First-Principles Investigation into the Elastic Anisotropy and Thermodynamic Properties of the L1₂-Type ScAl₃ Phase in Aluminum Alloys

Abstract

This study investigates the elastic anisotropy and thermodynamic properties of the L1₂-type ScAl₃ phase under extreme conditions (0–1500 K and 0–50 GPa) using first-principles calculations. The elastic constants were determined using a precise stress–strain method, with polycrystalline moduli derived via the Voigt–Reuss–Hill (VRH) approximation. A systematic analysis was conducted to characterize the elastic anisotropy of Young’s modulus, shear modulus, and Poisson’s ratio. Results demonstrate that ScAl₃ is mechanically stable and exhibits near-perfect elastic isotropy (A^U = 0.0001). Thermodynamic analysis via the quasi-harmonic Debye–Grüneisen model reveals that the phase maintains its structural integrity and significant heat resistance up to 1500 K, despite thermal softening. These findings provide theoretical insights into the physical nature of ScAl₃ intermetallics and offer quantitative guidance for the design and thermal treatment of Sc-reinforced aluminum alloys in high-temperature aerospace applications due to their superior combination of strength and toughness.

1. Introduction

In the development of next-generation aluminum alloys, Sc is recognized as the most potent micro-alloying element for enhancing strength and toughness [1,2]. Its addition facilitates microstructural refinement of the ingot [3] and elevates the recrystallization temperature [4]. Furthermore, Sc markedly enhances weldability, thermal stability [5], corrosion resistance [6], and neutron irradiation resistance [7]. Compared to traditional transition metals like Ti and Zr, Sc demonstrates superior strengthening efficiency per atomic percent while maintaining a density comparable to the Al matrix [8,9]. Quantitatively, the addition of 0.41 wt.% Sc in annealed Al alloys can generate a precipitation-strengthening increment of up to 350 MPa [10]. This is attributed to the formation of primary and secondary L1₂-type ScAl₃ nanoprecipitates [11], which exhibit a low lattice mismatch of 1.32–1.34% with the Al matrix, significantly better than the 2.04% mismatch observed for L1₂-TiAl₃ [8,10].

Prior studies have primarily focused on the phase stability, electronic structure, and optical properties of ScAl₃ [12,13]. However, a critical knowledge gap remains regarding its anisotropy of mechanical response and thermodynamic evolution under the extreme thermomechanical conditions—high temperature and high pressure—encountered during advanced aerospace manufacturing processes. Duan et al. [14] investigated the phase stability and electronic structure of ScAl₃ under high pressures of 0–40 GPa using first-principles calculations. Pan et al. [15] studied the temperature dependence (0–900 K) of the elastic properties of L1₂-type ScAl₃. While Zr is often co-added to form (Sc, Zr)Al₃ shells that inhibit Sc diffusion—with Zr diffusing four orders of magnitude slower than Sc at 300 °C [10]—the fundamental anisotropic nature of the ScAl₃ core remains the limiting factor for mechanical stability.

This study addresses this deficiency by providing the first comprehensive rigorous first-principles evaluation of the elastic anisotropy and thermodynamic evolution of L1₂-ScAl₃ up to 1500 K and 50 GPa using the quasi-harmonic Debye–Grüneisen model, offering the quantitative numbers necessary to refine hardening mechanisms in complex alloy systems. In particular, we examine the Grüneisen constant and the evolution of the thermal expansion coefficient and heat capacity under different pressures, and resolve the controversy in the previous literature regarding the description of the thermal expansion coefficient in the high-temperature region [11,16].

2. Computational Methods

As shown in Figure 1a, the L1₂-type ScAl₃ phase crystallizes in the Pm3m space group with an experimental lattice constant of a = 4.103 Å (ICSD #107878) [17,18]. Calculations were performed within the framework of density functional theory (DFT) using the CASTEP 20.11 [19] code with a plane-wave basis set. Periodic boundary conditions were applied to the cubic unit cell to simulate bulk properties. Non-spin polarized calculation was used due to a non-magnetic material of ScAl₃. The interaction between valence electrons and ion cores was described by ultrasoft pseudopotentials, while the GGA-PBEsol functional was employed for the exchange-correlation potential to allow for an accurate description of the exchange-correlation effects, which is optimized for densely packed solids [20]. A plane-wave cutoff energy of 600 eV and a 10 × 10 × 10 Monkhorst–Pack k-point grid were adopted. The convergence of the plane-wave cut energy and k-point sampling was rigorously tested to ensure energy accuracy. Gaussian smearing (width = 0.1 eV) was applied to the electronic occupations. Valence electron configurations were explicitly set to 3s²3p⁶3d¹4s² for Sc and 3s²3p¹. Full geometry optimization, including cell volume and atomic positions, was executed using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. Stringent convergence criteria were set to 5 × 10⁻⁶ eV/atom for energy, 0.01 eV/Å for maximum force, 0.02 GPa for maximum stress, and 5 × 10⁻⁴ Å for maximum displacement. Following optimization, the calculated equilibrium lattice constant was determined to be a = 4.0703 Å, which is in good agreement with the original experimental value of 4.103 Å [17]. The linear error is approximately 0.8% (with a corresponding volume error of ~2.4%), which falls within the standard acceptable range for GGA functionals [21]. To ensure the precision of the elastic constants (C_ij), the stress–strain method was implemented by applying a set of small finite strains to the optimized equilibrium unit cell and calculating the resulting stresses.

To determine the bulk properties and ensure numerical stability, 22 energy–volume (E-V) data points were calculated across a pressure range of 0–50 GPa and fitted to the third-order Birch–Murnaghan equation of state (EOS) [22,23]. The EOS fitting was adopted as the primary source for lattice parameters to mitigate discontinuities caused by fixed plane-wave cutoffs during single-point geometry relaxations. The optimized lattice parameter and bulk modulus (B₀) were 4.0907 Å and 92.303 GPa, respectively (Figure 1b). Additionally, the first (B′) and second (B″) pressure derivative of the isothermal bulk modulus are 3.719 and −3.90 × 10⁻² GPa⁻¹, respectively. The close agreement between calculated, fitted, and experimental values validates the reliability of the chosen computational parameters.

3. Results and Discussion

3.1. Elastic Properties of ScAl₃

Elastic constants (C_ij) characterize the resistance of a crystal to external stress [24]. For the cubic ScAl₃ phase, there are three independent elastic constants: C₁₁, C₁₂, and C₄₄ [25]. The calculated C_ij values using the stress–strain method are summarized in Table 1. The mechanical stability of the crystal was verified using the Born stability criteria for cubic crystal systems [26]: C₄₄ > 0, C₁₁ − C₁₂ > 0, and C₁₁ + 2C₁₂ > 0. All calculated elastic constants satisfy these requirements, confirming that the ScAl₃ is mechanically stable. Polycrystalline elastic moduli, including the bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (ν), were derived using the Voigt–Reuss–Hill (VRH) approximation. The VRH average, being the arithmetic mean of the Voigt (uniform strain) and Reuss (uniform stress) bounds, provides the most reliable theoretical estimate for polycrystalline aggregates [27]:

$$B_H={\displaystyle \frac{1}{2}}\left(B_V+B_R\right) G_H={\displaystyle \frac{1}{2}}\left(G_V+G_R\right),$$

(1)

The calculated polycrystalline elastic moduli of L1₂-type ScAl₃ are summarized in Table 1, with a determined bulk modulus (B) of 91.67 GPa, shear modulus (G) of 74.30 GPa, Young’s modulus (E) of 175.48 GPa, and Poisson’s ratio (ν) of 0.181. Notably, the bulk modulus derived from the elastic constants is in excellent agreement with the value (B₀) obtained from the third-order Birch–Murnaghan equation of state fitting, confirming the consistency of our calculations in estimating the compressibility of ScAl₃ [28]. To evaluate the ductile-to-brittle transition, the calculated Pugh ratio (G/B) of ScAl₃ is 0.81. While traditional models suggest that a Pugh ratio > 0.57 indicates intrinsic brittleness, it is important to note that the Pugh model cannot provide an absolute basis for the behavior of novel materials [29]. Therefore, we characterize ScAl₃ as having a “brittle tendency” relative to the Al matrix, which is further supported by the relatively low Poisson’s ratio. The calculated Vickers hardness (H_V) is approximately 15.30 GPa [30], indicating the significant hardness of this intermetallic phase, likely originating from the covalent character of the Sc-Al bonds. This brittle character tendency, combined with a low Poisson’s ratio and high Vickers hardness, underlines the role of ScAl₃ as a rigid, non-deforming reinforcement that enhances the strength of the Al matrix via Orowan strengthening. The degree of elastic anisotropy in the crystalline material is typically quantified using the universal anisotropy index (A^U), defined by the following expression [26]:

$$A^U=5{\displaystyle \frac{G_V}{G_R}}+{\displaystyle \frac{B_V}{B_R}}-6,$$

(2)

The calculated universal anisotropy index (A^U = 0.0001) indicates that ScAl₃ exhibits near-perfect elastic isotropy. This near-zero value suggests that the bonding environment in the L1₂ structure is highly symmetric. This is further visualized through the 2D projections of Young’s modulus, linear compressibility, shear modulus, and Poisson’s ratio in the [001], [010], and [100] directions, respectively [31], as shown in Figure 2. The nearly circular profiles in these projections confirm the minimal orientation dependence of the elastic parameters, consistent with the calculated A^U. Despite this overall isotropy, the slight variation in Poisson’s ratio anisotropy indices (1.031) in Table 2 highlights subtle orientation dependencies that may influence dislocation mobility during deformation. This indicates that the mechanical stress is distributed uniformly across the Al–matrix interface, minimizing local stress concentrations and enhancing the structural integrity of aerospace components.

Table 1. The elastic constants (C_ij) and polycrystalline elastic properties of L1₂-type ScAl₃.

Elastic Properties	C11/ GPa	C12/ GPa	C44/ GPa	BH/ GPa	GH/ GPa	G/B	E/ GPa	ν	HV/ GPa [30]	AU	Refs.
Cal.	191.29	41.86	74.02	91.67	74.30	0.81	175.48	0.181	15.30	0.0001	This work
Exp.	182.6	45.9	68.4	91.5	68.4	0.75	164.2	0.20	13.16	—	Ref. [24]
Cal.	188.0	43.7	71.4	91.8	71.9	0.78	170.7	0.19	14.38	—	Ref. [25]
Cal.	181.5	37.8	71.0	85.7	71.6	0.84	167.6	0.17	15.43	—	Ref. [28]

Table 1. The elastic constants (C_ij) and polycrystalline elastic properties of L1₂-type ScAl₃.

Elastic Properties	C11/ GPa	C12/ GPa	C44/ GPa	BH/ GPa	GH/ GPa	G/B	E/ GPa	ν	HV/ GPa [30]	AU	Refs.
Cal.	191.29	41.86	74.02	91.67	74.30	0.81	175.48	0.181	15.30	0.0001	This work
Exp.	182.6	45.9	68.4	91.5	68.4	0.75	164.2	0.20	13.16	—	Ref. [24]
Cal.	188.0	43.7	71.4	91.8	71.9	0.78	170.7	0.19	14.38	—	Ref. [25]
Cal.	181.5	37.8	71.0	85.7	71.6	0.84	167.6	0.17	15.43	—	Ref. [28]

Table 2. The elastic anisotropy of ScAl₃, including maximum/minimum moduli and their respective anisotropy indices.

	Young’s Modulus		Linear Compressibility		Shear Modulus		Poisson’s Ratio
Value	Emin/GPa 174.96	Emax/GPa 176.26	βmin/TPa−1 3.6364	βmax/TPa−1 3.6364	Gmin/GPa 74.018	Gmax/GPa 74.715	νmin 0.1785	νmax 0.1841
Anisotropy	1.007		1.0000		1.009		1.031
Axis	−0.5774 0.5774 0.5773	−0.0000 −0.0000 1.0000	0.7934 0.0000 0.6088	0.0000 0.0000 1.0000	0.0000 0.0000 1.0000	0.7071 −0.0002 0.7071	0.7071 −0.0003 −0.7071	0.7071 −0.0002 0.7071
Second axis					1.0000 0.0002 −0.0000	0.7071 −0.0006 −0.7071	0.0002 1.0000 −0.0002	0.7071 −0.0006 −0.7071

Table 2. The elastic anisotropy of ScAl₃, including maximum/minimum moduli and their respective anisotropy indices.

	Young’s Modulus		Linear Compressibility		Shear Modulus		Poisson’s Ratio
Value	Emin/GPa 174.96	Emax/GPa 176.26	βmin/TPa−1 3.6364	βmax/TPa−1 3.6364	Gmin/GPa 74.018	Gmax/GPa 74.715	νmin 0.1785	νmax 0.1841
Anisotropy	1.007		1.0000		1.009		1.031
Axis	−0.5774 0.5774 0.5773	−0.0000 −0.0000 1.0000	0.7934 0.0000 0.6088	0.0000 0.0000 1.0000	0.0000 0.0000 1.0000	0.7071 −0.0002 0.7071	0.7071 −0.0003 −0.7071	0.7071 −0.0002 0.7071
Second axis					1.0000 0.0002 −0.0000	0.7071 −0.0006 −0.7071	0.0002 1.0000 −0.0002	0.7071 −0.0006 −0.7071

3.2. Thermodynamic Properties of ScAl₃

The phonon dispersion relations and phonon density of states for ScAl₃ were obtained using the density functional perturbation theory, and are presented in Figure 3. For the L1₂-type unit cell containing four atoms, the phonon spectrum consists of 12 branches: three acoustic and nine optical modes. The absence of negative or imaginary frequencies throughout the Brillouin zone rigorously confirms the dynamical stability of the L1₂-ScAl₃ phase [11]. The calculated phonon density of states using a dense 30 × 30 × 30 q-mesh exhibits smooth quadratic growth in the low-frequency acoustic region.

Since the melting point of ScAl₃ is approximately 1593 K [32], the thermodynamic response under extreme conditions was investigated using the quasi-harmonic Debye–Grüneisen model, which was implemented using the GIBBS2 code to bridge the gap between zero-temperature DFT data and finite-temperature behavior [33,34]. As shown in Figure 4, the isothermal bulk modulus (B_T) exhibits a monotonic decrease with increasing temperature, indicative of thermal softening due to intensified atomic vibrations. At zero pressure, B_T decreases slowly from approximately 92.30 GPa at 0 K to about 74.01 GPa at 1500 K, indicating that ScAl₃ possesses excellent heat resistance and thermal stability, thereby effectively enhancing the high-temperature creep resistance of aluminum alloys during prolonged service when ScAl₃ nanoprecipitates are highly effective in pinning grain boundaries. Conversely, B_T scales positively with pressure, reflecting the lattice stiffening under pressure.

The variation in the Debye temperature (Θ_D) of ScAl₃ as a function of temperature and pressure is displayed in Figure 5. The Θ_D provides insight into the bond strength and hardness. At zero pressure, Θ_D for ScAl₃ is approximately 629 K, which is close to the value (625.77 K) calculated by Duan et al. [14]. While Θ_D decreases slightly with temperature, it exhibits a much stronger sensitivity to pressure, increasing significantly as the lattice is compressed.

The linear thermal expansion coefficient (α) shows a rapid non-linear increase below 300 K, followed by a near-linear expansion at higher temperatures (Figure 6). At 300 K and 0 GPa, α is approximately 1.4 × 10⁻⁵ K⁻¹, which is in excellent agreement with the experimentally observations of (1.43–1.6) × 10⁻⁵ K⁻¹ [11,16] for bulk alloy system and aligns with predictions from other high-accuracy first-principles calculations [35]. The results indicate that applied pressure significantly suppresses thermal expansion, particularly with the effect being more pronounced at elevated temperatures. These quantitative insights offer direct guidance for optimizing high-pressure manufacturing processes, such as isothermal forging and extrusion, where the phase’s stability is paramount.

The Grüneisen constant (γ), which characterizes anharmonic effects and is relevant with the lattice thermal conductivity, increases with temperature and decreases with pressure, as illustrated in Figure 7a. Figure 7b illustrates the temperature dependence of entropy (S) for ScAl₃ under various pressures. At zero pressure, S increases sharply with temperature below 300 K, transitioning to a more moderate growth rate above 400 K. This temperature-dependent behavior remains qualitatively consistent under elevated pressures. Conversely, at a constant temperature, S exhibits a monotonic decrease with increasing pressure. This reduction is primarily attributed to pressure-induced volume contraction, which suppresses the amplitude of atomic vibrations and subsequently reduces the vibrational entropy. The heat capacities (C_V, C_P) were also evaluated (Figure 7c,d). At low temperatures, C_V follows the T³ law, while at high temperatures, it converges toward the Dulong–Petit limit of approximately 99.77 J/(mol·K). The observed increase in C_P at high temperatures compared to C_V is attributed to the PdV work associated with thermal expansion. Overall, the quasi-harmonic Debye–Grüneisen model retains high physical reliability in describing the thermodynamic properties of ScAl₃ up to 1500 K, and can accurately capture the fundamental nonlinear evolution of its volume and heat capacity with temperature.

4. Conclusions

Our first-principles investigation provides a quantitative foundation for understanding the mechanical and thermodynamic limits of the L1₂-type ScAl₃ phase. Geometry optimization using the PBEsol functional yielded lattice parameters in close agreement with experimental data. By resolving the discrepancy in lattice constant errors with the third-order Birch–Murnaghan equation of state, we established a more robust physical model. The predicted moduli (B = 91.67 GPa, G = 74.30 GPa, E = 175.48 GPa) and the Pugh ratio (0.81) characterize ScAl₃ as a mechanically stable, high-strength, high-hardness intermetallic compound with a tendency toward brittleness and exceptional elastic isotropy. The thermodynamic analysis via the quasi-harmonic Debye–Grüneisen model provided the temperature and pressure dependencies of key parameters (B_T, Θ_D, α, C_V, and Cp), revealing that ScAl₃ maintains high thermal stability (isothermal bulk modulus of ~74 GPa at 1500 K), which is pivotal for maintaining the creep resistance of aluminum alloys in high-temperature aerospace applications. The small thermal expansion mismatch between ScAl₃ and Al at high temperatures ensures interfacial stability. This work provides the necessary input parameters for multiscale hardening models, enabling more accurate predictions of the yield strength and toughness of complex Al-Sc alloy systems.