S_{33}\right)-4\left(S_{12}+S_{13}+S_{23}\right)+3\left(S_{44}+S_{55}+S_{66}\right)\right]$$

(7)

where C_ij is the elastic constant and S_ij = 1/C_ij, G_V (G_R) and B_V (B_R) are the shear modulus and bulk modulus computed from Voigt (Reuss) model. Therefore, the calculation of their average value gives the final values of B and G [30]:

$$B=\left(B_V+B_R\right)/2$$

(8)

$$G=\left(G_V+G_R\right)/2$$

(9)

The Young’s modulus (E) and Poisson’s ratio (σ) of Al₁₃(Fe,M)₄ can be obtained as follows:

$$E=9BG/\left(3B+G\right)$$

(10)

$$\sigma =\left(3B-2G\right)/\left(6B+2G\right)$$

(11)

The intrinsic brittleness of Fe-containing intermetallics substantially hinders the application of Al-Mg-Fe alloys. Due to the lack of experimental values, we have compared the calculated values with other work, which shows a good agreement as illustrated in

Table 5. Bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, B/G, and intrinsic hardness of Al₁₃(Fe,M)₄ phases with different elements doping, and the unit is GPa.

Species	B	G	E	σ	B/G	HVC	HVT	KIC (MPa·m1/2)
Al13Fe4	116.11	62.24	158.41	0.273	1.87	7.81	8.44	1.34
Cal. Al13Fe4 1	123	76	189	0.243	1.618	-	-	-
Cal. Al13Fe4 2	121.886	86.942	210.723	0.211	1.40	-	-	-
Al13(Fe,Co)4	118.43	72.8	181.25	0.245	1.63	10.9	11.01	1.46
Al13(Fe,Cu)4	118.42	49.17	129.58	0.317	2.41	3.98	5.34	1.20
Al13(Fe,Cr)4	117.77	72.67	180.82	0.244	1.62	10.95	11.05	1.45
Al13(Fe,Mn)4	120.64	74.56	187.48	0.236	1.61	11.12	11.23	1.49
Al13(Fe,Ni)4	111.68	60.87	154.53	0.269	1.83	7.88	8.46	1.30

^1—Calculated values of Yulong Li et al. [37]. ^2—Calculated values of Liu He et al. [38].

. It is, hence, indispensable to investigate the enhancement of its ductility via doping of alloying elements. The mechanical properties evolution of Al₁₃(Fe,M)₄ phases with Co, Cu, Cr, Mn, and Ni and doping have been summarized in Table 4 and Figure 5. The critical values of Poisson’s ratio (0.26) and Pugh ratio B/G (1.75) have been broadly employed as indicators representing the ductility of materials. A larger Poisson’s ratio (σ) signifies the higher softness of the material [31]. The σ and B/G values for Al₁₃Fe₄, Al₁₃(Fe,Cu)₄, and Al₁₃(Fe,Ni)₄ are 0.273, 0.317, 0.269, and 1.87, 2.41, 1.83, respectively, which means a better ductility of these phases.

The capacity of Al₁₃(Fe,M)₄ phases to stop cracks from spreading is characterized by their fracture toughness (KIC), which is crucial for the service safety of Al-Mg-Fe alloys. Ogata and Niu have developed a model for determining fracture toughness that can be applied to several materials by examining the relationship between the elastic characteristics of materials and their fracture toughness. During the deformation of a material, bonds break and reform resulting in the displacement of atoms and slipping of atomic planes, and materials with high fracture toughness usually exhibit high ductility and yield at high critical strain. Therefore, B/G is in positive correlation with fracture toughness K_IC [32]. This paper uses the following, reasonably straightforward calculation for fracture toughness [33,34,35,36]:

$${\mathrm{K}}_{\mathrm{IC}}={\mathrm{V}}_0^{1/6}·G·{\left(B/G\right)}^{1/2}$$

(12)

where G and B, respectively, denote the shear and bulk moduli (in MPa) of Al₁₃(Fe,M)₄, V₀ is the volume per atom (in m³), and the unit of K_IC is MPa m^1/2. By comparison of the K_IC in Table 5, it was found that Co, Cr, and Mn are prospective candidates for the fracture toughness optimization of Al₁₃Fe₄.

The mechanical modulus value and variation of Al₁₃(Fe,M)₄ phases with the doping of different alloying elements are listed in Table 5 and Figure 6. The calculated Young’s modulus value of Al₁₃Fe₄ is 158.41 GPa. With the doping of Co, Cr, and Mn, it is increased to 181.25, 180.82, and 187.48 GPa. However, for the doping of Cu and Ni, the calculated Young’s modulus value of Al₁₃(Fe,M)₄ decreased to 129.58 and 154.43GPa, respectively. The shear and bulk moduli both exhibit the same fluctuation trend. Young’s modulus can reveal the degree of atom bonding in crystals. In general, the strength of a chemical bond increases with increasing Young’s modulus values. The mechanical modulus evolution can therefore be attributed to the variation of chemical bonding strength in Al₁₃(Fe,M)₄, which is also validated by the TDOS in Figure 4. It can be concluded that the chemical bonding strength of Al-M increases in the sequence of Al-Cu, Al-Ni, Al-Fe, Al-Cr, Al-Co, and Al-Mn, which can be appropriately elaborated using the mean bond population variation in Figure 6.

An essential index for the mechanical assessment of Al₁₃Fe₄ is its intrinsic hardness. In this study, two alternative models—model Chen’s (H_VC) and Tian’s model (H_VT)—are used to predict the hardness evolution of Al₁₃(Fe,M)₄ with Co, Cu, Cr, Mn, and Ni doping [39].

$$H_{VC}=2{(k^{-2}G)}^{0.585}-3$$

(13)

$$H_{VT}=0.92k^{-1.137}{G}^{0.708}$$

(14)

The Pugh ratio, k, is defined as k = B/G. Table 5 provides a summary of the computed outcomes. As shown in Figure 6, the computed results of the elastic moduli, B/G, and H_V are deducted from the corresponding Al₁₃Fe₄ values to more clearly illustrate the differences in the mechanical characteristics of Al₁₃(Fe,M)₄ (i.e., Al₁₃Fe₄ is taken as the reference state). The altered mechanical characteristics of Al₁₃(Fe,M)₄ brought on by the doping of alloying elements are then discovered. The G/B and values fluctuation show that Cu doping can increase Al₁₃Fe_4′s ductility while lowering its elastic modulus and hardness. Co, Cr, and Mn can increase the elastic modulus and hardness prominently.

The mean bond population and bond length variation of Al-Fe and Al-M in Al₁₃(Fe,M)₄ phases with Co, Cu, Cr, Mn, and Ni doping are depicted in Figure 6. Mulliken atomic populations are largely based on first-order electron density functions within a linear combination of atomic orbitals–molecular orbital (LCAO-MO) theory [39]. Mulliken population analysis assesses the distribution of electrons in several fractional means among the different parts of atomic bonds. In addition, the overlap population abides by a good relationship with the covalency of bonding and bond strength. More description is given in [21,22]. Mean bong population is an analysis of the electronic occupancy state in atomic orbits, which can be calculated as follow:

$$n_m\left(AB\right)={\displaystyle \sum _{k}W\left(k\right)}{\displaystyle \sum _u^{onA}{\displaystyle \sum _v^{onB}{P}_{uv}}}\left(k\right)S_{uv}\left(k\right)$$

(15)

where W(k) is the weight, k is the wave vector, u and v are the electron orbitals, P(k) is the density matrix of the corresponding electron orbitals, and S(k) is the overlap matrix. The average bond length and the average number of bonds can be obtained by the following equation:

$$\overline{L}(AB)=\frac{{\displaystyle \sum _i{L}_i{N}_i}}{{\displaystyle \sum _i{N}_i}}$$

(16)

$$\overline{n}(AB)=\frac{{\displaystyle \sum _i{n}_{i}A{B}_{N_i}}}{{\displaystyle \sum _i{N}_i}}$$

(17)

where N_i is the total number of different bonds in the unit cell, L_i is the bond length of different bonds, and n_i is the population of different bonds. The bond length and population of the Al-Fe and Al-M bonds in Al₁₃Fe₄ were changed by the alloying elements. Larger and more positive population values typically suggest the existence of stronger covalent bonds. For every Fe atom that is replaced by an alloying atom in Al₁₃Fe₄, one Al-M bond is replaced by a pair of original Al-Fe bonds. The bond number and bond length of the Al-M bonds in Al₁₃Fe₄ were changed by the alloying metals. Al₁₃(Fe,M)₄ has all Al-M bonds, although the chemical linkages Al-Co, Al-Cr, and Al-Mn have the largest mean bond populations (Figure 6), indicating a stronger bonding strength. This precisely explains the improved modulus value of Al₁₃(Fe,Co)₄, Al₁₃(Fe,Cr)₄, and Al₁₃(Fe,Mn)₄ among these Al₁₃(Fe,M)₄ phases. For Al₁₃(Fe,Cu)₄ and Al₁₃(Fe,Ni)₄, the smaller mean bond population of Al-Cu and Al-Ni indicates their weaker bonding strength, leading to the smaller mechanical modulus and hardness.

Since crack nucleation and propagation are directly related to elastic anisotropy, Al₁₃Fe_4′s mechanical characteristics anisotropy is also crucial for the performance of Al-Mg-Fe alloys. The Young’s modulus, hardness, and Poisson ratio surface contours of -Al₁₃Fe₄ with Co, Cu, Cr, Mn, and Ni doping are depicted in Figure 7, Figure 8 and Figure 9, respectively, to illustrate the elastic anisotropy of Al₁₃(Fe,M)₄. The mechanical surface contours of isotropic materials are a perfect sphere. Al₁₃Fe₄ exhibits high Young’s modulus anisotropy, according to Figure 7a, since the surface contours depart from a perfect sphere. Young’s modulus values along [001] directions are higher than other directions for all Al₁₃(Fe,M)₄ phases. The modulus anisotropy of Al₁₃(Fe,M)₄ phases can be predicted using the computed C_ij in Table 4. C₁₁ and C₃₃ represent a crystal’s capacity to withstand axial strain in the [100] and [001] axes, respectively [17,21,35,36,37,38]. The modulus of Al₁₃(Fe,M)₄ phases along the [100] direction is smaller than that in the [001] directions, as indicated by C₁₁ being smaller than C₃₃. The results are confirmed by modulus surface contours in Figure 7. Doping Al₁₃Fe₄ with elements such as Co, Cu, Cr, and Ni results in a strengthening of the material’s elastic anisotropy (Figure 7b–d,f). Mn is an interesting candidate for use as an alloying element because of its potential to reduce the elastic anisotropy of Al₁₃Fe₄ (see Figure 7e). This, in turn, would be beneficial to the mechanical characteristics of Al-Mg-Fe alloys (Figure 7f). As can be seen in Figure 8, the bulk modulus of these Al₁₃(Fe,M)₄ phases exhibit a clear anisotropy in their behavior. The value of all Al₁₃(Fe,M)₄ phases is greatest along the direction indicated by the notation [011]. Doping with Co and Cu led to a reduction in the elastic anisotropy (Figure 8b,c), whereas doping with Ni resulted in an extensive strengthening of the elastic anisotropy (Figure 8f).

The Poisson ratio surface contours of Al₁₃(Fe,M)₄ with various alloying elements doped are plotted in Figure 9 to more clearly illustrate the characteristics of mechanical anisotropy. It is obvious that Al₁₃(Fe,Ni)₄ shows the strongest Possion ratio anisotropy, and the descending order of Poisson ratio anisotropy for Al₁₃(Fe,M)₄ phases is Al₁₃(Fe,Ni)₄ > Al₁₃(Fe,Mn)₄ > Al₁₃Fe₄ > Al₁₃(Fe,Cr)₄ > Al₁₃(Fe,Co)₄ > Al₁₃(Fe,Cu)₄.

4. Conclusions

In conclusion, using first-principles calculations based on DFT, the impact of Co, Cr, Cu, Mn, and Ni doping on the mechanical characteristics and electronic structure of Al₁₃Fe₄ has been comprehensively studied. The following are the primary conclusions:

(1)

The substitution of Fe by M (M = Co, Cr, Cu, Mn, and Ni) in Al₁₃Fe₄ is energetically favourable at 0 K due to the negative mixing enthalpy and cohesive energy.

(2)

With the doping of Co, Cr, and Mn, the Young’s modulus values are increased to 181.25, 180.82, and 187.48 GPa, the hardness values are increased to 10.9, 10.95, and 11.12 GPa, owing to the formation of stronger Al-Co, Al-Cr, and Al-Mn chemical bonds, the existence of which is validated by electronic structure calculation conducted in this study.

(3)

Co, Cr, and Mn are potential candidate elements for the mechanical optimization of Al₁₃Fe₄ for the simultaneous improvement of mechanical modulus, hardness, and fracture toughness (K_IC from 1.34 to 1.46, 1.45, and 1.49).