The Effect of Point Defects on Young’s Modulus of the Off-Stoichiometric Al₃X (X = Li, Sc, and Zr) Phases: A First-Principles Study

Abstract

L1₂-Al₃X (X = Li, Sc, and Zr) precipitates are the main strengthened phases of high-strength aluminum alloys and are critical for aerospace structural materials. Point defects and substitutional ternary elements change the mechanical properties of Al₃X. In this paper, the effect of point defects, including vacancy, antisite, and substitutional element addition defects on the elastic modulus of the off-stoichiometric Al₃X (X = Li, Sc, and Zr) phase were investigated by using first-principle calculations. The formation enthalpies of the defective Al₃X alloy and isolated point defects in Al₃X were calculated, and the results showed that the defects have an effect on the structure and elasticity of the off-stoichiometric Al₃X phases. The lattice distortion, elastic constants, and elastic moduli were further investigated. It was found that the point defects increased the Young’s modulus for Al₃Zr, and the doping of Er improved the Young’s modulus for off-stoichiometric Al₃Li and Al₃Sc. Adjusting the position of vacancies can improve the elastic modulus. In addition, the doping of substitutional elements (especially Sc, Ti, Zr, Hf, Ta, Mn, Ir, and Cf) can greatly increase the Young’s modulus of off-stoichiometric Al₃Li.

1. Introduction

L1₂-Al₃X (X = Li, Sc, and Zr) precipitates in aluminum alloys are beneficial for improving elastic properties [1,2,3,4,5]. Many studies have shown that defects and foreign element doping play an important role in elastic properties [6,7,8,9,10]. Although many point defects in Al₃X have been reported in experiments by researchers, the underlying kinetic transformation process is still unclear, primarily due to the difficulties in imaging the vacancies. Moreover, it is important to understand the effect of different point defect types on the elastic properties of structural materials.

Recently, based on Density Functional Theory (DFT), the first-principle calculations have provided another dimension of investigating the point defects computationally. For example, Dong et al. [11] found that point defects positively affect the brittle behavior of Hf₅BSi₃. Jiang et al. [12] investigated the effect of point defect on the structures and mechanical properties of defective BiCuOTe, and they found that point defects improved the elastic compliance but weakened the compression resistance of BiCuOTe. Fan et al. [13] suggested that Al matrix with vacancy and substitutional rare element addition presented the ductility and nonuniform. Muzy et al. [14] studied the point defects by thermostimulated luminescence, and it was found that the dislocations in the specimens were pinned due to point defects, while the samples with more dislocations became more resistant to plastic deformation in Verneuil crystals. Ouahrani et al. [15] studied the effect the point defects on Cu₂WS₄ single layer and found that the vacancy changed to the energy structure. Wang et al. [16] found that point defects cause lattice distortion for Al₃Sc and Al₃Lu, which reduces the elastic properties. In addition, the microalloying of rare earth elements has been recently reported to have an impact on the precipitation structure and, thus, its elastic properties [17,18,19,20,21,22]. Zhang et al. [17] proposed that the addition of Sc to Al-Cu alloys promoted the nucleation of Al grains by formation Al₃Sc, reduced the average size of θ′ precipitates, and increased the number density of θ′ precipitates, thereby improving the yield strength and tensile strength of Al alloys. Ying et al. [8] and Wang et al. [9] also suggested that the addition of Sc improved the grain refinement and thus improved the mechanical properties of the Al alloy. Moreover, it was found that the addition of Sc, Zr to aluminum alloys produced the Al₃(Sc, Zr) particles with core-shell structure, which refined the grain size [2], and improved the tensile strengths [3] and yield strengths [18,19]. Yi et al. [23] found that the doping of Sc had positive effects on the enhancement of creep resistance in the Al-Ce alloy. Ce promoted the formation of second phases and grain refinement [24,25], and Ekaputra et al. [20] found that the combination of the Al₁₁Ce₃ and L1₂-Al₃(Er, Sc, Zr) precipitates increased the strength of the aluminum alloy. It was found that the addition of Er to the Al-Sc alloys promoted the formation of an Al₃(Er, Sc) structure, improving the elastic properties of Al alloys [21,22]. Fang et al. [26,27,28] revealed that Yb additions could enhance the fracture toughness of Al-Zn-Mg-Cu alloy, and they also found that Pr could improve the strength, fracture toughness, and ductility of Al-Zn-Mg-Cu-Zr alloy [29]. Tian et al. [30] investigated the effects of Sc on the elastic properties of Al₃Li by first-principles calculations and found that the doping of Sc enhanced the shear deformation resistance and stiffness of Al₃Li. Compared with rare earth elements, generally, Er exhibits a relatively large diffusivity in α-Al [31] and thus shows a robust defect formation activity. Notably, full information on the influences of Er on the elastic properties is of great value for the development and design of materials. There are a large number of vacancies in Al₃X in Al alloys, and previous experiments reported that the doping of Er led to the formation of an Al₃(X, Er) structure [20,21,22,32,33]. However, the intermediate steps of how the solute atom Er substitutes the vacancy in Al₃X to form a new structure and how they change the modulus in this paper are not systematically studied. The formation of the intermediate states as precipitation is experimentally difficult to observe due to the small amounts, while the first-principle calculations can analyze the structural properties of energy [34].

In general, the point defects of intermetallic compounds are complex, and the various types of vacancy, antisite, and substitute point defects co-exist, causing them to deviate from their stoichiometric ratios. The point defect with off-stoichiometric ratios both have physical significance in Al₃X phases. In this work, we investigated the effects of point defects and the alloying element Er on the structural stability and elastic properties of off-stoichiometric Al₃X while considering the new structures as pure precipitations by using first-principle calculations. The effects of complex defect forms, such as double, triple, quadruple, and septuplet defects, are left for further study. Although we just studied the effect of only one Er atom on Al₃X, knowledge of the first step of Er addition on the Al₃X is of great importance for understanding the nucleation of Al₃(X, Er), which may be useful in the design of guided experiments for the development of new materials. To prove the reliability of our model structures, we measured Young’s modulus (E) of Al₃Sc particles by a nanoindentation experiment. The occupation behavior and the elastic influence of the Er-added point defects in Al₃X were further analyzed. This lays a foundation for the application of point defects and Er in aluminum alloys and could provide theoretical data and directions for the development of high-performance Al alloys.

2. Computational Methods

In this paper, first-principle calculations based on the Density Functional Theory (DFT) [35,36] of the effects of point defects and the Er addition on structural stability and elastic properties of off-stoichiometric Al₃X (X = Li, Sc, and Zr) were performed by using the Vienna Ab-initio Simulation Package (VASP) [37,38,39,40,41]. In this study, the projector-augmented wave (PAW) [42] method was used to deal with the electron-core interactions. For the electron exchange-correlation, the Perdew–Burke–Ernzerh (PBE) functional [43] based on generalized gradient approximation (GGA) [44] was employed throughout the predictions. The Perdew–Wang parameterization (PW91) [45,46] and the local density approximation (LDA) [47,48] were only used to reproduce the previous reports and have been found to be less accurate than PBE. The plane wave cutoff energy was set to 450 eV, and the energy and force convergence criteria were less than 1.0 × 10⁻⁶ eV and 1.0 × 10⁻² eV/Å, respectively. In total, there are 32 atoms in a 2 × 2 × 2 supercell for the calculation of defective structures and doped structures. The K-points were selected to be 9 × 9 × 9 for 32-atom defect phases in order to sample the Brillouin region. The optimized cutoff energy and K-points were validated by taking the residual energy less than 3 × 10⁻³ eV.

To check the effect of supercell size, we took the unit cell of the Al₃X phase as an example to perform a convergence test by considering the supercell of 2 × 2 × 2 (32 atoms), 3 × 3 × 3 (108 atoms), and 4 × 4 × 4 (256 atoms). The formation enthalpies for these different supercells were calculated, and the difference among them is less than 0.01 eV/atom. Therefore, a 2 × 2 × 2 supercell was used for calculation. We have performed the total energies and the formation enthalpies of the optimized structures in Table S1. The formation enthalpies for pure Al₃X supercell and defected Al₃X structure with point defect could be calculated as [49,50]

$$∆H=(E_{tot}-a{E}_{Al}-b{E}_X)/(a+b)$$

(1)

where $E_{tot}$ is the total energy of the pure or defective off-stoichiometric Al₃X, and X represents the atoms of Li, Sc, or Zr. $E_{Al}$ and $E_X$ represent the energy of each element at the bulk states of Al and X bulk materials, respectively. a and b are the atom numbers of Al and X in the pure or defective Al₃X, respectively. According to the Wagner-Schottky model, the formation enthalpy as a function has a linear relationship with the point defect concentrations,

$$∆H=∆H_{{Al}_{24}{X}_8}+\sum _d{H}_d{x}_d$$

(2)

where $x_d$ is the atomic concentration in point defects. $∆H_{{Al}_{24}{X}_8}$ is the formation enthalpy (∆H) of the full-ordered stoichiometric Al₂₄X₈ supercell, and the subscript d represents the point defect types. $H_d$ is the formation enthalpy of the point defect in stoichiometric Al₃X. The point defect formation enthalpy (H_d) was also calculated by finite differencing [51,52],

$$H_d=\frac{\partial \Delta H}{\partial x_d}\approx \frac{∆H_d-∆H_{{Al}_{24}{X}_8}}{x_d}$$

(3)

$$x_d=1/(a+b)$$

(4)

where $∆H_d$ is the formation enthalpy of defective Al₃X structures. The phase stability of off-stoichiometric Al₃X with doping Er could be evaluated by the formation enthalpy (ΔH),

$$∆H=(E_{tot}-a{E}_{Al}-b{E}_X-E_{Er})/(a+b+1)$$

(5)

where $E_{tot}$ is the total energy of the doped off-stoichiometric Al₃X, and $E_{Er}$ represent the energy of the Er element at the bulk state. a and b are the atom numbers of Al and X in the doped off-stoichiometric Al₃X.

The elastic constants (C_ij) of the crystal are a measure of the resistance to external pressure, playing an important role in mechanical and dynamical behaviors. In this work, the elastic constants (C_ij) were obtained by the coefficient of the stress-strain curve [53]. There are two methods of calculating elastic constants from the first principles: the stress–strain and the energy–strain. In the condition of small strains, there was a linear relationship satisfying Hooke’s law between strain ε = (ε₁, ε₂, ε₃, ε₄, ε₅, ε₆) and stress σ = (σ₁, σ₂, σ₃, σ₄, σ₅, σ₆):

$${\mathsf{\sigma }}_i={\sum }_{j=1}^6{C}_{ij}{\mathsf{\epsilon }}_j$$

(6)

where ${\mathsf{\sigma }}_i$ and ${\mathsf{\epsilon }}_j$ represented strain and stress with six independent components respectively. C_ij represented the elastic constant with subscripts using the Voigt notation. For the stress–strain method, based on the first-order derivative of the stresses, the corresponding stresses are calculated in terms of different strains. The primary term coefficients can be obtained by fitting through Equation (5) to obtain the elastic constants C_ij. There is an energy variation of the equilibrium lattice structure under applied minor strains. In a second-order approximation, the elastic energy $∆\mathrm{E}\left(\mathrm{V},\left\{{\epsilon }_i\right\}\right)$ of a solid under strain can be given by Taylor expansion:

$$∆\mathrm{E}\left(\mathrm{V},\left\{{\epsilon }_i\right\}\right)=\mathrm{E}\left(\mathrm{V},\left\{{\epsilon }_i\right\}\right)-\mathrm{E}({\mathrm{V}}_0,0)=\frac{V_0}{2}{\sum }_{j=1}^6{C}_{ij}{\mathsf{\epsilon }}_i{\mathsf{\epsilon }}_j$$

(7)

where $\mathrm{E}\left(\mathrm{V},\left\{{\epsilon }_i\right\}\right)$ and $\mathrm{E}({\mathrm{V}}_0,0)$ are the total energy of the structure with and without applied strain, respectively, and ${\mathrm{V}}_0$ is the equilibrium volume. For the energy–strain method, the energy variation of the solid under the different strains is calculated, and the elastic constants are derived from the second-order derivatives of the energy versus strain curves by fitting Equation (6). For each strain, there are six independent stress components but only one energy to fit, obtaining the elastic constants. Compared to the energy–strain method, the stress–strain method requires a smaller set of distortions and is more efficient [54,55,56,57]. In this paper, the calculations of the elastic constants have been performed using the stress–strain approach. When a train ε is applied to deform the crystal, the lattice vectors between the distorted and equilibrium cells can be expressed as:

$$\left[\begin{array}{c}{a}^{\prime }\\ b^{\prime }\\ c^{\prime }\end{array}\right]=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]·(I+ϵ)$$

(8)

where $I$ is the unit matrix. The strain tensor $ϵ$ and elastic constants $C_{ij}$ of the fcc structure Al₃X can be described by

$$\mathsf{ϵ}=\left[\begin{array}{ccc}{e}_1& \frac{e_6}{2}& \frac{e_5}{2}\\ \frac{e_6}{2}& e_2& \frac{e_4}{2}\\ \frac{e_5}{2}& \frac{e_4}{2}& \frac{e_3}{2}\end{array}\right]$$

(9)

$$C_{ij}=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{12}\\ \begin{array}{c}{C}_{12}\\ C_{12}\\ 0\end{array}& \begin{array}{c}{C}_{11}\\ C_{12}\\ 0\end{array}& \begin{array}{c}{C}_{11}\\ C_{11}\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\end{array} \begin{array}{ccc}\begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}\\ C_{44}& 0& 0\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}{C}_{44}\\ 0\end{array}& \begin{array}{c}0\\ C_{44}\end{array}\end{array}\right]$$

(10)

then substituting the Equation (9) into Equation (5), the three-independent elastic constants (C₁₁, C₁₂, C₄₄) of cubic crystal can be calculated by

$$\left(\begin{array}{c}\begin{array}{c}{\mathsf{\sigma }}_1\\ {\mathsf{\sigma }}_2\end{array}\\ {\mathsf{\sigma }}_3\\ \begin{array}{c}{\mathsf{\sigma }}_4\\ {\mathsf{\sigma }}_5\\ {\mathsf{\sigma }}_6\end{array}\end{array}\right)=\left(\begin{array}{ccc}{C}_{11}& C_{12}& C_{12}\\ \begin{array}{c}{C}_{12}\\ C_{12}\\ 0\end{array}& \begin{array}{c}{C}_{11}\\ C_{12}\\ 0\end{array}& \begin{array}{c}{C}_{11}\\ C_{11}\\ 0\end{array}\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}0\\ 0\end{array}\end{array} \begin{array}{ccc}\begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}& \begin{array}{c}0\\ 0\\ 0\end{array}\\ C_{44}& 0& 0\\ \begin{array}{c}0\\ 0\end{array}& \begin{array}{c}{C}_{44}\\ 0\end{array}& \begin{array}{c}0\\ C_{44}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{\mathsf{\epsilon }}_1\\ {\mathsf{\epsilon }}_2\end{array}\\ {\mathsf{\epsilon }}_3\\ \begin{array}{c}{\mathsf{\epsilon }}_4\\ {\mathsf{\epsilon }}_5\\ {\mathsf{\epsilon }}_6\end{array}\end{array}\right)$$

(11)

All crystal structures need to satisfy the mechanical criterion in this work. And the criterion to evaluate the mechanical stability of pure, defective, and doped Al₃X could be expressed as [58,59,60]:

$$C_{11}-C_{12}>0,C_{11}>0, C_{44}>0,C_{11}+{2C}_{12}>0$$

(12)

According to the Voigt–Reuss–Hill (VRH) approach, the bulk modulus $B$, shear modulus $G$, Young’s modulus $E$, and Poisson’s ratios $v$ of cubic structures are written as [61]:

$$E=9\mathit{GB}/(3B+G),v=(3B-2G)/2(3B+G)$$

(13)

$$B_V=B_R=(C_{11}+2C_{12})/3$$

(14)

$$G=(G_V+G_R)/2$$

(15)

$$G_V=\left(C_{11}-C_{12}+3C_{44}\right)/5$$

(16)

$$G_R=5(C_{11}-C_{12})C_{44}/[4C_{44}+3(C_{11}-C_{12}\left)\right]$$

(17)

where, the subscripts V and R represent Voigt and Reuss approximations, respectively.

The number of independent elastic constants depends on the symmetry of crystals. The lower the symmetry, the more the independent elastic constants. For example, there are three independent elastic constants for Al₃X cubic crystals, namely C₁₁, C₁₂, and C₄₄ [62,63]. For the defective off-stoichiometric Al₃X, the point defect changes the composition of the chemical formula and the symmetry of the crystals, and the ratio of Al to X is no longer 3:1 (as shown in Table S2). These structures, without the L1₂ structure, will have more independent elastic constants, so the elastic moduli were calculated with the help of the VASPKIT program [64]. According to the coefficient of the stress–strain curve, VASPKIT can identify the symmetry of the defective supercell structure and obtain the corresponding stiffness tensor matrix, acquiring independent elastic constants as well as polycrystalline elastic modulus.

3. Experimental Procedure

To verify our theoretical calculation, we investigated the commercial Al-2Sc binary alloy (as cast) with a 2% mass fraction of Sc. The Al-2Sc cast has a large size of primary Al₃Sc particles at a low solidification rate [65]. Using a continuous stiffness model [66], Young’s modulus E of Al₃Sc particles was measured by nanoindentation experiments. All measurements were done on the Keysight G200 Nanoindenter, with a diamond Berkovich-type tip at a room temperature of 23.7 °C. In this work, the simulation calculations were performed at T = 0 k, and the room temperature in the experiment can correspond with our calculation conditions to verify our calculation results. Fused quartz was chosen as a reference sample for calibration of the tip area function. To keep the reliability of the experimental results, the test selected peak loads ranges from 1 to 10 mN, and the indentations were made on at least five different Al-2Sc particles for each peak load, and the nanoindentation depth stabilized between 50 nm and 150 nm while the Al₃Sc particle size was intentionally annealed to the size over 150 nm. Data were analyzed by using NanoSuite 6.40.458 software (Copyright 2015 Keysight Technologies, Santa Rosa, CA, USA) to determine the starting position of the indenter contacting the specimen surface to calibrate the load indentation curve. The modulus of elasticity was measured from the contact area versus the continuous stiffness.

4. Results and Discussion

In this paper, the structural stability and elastic properties of defective and Er-doping off-stoichiometric Al₃X phases were calculated. First, the bulk properties (lattice constants and bulk modulus) of L1₂-Al₃X were calculated. Then, a nanoindentation experiment on the primary Al₃X phase was performed. Finally, the stability and elastic properties of defective and doped off-stoichiometric Al₃X were obtained, clarifying the effects of the point defects and Er additions.

4.1. The Bulk Properties

The L1₂-Al₃X precipitates are face-centered cubic crystal structures (see Figure 1) [5,67,68,69,70,71]. To study the ground-state properties of Al₃Li, Al₃Sc, and Al₃Zr, their equilibrium lattice configurations using different methods for the electron exchange-correlation functionals were calculated and compared to previous experiments [72,73,74,75,76,77,78]. The calculated ground-state properties are shown in

Table 1. Calculated lattice parameters (a₀, Å) and bulk modulus (B₀, GPa) of Al₃X.

Phases	Lattice Parameter and Bulk Modulu	PBE	LDA	PW91	Cal. Ref	Expt. Ref
Al3Li	a0 (Å)	4.027	3.967	4.032	3.949LDA [79]	4.010 [75]
	B0 (GPa)	64.032	70.07	62.40	62.9GGA [76] 72.3LDA [76]	66.0 [76]
Al3Sc	a0 (Å)	4.103	4.034	4.105	4.038LDA [78]	4.101 [77]
	B0 (GPa)	89.28	100.24	88.03	86.6PW91 [80] 87.3PBE [80]	91.7 [78]
Al3Zr	a0 (Å)	4.106	4.041	4.110	4.107PBE [81] 4.110PW91 [82] 4.097 [82]	-
	B0 (GPa)	102.16	113.45	100.46	103.45PW91 [80] 106.0PBE [80]	-

. The lattice constants (a₀) and bulk modulus (B₀) agree well with the experimental data and prior calculations [75,76,77,78,79,80,81,82]. There were no experimental data of a₀ and B₀ for Al₃Zr, so the comparisons of Al₃Li and Al₃Sc to experiments were performed. It is clearly shown that our calculations using PBE have a better match with the results of previous experiments compared to those using LDA or PW91. Although the calculation overestimates (underestimates) the a₀ (B₀) of Al₃Li by 0.4% (3%) and Al₃Sc by 0.05% (3%), the accuracy is within the reliability range of computational models [76,78,79,80,83,84].

Before we discuss the optimal doping position of Er in off-stoichiometric Al₃X, it is necessary to have a good understanding of the nature of point defects in Al₃X. The point defects lead to small deviations in the stoichiometric ratios of the chemical compositions with respect to the fully ordered L1₂-Al₃X, formatting the Al-rich and X-rich off-stoichiometric Al₃X. For the 32-atom Al₃X, there are four possible types of point defects, as shown in Figure 2(a1–a4), designated as V_Al, X_Al, V_X, and Al_X, respectively. Considering the large number of free Al atoms from the aluminum matrix filling around the Al₃X crystals, the different types of point defects formed were discussed. For Al-rich off-stoichiometric Al₃X, the X atom in Al₃X migrates and leaves V_X. The vacancy would be occupied by the Al atom from inside the Al₃X or Al matrix, forming Al_X. For X-rich off-stoichiometric Al₃X, the Al atom migrates and reforms the vacancy of V_Al; the X atom from Al₃X could occupy the vacancy to form an antisite X_Al. The atoms were allowed to occupy their ideal lattice positions in the off-stoichiometric Al₃X supercell at the beginning, and then local relaxation of atoms around the point defect was performed in order to enable all atoms to move to their equilibrium positions. To better understand the lattice structure, the mean distance of n-th nearest neighbors (nNN) of atomic pairs in Al₃X were counted and shown in Figure 2b. The point defects caused the displacement of atoms, and the specific data is shown in Table S3.

In general, the lower the alloy formation enthalpy of defective Al-rich or X-rich off-stoichiometric Al₃X supercells are, the better their stability would be [49,85]. Fully relaxed formation enthalpies of ordered stoichiometric Al₃X, Al-rich, and X-rich off-stoichiometric Al₃X were listed in

Table 2. The formation enthalpies of stoichiometric Al₃X and defective off-stoichiometric Al₃X structures, and the point defect formation enthalpies of off-stoichiometric Al₃X structures.

Defect		∆H (eV/atom)					Hd (eV/Defect)
		Perfect	VAl	XAl	VX	AlX	VAl	XAl	VX	AlX
Al24Li8	Cal.	−0.100	−0.074	−0.101	−0.074	−0.081	0.799	−0.040	0.809	0.606
	Ref. [87]	−0.100	−0.092	−0.101	−0.092	−0.094	0.791	−0.088	0.823	0.583
	Ref. [62]	−0.080	−0.054	−0.073	−0.053	−0.060	-	-	-	-
Al24Sc8	Cal.	−0.454	−0.421	−0.435	−0.393	−0.374	1.006	0.594	1.892	2.552
	Ref. [16]	−0.463	−0.428	−0.443	−0.399	−0.381	1.490	0.584	2.402	2.575
Al24Zr8	Cal.	−0.488	−0.474	−0.481	−0.450	−0.418	0.430	0.215	1.169	2.238
	Ref. [86]	-	-	-	-	-	0.410	0.320	1.060	2.070

and Figure 3. It can be seen that the formation enthalpies of defective off-stoichiometric Al₃X supercell alloys were negative, indicating their thermodynamic stability [86]. The results of Al₃Li, Al₃Sc, and Al₃Zr in Table 2 were also consistent with previous theoretical calculations [16,62,86,87]. For X-rich off-stoichiometric Al₃X alloys, the formation enthalpies of Al antisite (X_Al) (Al₃Li—0.101 eV/atom, Al₃Sc—0.435 eV/atom, and Al₃Zr—0.481 eV/atom) are smaller than that of Al vacancy, meaning that Al antisite defect (X_Al) are stable. In addition, the results showed that the stable point defects in L1₂ Al₃X are antisite on both Al-rich and X-rich off-stoichiometry. For off-stoichiometric Al₃Sc and Al₃Zr, the results showed that the stable point defects are Al antisites in X-rich and X vacancies in X-rich off-stoichiometry.

The point defect changed the distance of the surrounding atoms. To quantify and distinguish the structural properties of the defective off-stoichiometric Al₃X, we calculated radial distribution functions (RDFs) of the Al-Al, X-X, and Al-X atomic pairs within these defective structures and took a comparison. The results are shown in Figure 4(1a–3d). The RDF as a key quantity for evaluating the local distribution of atomic pairs can be expressed as [88]:

$$g\left(r\right)=\frac{n\left(r\right)}{{\rho }_{0}V}\approx \frac{n\left(r\right)}{4\pi r^2{\rho }_0{\delta }_r}$$

(18)

where $r$ is the cutoff radii from the center atoms, ${\delta }_r$ is the spherical shell thickness,$n\left(r\right)$ is the number of particles inside the spherical shell, and ${\rho }_0$ is the median density of the ideal crystal. As seen in Figure 4(1a–3d), there is a sharp peak near the first nearest neighbor (1NN) for all defective structures, which indicates the density of the first nearest neighbor atomic pairs. The low peaks in the third and fifth nearest neighbors indicate a weak intensity. The calculation results are very similar for the vacancy (V_Al and V_X), both having two peaks in the 1NN–5NN. For the X_Al antisite, there are three peaks in the 1NN and 3NN. In brief, there are relatively weak changes in atom displacements relative to the ideal position.

The site preference of Er in off-stoichiometric Al₃X was calculated by taking four occupation sites in off-stoichiometric Al₃X: (i) the Er atom directly occupying the Al site (Er_Al); (ii) the Er atom replacing the Al site meanwhile the Al atom occupying X site (Er_Al + Al_X); (iii) the Er directly taking up the X site (Er_X); and (iv) the Er replacing the X site meanwhile X atom occupying the Al site (Er_X + X_Al). The four occupation positions of Er occupying 32-atom Al₃X are shown in Figure 5a–d.

The site preference of the Er element in off-stoichiometric Al₃X was evaluated by calculating the enthalpy at absolute temperature. According to Equation (5), the enthalpies were listed in

Table 3. Formation enthalpies ΔH (eV/atom) of doping Er in Al₃X.

Phase	$∆\mathit{H}$ (eV/atom)
	ErAl	ErX + XAl	ErX	ErAl + AlX
Al24Li8	−0.125	−0.189	−0.171	−0.112
Al24Sc8	−0.533	−0.554	−0.585	−0.452
Al24Zr8	−0.598	−0.617	−0.639	−0.522

, and all doped Al₃X phases were stable due to their negative formation enthalpies. For X-rich off-stoichiometric Al₃Li, it was found that the formation enthalpy of off-stoichiometric Al₃Li (Er_Li + Li_Al) (−0.189 eV/atom) is lower than off-stoichiometric Al₃Li (Er_Al) (−0.125 eV/atom). For Al-rich off-stoichiometric Al₃Li, the formation enthalpy of Al₂₄Li₇Er (Er_Li) (−0.171 eV/atom) is lower than Al₂₄Li₇Er (Er_Al + Al_Li) (−0.112 eV/atom), revealing that the Er atom prefers to substitute the Li atom. Meanwhile, improving the point defect structure of Er indirectly occupying the Li site is more stable compared to direct occupancy. As a result, the lattice parameter is 4.059 Å. For off-stoichiometric Al₃Sc and Al₃Zr, the formation enthalpies show that the Er atom prefers to occupy the Sc (Zr) site in Al-rich and Sc (Zr)-rich off-stoichiometric Al₃Sc (Al₃Zr) supercells directly, increasing the lattice parameter to 4.123 Å (4.119 Å). The RDF diagram showed the change in crystal structure after Er doping (as shown in Figure 6). For the Er-Al and Er-Li bonds, there are sharp peaks in 1NN, 2NN, 3NN, and 4NN. In particular, there are additional Er-Al peaks at 1NN and 3NN and additional Er-Li peaks at 2NN and 4NN for the Er_X + X_Al, indicating the doping of Er changed the structure and increased the combination of bonds. The atom radius of Er (1.75 Å) [89] is similar to that of Sc (1.57 Å) [90] and Zr (1.60 Å) [91], while the atomic radius of Al (1.43 Å) is close to that of Li (1.52 Å) [92]. Therefore, Er is likely to occupy the Sc (Zr) sites, while Al prefers to occupy the Li sites. The doping of the Er element induces a local stress field and tends to replace the similar-sized atoms [93], which is the essential condition for the stabilization of the doped phases. Therefore, Er prefers to occupy the X atom site, not to occupy the Al atom site.

4.2. Elastic Modulus

The elastic properties of materials reflect their strain resistance, which plays the role of one of the most important mechanical properties [94,95,96]. In fact, these properties are very important for the industrial applications of engineering materials. To study the effect of the point defects and Er doping on the elastic properties of these phases, the elastic constants (C_ij), elastic modulus (B, G, and E), υ, and B/G of pure, defective, and doped off-stoichiometric Al₃X at 0 K were calculated. The values of modulus reflect the bonding strength between atoms in crystals. The ratio of B/G indicates the size of the plasticity range, and a high value is associated with malleability and a low value with brittleness [62,97]. Meanwhile, the corresponding previous experimental and calculated data in the literature [30,76,78,79,80,81,82,98,99,100] of pure Al₃X were used to verify the accuracy of our calculations, as shown in

Table 4. The calculated bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratios v, and ratios of the bulk modulus to shear modulus (B/G) of Al₃X in comparison to experiments.

Phase	Species	B	G	E	v	B/G
Al24Li8	Perfect	63.8	42.7	104.7	0.226	1.495
	Ref. [30]	62.8	42.3	103.6	0.225	1.484
	Ref. [98]	62.7	42.7	104.4	0.222	1.468
	Ref. [76]	62.9	49.0	116.6	0.191	1.284
	Ref. [79]	66.0	43.0	105.9	0.232	1.535
	VAl	68.2	38.7	97.7	0.261	1.760
	LiAl	67.4	40.3	100.8	0.251	1.673
	VLi	65.2	43.9	107.6	0.225	1.484
	AlLi	67.0	39.2	98.3	0.255	1.711
	ErAl	61.4	38.3	95.0	0.242	1.605
	ErLi + LiAl	63.2	44.5	108.1	0.215	1.421
	ErLi	66.6	44.5	109.1	0.227	1.498
	ErAl + AlLi	64.0	37.5	94.2	0.255	1.707
Al24Sc8	Perfect	88.2	69.4	164.9	0.189	1.272
	Ref. [76]	91.8	71.7	170.7	0.190	1.280
	Ref. [78]	92.3	78.7	183.9	0.168	1.173
	Ref. [99]	85.9	72.2	169.1	0.171	1.190
	Ref. [80]	87.3	71.2	168.0	0.179	1.226
	Ref. [100]	86.0	74.5	173.5	0.164	1.154
	Ref. [78]	91.7	68.2	163.9	0.202	1.345
	VAl	89.9	66.4	159.8	0.204	1.354
	ScAl	87.9	67.0	160.2	0.196	1.312
	VSc	84.3	68.2	161.1	0.182	1.237
	AlSc	85.1	65.6	156.6	0.193	1.297
	ErAl	89.9	66.4	159.8	0.204	1.354
	ErSc + ScAl	97.8	72.8	174.9	0.202	1.344
	ErSc	93.2	71.1	170.1	0.196	1.310
	ErAl + AlSc	85.1	65.6	156.6	0.193	1.297
Al24Zr8	Perfect	103.0	65.8	162.7	0.237	1.567
	Ref. [80]	106.0	65.9	163.8	0.243	1.608
	Ref. [82]	101.4	62.9	156.4	0.243	1.612
	Ref. [82]	99.6	82.1	193.1	0.177	1.213
	Ref. [81]	102.6	66.7	164.5	0.233	1.538
	VAl	110.0	70.1	173.4	0.237	1.570
	ZrAl	123.5	65.0	165.9	0.276	1.900
	VZr	100.4	70.8	172.1	0.214	1.417
	AlZr	100.5	61.6	153.5	0.245	1.630
	ErAl	99.7	62.3	154.7	0.241	1.600
	ErZr + ZrAl	100.2	65.0	160.4	0.233	1.541
	ErZr	100.3	65.1	160.6	0.233	1.541
	ErAl + AlZr	96.3	63.1	155.4	0.231	1.527

and Table S1.

As listed in Table 4, the calculated results of bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratios v, and ratios of the bulk modulus to shear modulus B/G for the cubic phase Al₃Li are 63.8 GPa, 42.7 GPa, 104.7 GPa, 0.226, and 1.495, respectively, which are close to the experimental results [79] (66.0 GPa, 43.0 GPa, 105.9 GPa, 0.23, and 1.535, respectively). Additionally, our calculation values of Al₃Li are nearly the same as previous calculations [30,76,98], as shown in the Table S4. For the pure Al₃Sc phase, our calculation data of bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratios v, and ratios of the bulk modulus to shear modulus B/G are 88.2 GPa, 69.4 GPa, 164.9 GPa, 0.189, and 1.272, respectively, which agrees well with the experimental data [78] and calculated data [76,78,80,99,100]. To further ensure the reliability of the calculated data, as described in Section 3, we measured and calculated Young’s modulus E by nanoindentation experiments. Overall, thirteen effective curves of Young’s modulus and indentation depth were obtained by measuring different Al-2Sc particles. As shown in Figure 7, the E of Al₃Sc was determined as 170.1 ± 3.9 (GPA), which was in good agreement with our calculated value (171.1 GPA) and Woodward’s value (164.9 GPa). Similarly, the calculated results of elastic properties of pure Al₃Zr were close to previous reports [80,81,82]. Those data further confirmed the validity of our model and the reliability of our calculation results.

To study the influence of defect points and Er doping, we further calculated and compared the elastic properties of pure, defective, and doped Al₃X (X = Li, Sc, and Zr). The defective and doped Al₃X all satisfied the mechanical stability according to Equation (12). The results showed that the defective Al₃X is not only energetically stable but also mechanically stable. As seen in Table 4, combining the elastic modulus (B, G, and E), Poisson’s ratios, and ratios of the bulk modulus to shear modulus of pure and defective off-stoichiometric Al₃X, we could find that: (i) compared with other types of defects (V_Al, Li_Al, and Al_Li), the V_Li defect could improve the elastic modulus (B, G, E) while maintaining Poisson’s ratios, and ratios of the bulk modulus to shear modulus similar to pure Al₃Li. Since Li_Al in Li-rich and Al_Li in Al-rich off-stoichiometric Al₃Li were relatively stable, we speculated that the existence of point defects would increase the bulk modulus, Poisson’s ratios, and ratios of the bulk modulus to shear modulus of off-stoichiometric Al₃Li. Those results mean that defective Al₃Li has a higher resistance to deformation and ductility than pure Al₃Li [94,101]; (ii) the existence of point defect (except for V_Sc) increased Poisson’s ratios and ratios of the bulk modulus to the shear modulus of off-stoichiometric Al₃X, meanwhile decreasing the elastic modulus (except for bulk modulus of V_Al). Because the Sc_Al in Sc-rich and V_Sc in Al-rich off-stoichiometric Al₃Sc were stable, we predicted that the point defects would reduce the bulk modulus, shear modulus, and Young’s modulus. The results showed that point defects decrease the resistance to the strain of Al₃Sc crystal, making it easy to undergo deformation; and (iii) the presence of V_Al and Zr_Al increases the elastic modulus and Poisson’s ratios and ratios of the bulk modulus to the shear modulus of off-stoichiometric Al₃Zr. Al_Zr decreased its elastic modulus while increasing Poisson’s ratios and ratios of the bulk modulus to the shear modulus. Additionally, V_Zr decreased its Poisson’s ratios, ratios of the bulk modulus to shear modulus, and its bulk modulus. Because Zr_Al in X-rich and V_Sc in Al-rich off-stoichiometric Al₃Zr were stable, we inferred that the point defects would enhance the Young’s modulus of off-stoichiometric Al₃Zr. The results suggested that defective Al₃Zr were less prone to deformation than pure stoichiometric Al₃Zr. The lattice parameter of Al bulk is 4.038 Å. The lattice mismatch of defective Al₃X with Al bulk became larger, and that of defective off-stoichiometric Al₃Zr with Al bulk became small, which is inversely related to the elastic modulus. Compared the elastic properties of pure Al₃X with that of Er-doping off-stoichiometric Al₃X, we could conclude that: (i) the substitution of Al to the Er site led to the decrease of elastic modulus while increasing Poisson’s ratios, and ratios of the bulk modulus to shear modulus of off-stoichiometric Al₃Li. The substitution of Li to Er and Al to Li sites increased the shear modulus, Young’s modulus, Poisson’s ratios, and ratios of the bulk modulus to the shear modulus of off-stoichiometric Al₃Li. The substitution of Li to Er site increases the elastic modulus (B, G, E), Poisson’s ratios, and ratios of the bulk modulus to shear modulus of off-stoichiometric Al₃Li. The substitution of Al to Er and Li to Al sites increases the bulk modulus, Poisson’s ratios, and ratios of the bulk modulus to shear modulus of off-stoichiometric Al₃Li. Because the Er atom preferred to occupy the Er_Li + Li_Al site, we could conclude that the doping of Er increased the shear modulus and Young’s modulus of off-stoichiometric Al₃Li. The results indicated that the Al₃Li with doping Er has a higher resistance to deformation and that of shear strain; (ii) the substitution of Al to Er site increases the bulk modulus, Poisson’s ratios, and ratios of the bulk modulus to the shear modulus of off-stoichiometric Al₃Sc. The substitution of Sc to Er and Al to Sc sites and Sc to Er sites both increase the elastic modulus of Al₃Sc, and the increment of occupying Er_Sc site was less than that of the Er_Sc + Sc_Al site relatively. Er_Al + Al_Sc decreases the elastic modulus, Poisson’s ratios, and ratios of the bulk modulus to the shear modulus. Because of the preference of the Er_Sc, we could find that the doping of the Er atom increased the bulk modulus, shear modulus, Young’s modulus, and ratios of the bulk modulus to the shear modulus while decreasing Poisson’s ratios of off-stoichiometric Al₃Sc. The results demonstrated that Er-doping improved resistance to strain and deformation while increasing plasticity; and (iii) the substitution position of Er_Al, Er_Zr + Zr_Al, Er_Zr, and Er_Al + Al_Zr all decreased the elastic modulus of off-stoichiometric Al₃Zr. The Er atom preferred to occupy the Er_Zr, so we could conclude that the doping of the Er atom decreased the elastic modulus, Poisson’s ratios, and ratios of the bulk modulus to the shear modulus of off-stoichiometric Al₃Zr. The results suggested that Er-doping improved the resistance to deformation and plasticity. The lattice mismatch of Er-doping off-stoichiometric Al₃X with Al bulk all became larger, indicating that Er has some impact on lattice mismatch and elastic modulus of off-stoichiometric Al₃X.

Combined with the relatively more stable structures of defective and Er-doping off-stoichiometric Al₃X, a more intuitive variation of the pure, defective, and Er-doping elastic properties is shown in Figure 8a–e. Compared with the clean Al₃Li, the bulk modulus of defective off-stoichiometric Al₃Li generally increases by 5.64%. The shear modulus and Young’s modulus of defective off-stoichiometric Al₃Li generally decrease by 5.62% and 3.72%, respectively. Poisson’s ratios and ratios of the bulk modulus to the shear modulus of defective off-stoichiometric Al₃Li generally increase by 11.1% and 11.91%, respectively. Those results meant that defective off-stoichiometric Al₃Li have smaller shear modulus and Young’s modulus but higher bulk modulus than pure Al₃Li. However, the bulk modulus of Er-doping off-stoichiometric Al₃Li decreases by 0.94%. Shear modulus and Young’s modulus of Er-doping off-stoichiometric Al₃Li increase by 4.22% and 3.25%, respectively. Poisson’s ratios and ratios of the bulk modulus to shear modulus of Er-doping off-stoichiometric Al₃Li decrease by 4.87% and 4.95%, respectively. Compared with the clean Al₃Sc, the bulk modulus, shear modulus, and Young’s modulus of defective off-stoichiometric Al₃Sc decrease 0.34%, 3.46%, and 2.85% respectively, and that of Er-doping off-stoichiometric Al₃Sc increase 5.67%, 2.45%, and 3.15% respectively; Poisson’s ratios and ratios of the bulk modulus to shear modulus of defective off-stoichiometric Al₃Sc increased by 3.74% and 3.14%, and that of Er-doping Al₃Sc increased by 3.64% and 2.99%. Those results indicated that defective off-stoichiometric Al₃Sc have lower elastic modulus but higher Poisson’s ratios and ratios of the bulk modulus to shear modulus than pure Al₃Sc, whereas the Er-doping off-stoichiometric Al₃Sc have higher elastic modulus and Poisson’s ratios and ratios of the bulk modulus to shear modulus. Compared with the Al₃Zr, the bulk modulus and Young’s modulus of defective off-stoichiometric Al₃Zr increased by 19.99% and 1.97%, and that of Er-doping off-stoichiometric Al₃Zr decreased by 2.62% and 1.29%; shear modulus of defective and Er-doping off-stoichiometric Al₃Zr decreased by 12.22% and 1.06% respectively; Poisson’s ratios of defective and Er-doping off-stoichiometric Al₃Zr increased by 16.46% and 21.25%; the ratios of the bulk modulus to shear modulus of defective and Er-doping off-stoichiometric Al₃Zr decreased by 1.69% and 1.66%. Those results meant that the defective off-stoichiometric Al₃Zr has a higher bulk modulus and Young’s modulus than pure Al₃Zr, whereas Er-doping off-stoichiometric Al₃Zr has a higher shear modulus.

We can increase the elastic modulus of the phase by adjusting the point defects. The effect of doping other solid solution atoms on the Young’s modulus remains to be further investigated, which is significant for screening the design of new materials with high modulus. For off-stoichiometric Al₃Li, the low of Young’s modulus with solid solution atoms was summarized, as shown in Figure 9. The results show that the Young’s modulus of off-stoichiometric Al₃Li tends to increase and then decrease with increasing atomic number at the same period. In particular, the atoms of elements, such as Sc, Ti, Zr, Hf, Ta, Mn, Ir, or Cf, can greatly increase the Young’s modulus.

5. Conclusions

In this paper, the point defects and the substitutional element additions to the off-stoichiometric Al₃X structure were investigated by using first-principle calculations. The formation enthalpies of defective and doped off-stoichiometric Al₃X were calculated, and the elastic properties of point defects and Er were analyzed. We found that the negative formation enthalpies of defective and doped off-stoichiometric Al₃X indicate thermodynamic stability. By comparing the defective structure with the pure Al₃X structures, the following conclusions can be drawn: (i) X_Al is easier to form in off-stoichiometric Al₃X, Er tends to occupy the Li site in off-stoichiometric Al₃Li (Er_Li + Li_Al) indirectly, and Er prefers to occupy the X sites; (ii) the existence of point defects would reduce Young’s modulus of off-stoichiometric Al₃Li and Al₃Sc, while they can increase Young’s modulus of the off-stoichiometric Al₃Zr phase; (iii) the substitution of the Er atom in the off-stoichiometric Al₃Sc phase could increase elastic modulus. These results suggest that point defects and micro-alloying Er could improve the elastic properties of the Al-Li alloys; (iv) for off-stoichiometric Al₃Li, the Young’s modulus of Al₃Li was improved by adding such substitutional atoms as Sc, Ti, Zr, Hf, Ta, Mn, Ir, or Cf.