First-Principles-Based Structural and Mechanical Properties of Al₃Ni Under High Pressure

Abstract

The structural, elastic, and thermal characteristics within the 0–30 GPa pressure range of Al₃Ni intermetallic compounds were extensively studied using first-principles computational techniques. Using structural optimization, lattice parameters and the variation in volume variation under diverse pressures were determined, and the trends in their structural alteration with pressure were identified. The computed elastic constants validate the mechanical stability of Al₃Ni within the applied pressure range and show that its compressive stiffness and shear resistance increase rapidly with increasing pressure. The Cauchy pressure variation implies that the metallic nature of Al₃Ni increases gradually with increasing pressure. Moreover, through analysis of Poisson’s ratio, the anisotropy factor, and the sound velocity, we ascertained that pressure attenuates the anisotropic attributes of the material, and Al₃Ni exhibits more pronounced isotropic characteristics and mechanical homogeneity under high-pressure conditions. The substantial increase in the Debye temperature further suggests that high pressure fortifies the lattice dynamic rigidity of the material. This current research systematically elucidated the stability of Al₃Ni under high-pressure conditions and the law of the transformation of it mechanical behavior, providing a theoretical foundation for its application under extreme circumstances.

1. Introduction

Aluminum–nickel eutectics have been examined in automotive applications as high-temperature, high-strength replacements for aluminum–copper-based cast aluminum alloys [1,2,3,4]. In contrast to the reinforcing phase in aluminum–copper alloys, the reinforcing phase, Al₃Ni, in aluminum–nickel alloys can resist roughening at temperatures up to approximately 400 °C, thereby leading to outstanding mechanical properties at elevated temperatures [5,6,7]. Nickel exhibits a diffusion coefficient comparable to that of copper and silicon; nevertheless, it has a substantially lower solid solubility in the aluminum matrix [8], which limits the quantity of the nickel solute atoms that contribute to the roughening of Al₃Ni. The intermetallic compound Al₃Ni can solidify in the form of fibers within the aluminum matrix [9]. The bar-like nature of Al₃Ni enables reinforcement via stress transfer and inhibits dislocation movement. However, the patterns of the variation in the structural and properties of Al₃Ni under stress conditions have not been comprehensively investigated, and the knowledge deficiency in this domain restricts the expansion in its practical applications in extreme environments.

In the domain of materials science, the pressure of the environment profoundly influences the structure and properties of materials [10,11]. A multitude of metals and intermetallic compounds experience significant alterations in their crystal structures, electronic properties, and mechanical properties under high-pressure conditions [12,13,14,15]. Consequently, by exploring the structural optimization and high-pressure mechanical properties of Al₃Ni, it is possible to create a theoretical foundation for its potential application under high-pressure conditions.

Existing research has predominantly focused on the structure and mechanics of Al₃Ni under ambient pressure, overlooking the variations in material properties under high-pressure conditions [16]. In the present study, conventional mechanical experiments could not be used for conducting detailed analyses of the materials under diverse pressures, thereby making numerical simulation methods crucial for high-pressure investigations. First-principles computational techniques, being parameter-free quantum mechanical computational approaches, have been extensively employed in materials science research due to their capacity to precisely predict material properties [17,18,19,20,21,22].

In the research on high-pressure properties, the unique and irreplaceable value of first-principles calculations has been demonstrated. For numerous materials such as MgB₂C₂ [23] and Mn₂MgGe [24] alloy, the long-period superstructures of h- and r-Al₂Ti [25], and the Si-doped RuGe compound [26], the evolutionary mechanisms of their structural, elastic, electronic, magnetic, thermodynamic, and other properties under high pressure are extremely complex and difficult to comprehensively and accurately analyze through traditional experimental methods. First-principles calculations, based on the fundamental principles of quantum mechanics and without relying on empirical parameters, can directly be used to delve into the electron structure level of materials through simulation and analysis, making it possible to reveal the microscopic details of these materials in a high-pressure environment [27,28]. They can not only accurately predict key characteristics such as changes in lattice parameters, alterations in elastic moduli, and adjustments in the electronic density of the state distribution of materials under high pressure but also explain the internal physical origins of various property changes at the atomic scale [29,30]. Through this calculation method, we can systematically screen and design materials with specific high-pressure performance at the theoretical level, which greatly reduces the cost of and guesswork in experimental exploration, thereby accelerating the research and development processes for new high-pressure functional materials and effectively promoting the further development and breakthroughs of materials science in the high-pressure field [31,32]. Hence, this systematic structural optimization and mechanical property analysis of Al₃Ni within the 0–30 GPa range based on first principles provides a reliable theoretical foundation for applying this material under high-pressure conditions.

We aimed to investigate in detail the structural changes and elastic constants correlated with the mechanics of Al₃Ni within the 0–30 GPa pressure range using the first-principles methodology. By computing and analyzing the elastic constants of Al₃Ni under various pressures, the essential mechanical properties of Al₃Ni under different pressure circumstances, such as Young’s ratio, shear ratio, bulk modulus and Poisson’s ratio, were inferred from the patterns of the variation in Al₃Ni’s mechanical behaviors under high-pressure conditions. The outcomes of this study not only bridge the knowledge gap regarding the mechanical characteristics of Al₃Ni but also offer theoretical backing for the selection and design of the materials of intermetallic compounds in high-pressure applications.

2. Methods of Calculation

In this study, density functional theory (DFT) [33], performed using Cambridge Serial Total Energy Package (CASTEP) code [34], was used for the calculations. Specifically, the Perdew–Burke–Ernzerh (PBE) [35] generalized gradient approximation (GGA) was used to describe the exchange–correlation energy functional. This approximation demonstrates satisfactory precision in characterizing the electronic interactions within a material. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) [36] algorithm was employed to relax all lattice parameters and atomic positions, in line with the total energy and force. The tolerance for convergence was configured as follows: total energy deviation was kept smaller than 1.0 × 10⁻⁵ eV/atom, the maximum ionic force was maintained below 0.03 eV/Å, and the stress deviation was held beneath 0.05 GPa, with maximum displacement between cycles under 5 × 10⁻⁴ Å and an ultra-soft pseudopotential (USPP) [37] description of the ion–electron interactions. The configuration of valence electrons with Ni corresponded to 3d⁸4s⁸, while that of Al corresponded to 3s²3p¹. The plane-wave energy cut-off was set at 360 eV. The convergence criterion was 1.0 × 10⁻⁶ eV/atom. The Monkhorst–Pack scheme [23] was used to construct the k-point grid for sampling the Brillouin zone. A k-point grid with dimensions of 8 × 6 × 10 was used. Prior to calculating the constants of elasticity for the Al₃Ni intermetallic compounds under specific pressure P, compound unit cells were optimized at corresponding pressures via the full relaxation of the volume, shape, and inner atomic locations until the interatomic forces were <1.0 × 10⁻² eV/Å.

The atomic arrangement of Al₃Ni was an orthorhombic crystal system with a PNMA space group, and a = 6.62 Å, b = 7.39 Å, and c = 4.81 Å lattice constants [38]. Each primitive cell encompassed 12 Al atoms and 4 Ni atoms, and the cell structure is illustrated in Figure 1.

For the hexagonal intermetallic compound Al₃Ni, nine independent single-crystal elastic constants existed, specifically including $C_{11}$, $C_{12}$, $C_{13}$, $C_{22}$, $C_{23}$, $C_{33}$, $C_{44}$, $C_{55}$, and $C_{66}$. For this research, the relationship between strain and stress was utilized to determine all the elastic constants based on optimized single cells under various pressures. The elastic constants were regarded as first-order derivatives of the stress in relation to the strain tensor. The elastic tensor was obtained by applying six finite deformations to the lattice and deducing the elastic constants from the strain–stress relationship [39]. When calculating the rigid ion elastic tensor, the relaxation of the ions was also considered. The contribution of the ions was determined by inverting the Hessian matrix of ions and multiplying the result with a tensor of internal strain [40]. The final elastic constants comprised the contributions from both the deformation of rigid ions and ion relaxation.

3. Results and Discussion

3.1. Structural Properties Under High-Pressure Conditions

Upon the completion of the geometrical optimization process, the most favorable structural model of Al₃Ni was obtained, which had appropriate cell characteristics of a = 6.669 Å, b = 7.356 Å, and c = 4.850 Å. Experimentally, Bradley et al. [38] determined the crystal parameters of Al₃Ni to be a = 6.859 Å, b = 7.352 Å, and c = 4.80 Å. Theoretically, N. S. Harsha Gunda et al. [41] computed the lattice parameters of orthorhombic Al₃Ni via first-principles simulations, yielding a = 6.620 Å, b = 7.39 Å, and c = 4.81 Å. Zheng et al. [16] theoretically calculated the parameters of the Al₃Ni lattice as a = 6.698 Å, b = 7.352 Å, and c = 4.801 Å. Shi et al. [42] calculated the cell parameters to be a = 6.565 Å, b = 7.257 Å, and c = 4.75 Å.

Table 1. The lattice constants a, b, and c (in Å) of Al₃Ni at zero pressure were calculated.

Method	a	b	c
Present	6.669	7.356	4.850
Exp. [38]	6.589	7.352	4.80
Theo. [41]	6.620	7.390	4.81
Theo. [16]	6.698	7.352	4.801
Theo. [42]	6.565	7.257	4.75

presents the lattice parameters we have calculated, along with a comparison to existing literature values. Through comparison, it was evident that the cell parameters obtained from the current calculation were similar to the existing experimentally and theoretically determined values. This proximity indicates the validity and correctness of our calculation method. Subsequently, the structure of the cell was optimized under pressures ranging from 0 to 30 GPa using the appropriate parameters at each pressure level.

Figure 2 presents the values of the unit cell parameters $a/a_0$, $b/b_0$, $c/c_0$, and $V/V_0$ under different pressures. A polynomial fitting was carried out on the ratios $a/a_0$, $b/b_0$, $c/c_0$, and $V/V_0$ in relation to the pressure, generating an equation that characterized their association, which is shown in Equation (1). In Equation (1), it can be observed that the constant term is approximately equal to 1, which implies a satisfactory fitting quality.

$$\begin{array}{cc}\hfill a/a_0& =0.99993-2.8800\times {10}^{-3}P+4.86766\times {10}^{-5}{P}^2-5.53964\times {10}^{-7}{P}^3\hfill \\ \hfill b/b_0& =1.00003-2.5100\times {10}^{-3}P+2.8616\times {10}^{-5}{P}^2-1.8892\times {10}^{-7}{P}^3\hfill \\ \hfill c/c_0& =0.99998-3.4700\times {10}^{-3}P+5.88562\times {10}^{-5}{P}^2-5.87492\times {10}^{-7}{P}^3\hfill \\ \hfill V/V_0& =0.99998-8.8100\times {10}^{-3}P+1.52766\times {10}^{-5}{P}^2-1.55016\times {10}^{-7}{P}^3\hfill \end{array}$$

(1)

3.2. Elastic Constants Within a Pressurized Environment

Elastic constants are physics quantities that establish the connection of stress and strain in an anisotropic medium. They characterize the elastic attributes of a material and the capacity of a crystal lattice to withstand applied stresses. Al₃Ni exhibits a crystalline orthorhombic structure with nine independently determined constants of elasticity, namely, $C_{11}$, $C_{12}$, $C_{13}$, $C_{22}$, $C_{23}$, $C_{33}$, $C_{44}$, $C_{55}$, and $C_{66}$. The elastic constants $C_{ij}$ of Al₃Ni at varying pressures, as shown in

Table 2. The elastic constants of Al₃Ni at varying pressures $C_{ij}$ (in GPa).

Pressure	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{13}$	${\mathit{C}}_{22}$	${\mathit{C}}_{23}$	${\mathit{C}}_{33}$	${\mathit{C}}_{44}$	${\mathit{C}}_{55}$	${\mathit{C}}_{66}$
0	194.188	99.825	92.714	188.249	85.107	159.507	82.487	70.186	50.690
5	191.437	93.151	85.004	207.704	86.025	169.722	91.692	78.453	61.053
10	247.767	124.152	119.637	233.570	108.005	204.375	97.675	85.127	70.922
15	299.935	118.541	131.479	292.731	115.682	260.294	85.198	79.370	79.709
20	296.481	120.618	126.124	326.432	116.447	264.949	93.454	84.887	89.939
25	308.355	127.146	141.222	344.547	123.846	287.486	99.168	91.471	97.117
30	333.877	156.503	159.168	370.778	153.446	369.997	120.139	115.574	111.860

, undergo substantial alterations with the increase in pressure. This implies that the materials’ mechanical properties are progressively improved in different directions under high-pressure conditions. The coefficients of elasticity $C_{11}$, $C_{22}$, and $C_{33}$ in the principal axis directions depict the strain responses of crystals when tensile stress is exerted in the [100], [010], and [001] crystal axis directions, respectively. Notably, all of these constants increase as the pressure rises. This suggests that Al₃Ni has enhanced resistance to deformation in these directions, especially $C_{33}$, which shows significant stiffness enhancement by increasing from 159.507 GPa to 369.997 GPa. Additionally, the transverse elastic constants $C_{12}$ and $C_{13}$ with $C_{23}$ describe the coupling between the [100] and [010], [100] and [001], and [010] and [001] directions, as well as increases with pressure, reflecting the enhanced interaction of Al₃Ni between these directions. Meanwhile, $C_{44}$, $C_{55}$, and $C_{66}$ describe the responses to shear stresses in the [100], [010], and [001] in the (010), (100), and (100) planes, respectively, and the results show that these constants also increase significantly at high pressures, especially $C_{66}$, which increases from 50.690 GPa to 111.860 GPa, indicating that the material’s shear stiffness significantly increases. Taken together, the constants of elasticity of Al₃Ni increase with pressure, and the differences in growth rates in different directions may further enhance its elastic anisotropy under high-pressure conditions.

For the orthorhombic crystal system of Al₃Ni, the hydrostatic criterion of mechanical stability is presented in Equation (2).

$$\begin{array}{cc}\hfill & {\tilde{C}}_{ii}>0,i=1\sim 6,\left|\begin{array}{c}{\tilde{C}}_{11}+{\tilde{C}}_{22}+{\tilde{C}}_{33}+2({\tilde{C}}_{12}+{\tilde{C}}_{13}+{\tilde{C}}_{23})\end{array}\right|>0,\hfill \\ \hfill & ({\tilde{C}}_{11}+{\tilde{C}}_{22}-2{\tilde{C}}_{12})>0,({\tilde{C}}_{11}+{\tilde{C}}_{33}-2{\tilde{C}}_{13})>0,({\tilde{C}}_{22}+{\tilde{C}}_{33}-2{\tilde{C}}_{23})>0\hfill \end{array}$$

(2)

where ${\tilde{C}}_{ii}=C_{ii}-P(i=1,3,4)$, ${\tilde{C}}_{12}=C_{12}+P$, and ${\tilde{C}}_{13}=C_{13}+P$. Evidently, Al₃Ni complies with the mechanical stability criterion at pressures between 0 GPa and 30 GPa. This implies that Al₃Ni is stable at pressures from 0 to 30 GPa.

3.3. Pressure-Induced Variations in Elastic Constants and Mechanical Properties

In industrial applications, macroscopic mechanical parameters such as Young’s ratio, shear ratio, bulk modulus, and Poisson’s ratio in crystals are referentially significant for elucidating the macroscopic mechanical characteristics of crystals. The techniques commonly utilized to obtain the shear ratio, Young’s ratio, and the bulk modulus from elastic constants are the Voigt approximation and the Reuss approximation. The Reuss approximation presumes a homogeneous distribution of stress to ascertain the lower bounds of the true shear ratio and bulk modulus of polycrystals. In contrast, the Voigt approximation assumes a consistent distribution of strain across polycrystalline structures for determining the upper bounds of the actual polycrystal’s shear ratio and bulk modulus. Hill recommends taking the average of two approximations, known as the Voigt–Reuss–Hill approximation [43].

In this way, for orthorhombic systems, the crystal bulk and shear moduli are, respectively, the average values of the bulk modulus according to the Voigt and Reuss ($B_V$, $B_R$) approximations and the shear modulus ($G_V$, $G_V$), as defined in Equation (3).

$$\begin{array}{cc}\hfill B_V& ={\displaystyle \frac{C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23})}{9}}\hfill \\ \hfill G_V& ={\displaystyle \frac{C_{11}+C_{22}+C_{33}+3(C_{44}+C_{55}+C_{66})-(C_{11}+C_{13}+C_{23})}{15}}\hfill \\ \hfill B_R& ={\Delta }^{-1}\left[\begin{array}{c}{C}_{11}(C_{22}+C_{33}-2C_{23})+C_{22}(C_{33}-2C_{13})+2C_{33}{C}_{12}\hfill \\ +C_{12}(2C_{23}-C_{12})+C_{13}(2C_{12}-C_{13})+C_{23}(2C_{13}-C_{23})\hfill \end{array}\right]\hfill \\ \hfill G_R& =15\left[\begin{array}{c}{C}_{11}(C_{22}+C_{33}+C_{23})+C_{22}(C_{33}+C_{13})+C_{33}{C}_{12}\hfill \\ -C_{12}(C_{23}+C_{12})-C_{13}(C_{12}+C_{13})-C_{23}(C_{13}+C_{23})\hfill \end{array}\right]/\Delta \hfill \\ \hfill \Delta & =C_{13}(C_{12}{C}_{23}-C_{13}{C}_{22})+C_{23}(C_{12}{C}_{13}-C_{23}{C}_{11})+C_{33}(C_{11}{C}_{12}-C_{12}^2)\hfill \end{array}$$

(3)

After computing the bulk modulus $B_V$ and shear modulus $G_V$ based on the Voigt formalism, as well as the bulk modulus BR and shear modulus GR following the Reuss formalism, the bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio ($\nu$) for the different pressures of the relevant precipitation phases were then ascertained using Equation (4).

$$B={\displaystyle \frac{B_V+B_R}{2}},G={\displaystyle \frac{G_V+G_R}{2}},E={\displaystyle \frac{9BG}{3B+G}},\nu ={\displaystyle \frac{3B-2G}{6B+2G}}$$

(4)

Figure 3 shows the changes in the volume bulk shear and Young’s moduli of Al₃Ni under different pressures. We can see that these moduli all show significant increases as the pressure increases, indicating that the overall mechanical properties of Al₃Ni are greatly enhanced under high-pressure conditions. The volume elasticity modulus rises steadily with increasing pressure, suggesting that the material’s compressive resistance is gradually enhanced under high-pressure conditions. This pattern implies that the structure of Al₃Ni becomes more compact as the pressure mounts, and the material acquires enhanced rigidity in terms of bulk compression. The shear modulus G also increases with an increase in pressure, suggesting that the shear resistance of the material is ameliorated under high-pressure conditions. The resistance to shear deformation in the Al₃Ni crystal structure intensifies with increasing pressure, leading to the elevated shear rigidity of the structure. This further implies that the bonding rigidity within the material is reinforced under high-pressure conditions, enabling it to more effectively withstand shear stresses. The most significant increase is observed in the Young’s modulus, E, indicating that the overall rigidity of Al₃Ni is considerably enhanced under high-pressure conditions. A high Young’s modulus implies that the material is more resistant to deformation via tension or compression under a high-pressure milieu, demonstrating more robust resistance to deformation. This tendency may be ascribed to the reduction in atomic spacing and the reinforcement of interatomic forces under high-pressure conditions, strengthening the material’s mechanical properties.

In accordance with Pugh’s rule [44], the $B/G$ value correlates with ductility and brittleness. Specifically, a higher $B/G$ implies the greater ductility of the material. If $B/G>1.75$, ductile behavior is expected; if not, the material is brittle. The $B/G$ of Al₃Ni alloy is presented in Figure 4a. The bulk-to-shear modulus ratio, $B/G$, exhibits irregular fluctuations in response to pressure. As depicted in Figure 4a, the Al₃Ni $B/G$ ratio remains consistently above 1.75 throughout the pressure range under investigation, signifying that Al₃Ni possesses favorable toughness within the studied pressure domain. Nevertheless, certain fluctuations in $B/G$ are detected with the increase in pressure. For instance, it drops to a minimum value of 1.89 at 5 GPa and subsequently recovers gradually to a relatively higher level at elevated pressures. Such fluctuations might stem from the rigidity alterations resulting from microscopic structural adjustments under diverse pressure conditions. The trend in Poisson’s ratio $\nu$ as a function of pressure is illustrated in Figure 4b. Poisson’s ratio $\nu$ characterizes the ratio of transverse to longitudinal deformation of the material and is typically used to assess the deformation properties of materials. The criterion proposed by Frantsevich et al. [45] stipulates that when Poisson’s ratio is $\nu <0.26$, the material predominantly displays brittle behavior; under the opposite conditions, it exhibits ductility. The Poisson’s ratio of Al₃Ni also exhibits fluctuations within the investigated pressure range. The overall variation spans from 0.27 to 0.30, suggesting that the deformation characteristics of Al₃Ni are marginally modified under different pressures. Poisson’s ratio attains its lowest value at 5 GPa and then progressively rebounds to a level approximate to 0.30. Such a trend implies the microscopic structural modulation of the material as the pressure escalates from low to high.

Figure 5 presents the alterations in the Cauchy pressures $C_{23}$-$C_{44}$, $C_{13}$-$C_{55}$, and $C_{13}$-$C_{66}$ for Al₃Ni under diverse pressures. In accordance with Pettifor’s rule [46], a larger positive value for the Cauchy pressure implies a larger quantity of metallic bonds within the material and, consequently, increased toughness; vice versa, a more significant negative Cauchy pressure value indicates an increased covalent bond number in the material and, thus, increased brittleness. Figure 5 shows that all three Cauchy pressures undergo fluctuating variations with the increase in pressure. The black curve, symbolizing $C_{23}$-$C_{44}$, attains a minimum value at a low pressure of 5 GPa and then progressively increases with the increasing pressure, reaching a high level of roughly 15 GPa and maintaining a relatively stable high value under the high-pressure regime (20–30 GPa). This pattern suggests that the metallicity of the $C_{23}$-$C_{44}$ of Al₃Ni intensifies with the increase in pressure, particularly in the medium- and high-pressure regions where more prominent metallic bonding characteristics are shown. The red curve, representing $C_{13}$-$C_{55}$, exhibits an overall trend similar to that of $C_{23}$-$C_{44}$, albeit with more pronounced fluctuations. As the pressure rises, $C_{13}$-$C_{55}$ displays significant oscillations from low to high pressure, signifying that the bonding properties of the material in this direction are more susceptible to the influence of pressure, demonstrating the dynamic modulation of the bonding properties with varying pressure. The blue curve, denoting $C_{12}$-$C_{66}$, also exhibits a certain degree of volatility under different pressures and reaches a maximum within the medium- and high-pressure ranges. Analogous to the other two Cauchy pressures, the increase in $C_{12}$-$C_{66}$ reflects the increase in the metallicity of the material in that direction, and the relatively minor fluctuation under high pressures implies that the bonding properties stabilize in that direction under high-pressure conditions. In general, the three Cauchy pressures of Al₃Ni display diverse fluctuation trends with the increase in pressure, indicating that the bonding properties and ductility of the material under different pressures experience complex alterations in all directions. The overall upward tendency of the Cauchy pressures reflects the reinforcement of the metallic bonding component of Al₃Ni under high pressures, improving ductility.

Hardness is an assessment of the ability of materials to withstand elastic deformation, plastic deformation, or damage upon the application of external forces. These characteristics are contingent upon various factors, including the material’s elastic constants, plastic behavior, elongation, and toughness. In theory, a polycrystal’s hardness (H) can be approximated using Equation (5), where G represents the shear modulus, and B represents the bulk modulus.

$$H=2{\left({\displaystyle \frac{G^3}{B^2}}\right)}^{0.585}-3$$

(5)

Figure 6 exhibits the variation in the hardness of Al₃Ni at different pressure levels. We can observe that the hardness of Al₃Ni progressively increases with the increase in pressure and presents an overall increasing tendency. Notably, under conditions of 20 GPa and above, the hardness significantly increases. In the low-pressure (0–10 GPa) range, the alteration in hardness is relatively mild and exhibits slight fluctuations. This phenomenon might be due to the crystal structure of Al₃Ni having not yet endured substantial compression or remodeling under low-pressure circumstances, thereby resulting in minimal hardness changes. Nevertheless, when the pressure is gradually enhanced above 15 GPa, the hardness begins a remarkable rise, signifying that the deformation resistance of Al₃Ni is substantially increased as the pressure rises. Particularly at 30 GPa, hardness attains its maximum value. The increase in hardness is due to the densification of the structure and the increase in interatomic forces under pressure. The reduction in the atom spacing of Al₃Ni under high-pressure conditions leads to stronger atomic bonding, which consequently improves the overall material stiffness and shear resistance. Such structural adjustments render the material less prone to deformation under high-pressure conditions, thereby increasing hardness.

3.4. Elastic Anisotropy Under Pressure

In science, elastic anisotropy represents the relationship between the directionality of a material’s physical properties, and the majority of materials display elastically anisotropic behavior. The study of intermetallic elastic anisotropy is closely related to the micro-fractures in materials. Elastic anisotropy characterizes the variation in shear deformation energy in diverse directions; that is, it serves as a metric for evaluating the resistance of the material to deformation by shear stress in various directions. Specifically, $A_{\left\{100\right\}}$ represents, in the (100) shear plane, the factor of shear anisotropy between the <011> and <010> orientations, which reflects shear deformation in the <011> and <010> directions on the (100) plane. $A_{\left\{010\right\}}$ is the factor of shearing anisotropy in the <101> and <001> directions of the (010) shear plane, which indicates the resistance to shear deformation in the directions <101> and <001> on the (010) plane. $A_{\left\{001\right\}}$ is the shear isotropic factor between the <110> and <010> directions of the (001) shear plane, reflecting the shear deformation resistance in the <110> and <010> directions on the (001) plane. The values of $A_{\left\{100\right\}}$, $A_{\left\{010\right\}}$, and $A_{\left\{001\right\}}$ must be equal to 1 for isotropic materials, and minor and major deviations indicate the degree of anisotropy. The elastic anisotropy factor can then be calculated based the elastic constants, and the expressions for the three elastic anisotropy factors are presented in Equation (6) [47].

$$A_{\left\{100\right\}}={\displaystyle \frac{4C_{44}}{C_{11}+C_{33}-2C_{13}}},A_{\left\{010\right\}}={\displaystyle \frac{4C_{55}}{C_{22}+C_{33}-2C_{23}}},A_{\left\{001\right\}}={\displaystyle \frac{4C_{66}}{C_{11}+C_{22}-2C_{12}}}$$

(6)

The compressive anisotropy factor $A^B$ and the shear anisotropy factor $A^G$ are computed with Equation (7):

$$A^B={\displaystyle \frac{B_V-B_R}{B_V+B_R}}\times 100\%,A^G={\displaystyle \frac{G_V-G_R}{G_V+G_R}}\times 100\%$$

(7)

The broad-spectrum anisotropy factor $A^U$ can be determined in accordance with Equation (8):

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

(8)

The scalar logarithmic Euclidean anisotropy index ($A^L$) can be calculated using Equation (9) [48]:

$$A^L=\sqrt{{\left[ln\left({\displaystyle \frac{B_V}{B_R}}\right)\right]}^2+5{\left[ln\left({\displaystyle \frac{G_V}{G_R}}\right)\right]}^2}$$

(9)

In Equations (7)–(9), $B_V$ and $G_V$ denote the bulk modulus and shear modulus, respectively, under Voigt notation, while $B_R$ and $G_R$ represent the bulk modulus and shear modulus, respectively, under Reuss notation. For isotropic materials, the values of $A^B$, $A^G$, $A^U$, and $A^L$ must be zero. Deviations greater than zero signify the extent of anisotropy.

Figure 7 depicts trends in the anisotropy factors of Al₃Ni under different pressures. In Figure 7a, $A_{\left\{100\right\}}$, $A_{\left\{010\right\}}$, and $A_{\left\{001\right\}}$, corresponding to different crystallographic orientations, exhibit a pronounced decreasing trend as the pressure increases. With a low pressure of 0 GPa, the highest condition, $A_{\left\{100\right\}}$, implies that the anisotropy of Al₃Ni is more prominent in the <011> and <010> directions on the (100) plane, while the values of $A_{\left\{010\right\}}$ and $A_{\left\{001\right\}}$ are relatively lower. These anisotropy factors decline rapidly with an increase in pressure and approach lower and more comparable values, especially around 10 GPa. This alteration indicates that the elastic disparity among the crystallographic directions of Al₃Ni diminishes as the pressure rises, the anisotropy progressively reduces, and the mechanical properties of the material become more uniform in all directions. Figure 7b presents the trends of the anisotropy percentages $A^B$ and $A^G$ when compressing and shearing as a function of pressure. The value of $A^G$ is relatively large at low pressures, signifying significant anisotropy in the shear modulus. However, as the pressure increases, $A^G$ decreases remarkably, approaching zero after 15 GPa, suggesting that anisotropy in the shear modulus almost vanishes and the material exhibits isotropy at high pressures. In contrast, the value of $A^B$ varies to a lesser extent throughout the entire pressure range, suggesting that the anisotropy of the bulk modulus is relatively low and that the influence of pressure is not as substantial. The variation in the broad-spectrum anisotropy index $A^U$ and the logarithmic-Euclidean anisotropy index $A^L$ with pressure is shown in Figure 7c. Both indices decrease with an increase in pressure, reaching a minimum at around 15 GPa and then approaching zero. This tendency implies that the overall anisotropy of Al₃Ni diminishes rapidly under high-pressure conditions, and the properties of the material converge in all directions, which further validates the enhancement in the material’s isotropy under high-pressure conditions. In general, the anisotropy factor of Al₃Ni decreases substantially as the pressure increases, especially above 15 GPa, where the anisotropy factor approaches zero, indicating that Al₃Ni gradually assumes an isotropic nature under high-pressure conditions. With the increase in pressure, the internal structure of the material may undergo adjustments and densification, attenuating the mechanical discrepancies in different crystallographic directions, thereby exhibiting a more homogeneous mechanical response under high-pressure conditions. This property of enhanced isotropy under high-pressure conditions offers favorable mechanical support for the application of Al₃Ni in high-pressure environments.

3.5. The Speed of Sound and Thermal Conductivity Under Pressure

The pure longitudinal ($V_l$) and transverse ($V_t$) sound velocities of orthorhombic Al₃Ni in the [100], [010], and [001] directions can be computed in accordance with the obtained elastic constant $C_{ij}$, following Brugger’s methodology [49]. Specifically, the sound velocity in the [100] direction is calculated using Equation (10):

$$\left[100\right]v_1=\sqrt{{\displaystyle \frac{C_{11}}{\rho }}},\left[010\right]v_{11}=\sqrt{{\displaystyle \frac{C_{66}}{\rho }}},\left[001\right]v_2=\sqrt{{\displaystyle \frac{C_{55}}{\rho }}}$$

(10)

The sound velocity in the [010] direction is calculated using Equation (11):

$$\left[010\right]v_l=\sqrt{{\displaystyle \frac{C_{22}}{\rho }}},\left[100\right]v_{t1}=\sqrt{{\displaystyle \frac{C_{66}}{\rho }}},\left[001\right]v_{t2}=\sqrt{{\displaystyle \frac{C_{44}}{\rho }}}$$

(11)

The sound velocity in the [001] direction is determined via Equation (12):

$$\left[001\right]v_l=\sqrt{{\displaystyle \frac{C_{33}}{\rho }}},\left[100\right]v_{t1}=\sqrt{{\displaystyle \frac{C_{55}}{\rho }}},\left[010\right]v_{t2}=\sqrt{{\displaystyle \frac{C_{44}}{\rho }}}$$

(12)

where $v_{t1}$ and $v_{t2}$ stand for the first transverse mode and second transverse mode, respectively, and $\rho$ is the mass density. Because these elastic constants determine sound velocities, their anisotropy reflects that of Al₃Ni. Moreover, the longitudinal ($V_L$) and transverse ($V_T$) sound velocities of Al₃Ni can be calculated from the obtained bulk modulus B and shear modulus G using Equation (13) [50]:

$$V_L=\sqrt{{\displaystyle \frac{3B+4G}{3\rho }}},V_T=\sqrt{{\displaystyle \frac{G}{\rho }}}$$

(13)

Here, $\rho$ represents density, B stands for the bulk modulus, and G denotes the shear modulus. Furthermore, the velocity of sound average is calculated using Equation (14) [50]:

$$V_M={\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{V_L^3}}+{\displaystyle \frac{2}{V_T^3}}\right)\right]}^{-{\displaystyle \frac{1}{3}}}$$

(14)

Figure 8 provides a comprehensive illustration of the variation in the sound velocity of Al₃Ni at different pressures. We observe from the figure that the sound velocity of Al₃Ni in all crystal directions increases with the increase in pressure. Notably, the increase in longitudinal sound velocity ($v_l$) is significantly more pronounced than that of the sound velocities ($v_{t1}$) and ($v_{t2}$). In the [100], [010], and [001] crystal directions, the longitudinal sound velocity increases from 6.0–6.5 × 10³ m/s at low pressures to approximately 8.0–9.0 × 10³ m/s at high pressures. This indicates that the compressive rigidity of Al₃Ni substantially increases with increasing pressure in all crystal directions. Concurrently, the transverse sound velocities ($v_{t1}$) and ($v_{t2}$) display a more moderate increase, yet they still reflect the gradual increase in shear stiffness. In (d), the overall trend in the longitudinal sound velocity ($V_l$) aligns with those of the individual crystal directions. The transverse sound velocity ($V_t$) and the mean sound velocity ($V_m$) also incrementally increase with pressure, signifying that the material experiences an overall improvement in its mechanical properties at high pressures. In particular, the average sound velocity ($V_m$) increases from 4.5 × 10³ m/s to 6.0 × 10³ m/s, which further validates the overall enhancement in the elastic rigidity and deformation resistance of Al₃Ni under high-pressure conditions.

Upon the acquisition of the velocity of sound, the Debye temperature may be obtained using Equation (15):

$${\Theta }_D={\displaystyle \frac{h}{k_B}}\left[{\displaystyle \frac{3n}{4\pi }}{\left({\displaystyle \frac{N_A\rho }{M}}\right)}^{{\displaystyle \frac{3}{V_M}}}\right]$$

(15)

where the parameters h, $k_B$, n, $N_A$, and M denote Planck’s constant, Boltzmann’s constant, the number of atoms in the molecular formula, Avogadro’s number, and the molecular weight, respectively.

Figure 9 presents the trend in Al₃Ni’s Debye temperature ($\Theta$) under varying pressures. We can discern that the Debye temperature exhibits a substantial upward trend as the pressure increases, increasing from approximately 520 K at 0 GPa to around 680 K at 30 GPa. This indicates that pressure significantly affects the dynamic lattice properties of Al₃Ni. The Debye temperature is a crucial parameter for characterizing the phonon behavior and thermal properties within crystals and is intimately associated with the elastic characteristics of the lattice, its rigidity, and the speed of sound. An increase in the Debye temperature implies that the lattice vibration frequency of Al₃Ni increases under high-pressure conditions, and the crystal becomes more rigid and exhibits enhanced responsiveness to thermal excitation. This phenomenon can be ascribed to the densification of the crystal structure and the reinforcement of interatomic interaction forces under pressure. Such densification and force enhancement increase the speed of sound (particularly in the longitudinal direction) with an increase in pressure, thereby causing the Debye temperature to rise.

4. Conclusions

In this study, a detailed examination of the structural, elastic, and mechanical properties of Al₃Ni pressurized from 0 to 30 GPa was conducted based on the first-principles framework. The outcomes demonstrate that the lattice parameters and volume of Al₃Ni progressively diminish with an increase in pressure, showing a conspicuous densification tendency. The elastic constants experience a significant increase with pressure, thereby validating the mechanical stability of the material and signifying that its compressive rigidity and shear resistance are substantially fortified under high-pressure conditions. The Cauchy pressure analysis indicates that the metallicity of the materials escalates with pressure. Simultaneously, the variations in Poisson’s ratio and the anisotropy factor suggest that high-pressure conditions elastically attenuate the anisotropic nature of the materials, and Al₃Ni exhibits more pronounced isotropic characteristics at high pressures. The sound velocity calculations show that both the longitudinal and transverse sound velocities increase with an increase in pressure, in accordance with the enhanced material stiffness. The remarkable elevation in the Debye temperature further reflects the augmented rigidity of the lattice dynamics of the material at high pressures. The current study systematically uncovered the transformation pattern of the high-pressure Al₃Ni structure and its properties, providing a theoretical foundation for its application in extreme circumstances and an essential reference point for the optimization of the properties and the design of intermetallic compounds.