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Article

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

1
School of Mechanical and Electrical Engineering, Xinyu University, Xinyu 338004, China
2
School of Mathematical Sciences and Physics, Jinggangshan University, Ji’an 343009, China
*
Author to whom correspondence should be addressed.
Crystals 2025, 15(1), 3; https://doi.org/10.3390/cryst15010003
Submission received: 7 December 2024 / Revised: 21 December 2024 / Accepted: 21 December 2024 / Published: 24 December 2024
(This article belongs to the Special Issue Microstructure and Properties of Metals and Alloys)

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 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.
a / a 0 = 0.99993 2.8800 × 10 3 P + 4.86766 × 10 5 P 2 5.53964 × 10 7 P 3 b / b 0 = 1.00003 2.5100 × 10 3 P + 2.8616 × 10 5 P 2 1.8892 × 10 7 P 3 c / c 0 = 0.99998 3.4700 × 10 3 P + 5.88562 × 10 5 P 2 5.87492 × 10 7 P 3 V / V 0 = 0.99998 8.8100 × 10 3 P + 1.52766 × 10 5 P 2 1.55016 × 10 7 P 3

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 i j of Al3Ni at varying pressures, as shown in Table 2, 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).
C ˜ i i > 0 , i = 1 6 , C ˜ 11 + C ˜ 22 + C ˜ 33 + 2 ( C ˜ 12 + C ˜ 13 + C ˜ 23 ) > 0 , ( C ˜ 11 + C ˜ 22 2 C ˜ 12 ) > 0 , ( C ˜ 11 + C ˜ 33 2 C ˜ 13 ) > 0 , ( C ˜ 22 + C ˜ 33 2 C ˜ 23 ) > 0
where C ˜ i i = C i i P ( i = 1 , 3 , 4 ) , C ˜ 12 = C 12 + P , and 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).
B V = C 11 + C 22 + C 33 + 2 ( C 12 + C 13 + C 23 ) 9 G V = C 11 + C 22 + C 33 + 3 ( C 44 + C 55 + C 66 ) ( C 11 + C 13 + C 23 ) 15 B R = Δ 1 C 11 ( C 22 + C 33 2 C 23 ) + C 22 ( C 33 2 C 13 ) + 2 C 33 C 12 + C 12 ( 2 C 23 C 12 ) + C 13 ( 2 C 12 C 13 ) + C 23 ( 2 C 13 C 23 ) G R = 15 C 11 ( C 22 + C 33 + C 23 ) + C 22 ( C 33 + C 13 ) + C 33 C 12 C 12 ( C 23 + C 12 ) C 13 ( C 12 + C 13 ) C 23 ( C 13 + C 23 ) / Δ Δ = 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 )
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 ( ν ) for the different pressures of the relevant precipitation phases were then ascertained using Equation (4).
B = B V + B R 2 , G = G V + G R 2 , E = 9 B G 3 B + G , ν = 3 B 2 G 6 B + 2 G
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 ν as a function of pressure is illustrated in Figure 4b. Poisson’s ratio ν 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 ν < 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 G 3 B 2 0.585 3
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 { 100 } 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 { 010 } 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 { 001 } 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 { 100 } , A { 010 } , and A { 001 } 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 { 100 } = 4 C 44 C 11 + C 33 2 C 13 , A { 010 } = 4 C 55 C 22 + C 33 2 C 23 , A { 001 } = 4 C 66 C 11 + C 22 2 C 12
The compressive anisotropy factor A B and the shear anisotropy factor A G are computed with Equation (7):
A B = B V B R B V + B R × 100 % , A G = G V G R G V + G R × 100 %
The broad-spectrum anisotropy factor A U can be determined in accordance with Equation (8):
A U = B V B R + 5 G V G R 6
The scalar logarithmic Euclidean anisotropy index ( A L ) can be calculated using Equation (9) [48]:
A L = ln B V B R 2 + 5 ln G V G R 2
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 { 100 } , A { 010 } , and A { 001 } , 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 { 100 } , implies that the anisotropy of Al3Ni is more prominent in the <011> and <010> directions on the (100) plane, while the values of A { 010 } and A { 001 } 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 i j , following Brugger’s methodology [49]. Specifically, the sound velocity in the [100] direction is calculated using Equation (10):
[ 100 ] v 1 = C 11 ρ , [ 010 ] v 11 = C 66 ρ , [ 001 ] v 2 = C 55 ρ
The sound velocity in the [010] direction is calculated using Equation (11):
[ 010 ] v l = C 22 ρ , [ 100 ] v t 1 = C 66 ρ , [ 001 ] v t 2 = C 44 ρ
The sound velocity in the [001] direction is determined via Equation (12):
[ 001 ] v l = C 33 ρ , [ 100 ] v t 1 = C 55 ρ , [ 010 ] v t 2 = C 44 ρ
where v t 1 and v t 2 stand for the first transverse mode and second transverse mode, respectively, and ρ 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 = 3 B + 4 G 3 ρ , V T = G ρ
Here, ρ 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 = 1 3 1 V L 3 + 2 V T 3 1 3
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 t 1 ) and ( v t 2 ). 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 t 1 ) and ( v t 2 ) 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):
Θ D = h k B 3 n 4 π N A ρ M 3 V M
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 ( Θ ) 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.

Author Contributions

Conceptualization, C.X. and X.Z.; methodology, X.Z.; software, Z.L. and Y.Z.; validation, Y.Z. and Z.C.; formal analysis, H.Y. and H.W.; investigation, C.X.; resources, X.Z.; data curation, X.Z.; writing—original draft preparation, B.Y.; writing—review and editing, X.Z.; visualization, C.X.; supervision, H.Y. and H.W.; project administration, Y.Z.; funding acquisition, Z.L. and X.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This study was funded by the Natural Science Foundation of Jiangxi Province (grant No. 20242BAB25032), the Science and Technology Program of the Education Office of Jiangxi Province (grant No. GJJ2401520), the PhD Start-up Fund of Natural Science Foundation of Jinggangshan University (grant JZB2329), and the Guiding Projects of the Science and Technology Programme in Ji’an City, Jiangxi Province (grant Nos. 20244-029653, 20244-029657).

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

References

  1. Czerwinski, F. Thermal stability of aluminum alloys. Materials 2020, 13, 3441. [Google Scholar] [CrossRef] [PubMed]
  2. Czerwinski, F. Thermal stability of aluminum-nickel binary alloys containing the Al-Al3Ni eutectic. Metall. Mater. Trans. 2021, 52, 4342–4356. [Google Scholar] [CrossRef]
  3. Suwanpreecha, C.; Pandee, P.; Patakham, U.; Limmaneevichitr, C. New generation of eutectic Al-Ni casting alloys for elevated temperature services. Mater. Sci. Eng. A 2018, 709, 46–54. [Google Scholar] [CrossRef]
  4. Pan, L.; Zhang, S.; Yang, Y.; Gupta, N.; Yang, C.; Zhao, Y.; Hu, Z. High-temperature mechanical properties of aluminum alloy matrix composites reinforced with Zr and Ni trialumnides synthesized by in situ reaction. Metall. Mater. Trans. A 2020, 51, 214–225. [Google Scholar] [CrossRef]
  5. Michi, R.A.; Toinin, J.P.; Seidman, D.N.; Dunand, D.C. Ambient-and elevated-temperature strengthening by Al3Zr-Nanoprecipitates and Al3Ni-Microfibers in a cast Al-2.9 Ni-0.11 Zr-0.02 Si-0.005 Er (at.%) alloy. Mater. Sci. Eng. A 2019, 759, 78–89. [Google Scholar] [CrossRef]
  6. Suwanpreecha, C.; Toinin, J.P.; Michi, R.; Pandee, P.; Dunand, D.C.; Limmaneevichitr, C. Strengthening mechanisms in AlNiSc alloys containing Al3Ni microfibers and Al3Sc nanoprecipitates. Acta Mater. 2019, 164, 334–346. [Google Scholar] [CrossRef]
  7. Pandey, P.; Makineni, S.K.; Gault, B.; Chattopadhyay, K. On the origin of a remarkable increase in the strength and stability of an Al rich Al-Ni eutectic alloy by Zr addition. Acta Mater. 2019, 170, 205–217. [Google Scholar] [CrossRef]
  8. Michi, R.A.; Plotkowski, A.; Shyam, A.; Dehoff, R.R.; Babu, S.S. Towards high-temperature applications of aluminum alloys enabled by additive manufacturing. Int. Mater. Rev. 2022, 67, 298–345. [Google Scholar] [CrossRef]
  9. Cantor, B.; Chadwick, G. The growth crystallography of unidirectionally solidified Al-Al3Ni and Al-Al2Cu eutectics. J. Cryst. Growth 1974, 23, 12–20. [Google Scholar] [CrossRef]
  10. Zhao, Z.; Zhang, H.; Yuan, H.; Wang, S.; Lin, Y.; Zeng, Q.; Xu, G.; Liu, Z.; Solanki, G.; Patel, K.; et al. Pressure induced metallization with absence of structural transition in layered molybdenum diselenide. Nat. Commun. 2015, 6, 7312. [Google Scholar] [CrossRef] [PubMed]
  11. Xiao, F.; Lei, W.; Wang, W.; Autieri, C.; Zheng, X.; Ming, X.; Luo, J. Pressure-induced structural transition, metallization, and topological superconductivity in PdSSe. Phys. Rev. B 2022, 105, 115110. [Google Scholar] [CrossRef]
  12. Ben sadallah, H.; Boulechfar, R.; Meradji, H.; Ghemid, S.; Khenioui, Y.; Lebga, N.; Khenata, R.; Bin-Omran, S.; Haq, B.U.; Kim, S.H. Phase stability and physical behaviour of Fe3Pd, FePd and FePd3 binary intermetallic compounds. Phys. B Condens. Matter 2024, 686, 416074. [Google Scholar]
  13. Yin, D.; Sahu, B.P.; Tsurkan, P.; Popov, D.; Dongare, A.M.; Velisavljevic, N.; Misra, A. High-pressure phase transitions in a laser directed energy deposited Fe-33Cu Alloy. Acta Mater. 2024, 268, 119797. [Google Scholar] [CrossRef]
  14. Ciftci, Y.O.; Çatıkkaş, B. Pressure effects on the structural, electronic, elastic, optical, and vibrational properties of YMg intermetallic compounds: A first-principles study. Phys. Scr. 2024, 99, 065981. [Google Scholar] [CrossRef]
  15. Wang, P.; Wang, Y.; Qu, J.; Zhu, Q.; Yang, W.; Zhu, J.; Wang, L.; Zhang, W.; He, D.; Zhao, Y. Pressure-induced structural and electronic transitions, metallization, and enhanced visible-light responsiveness in layered rhenium disulphide. Phys. Rev. B 2018, 97, 235202. [Google Scholar] [CrossRef]
  16. Zheng, B.; Zhao, L.; Hu, X.; Dong, S.; Li, H. First-principles studies of Mg17Al12, Mg2Al3, Mg2Sn, MgZn2, Mg2Ni and Al3Ni phases. Phys. B Condens. Matter 2019, 560, 255–260. [Google Scholar] [CrossRef]
  17. Li, Z.; Graziosi, P.; Neophytou, N. Efficient first-principles electronic transport approach to complex band structure materials: The case of n-type Mg3Sb2. Npj Comput. Mater. 2024, 10, 8. [Google Scholar] [CrossRef]
  18. Song, J.; Jiang, M.; Li, H.; Wan, C.; Chu, X.; Zhang, Q.; Chen, Y.; Wu, X.; Zhang, X.; Liu, J. First-Principles Computational Insights into Silicon-Based Anode Materials: Recent Progress And Perspectives. Surf. Rev. Lett. 2024, 31, 2430006. [Google Scholar] [CrossRef]
  19. Peng, J.; Zhao, Y.; Wang, X.; Zeng, X.; Wang, J.; Hou, S. Metal-organic frameworks: Advances in first-principles computational studies on catalysis, adsorption, and energy storage. Mater. Today Commun. 2024, 40, 109780. [Google Scholar] [CrossRef]
  20. Li, K.; Choudhary, K.; DeCost, B.; Greenwood, M.; Hattrick-Simpers, J. Efficient first principles based modeling via machine learning: From simple representations to high entropy materials. J. Mater. Chem. A 2024, 12, 12412–12422. [Google Scholar] [CrossRef]
  21. Pimachev, A.K.; Neogi, S. First-principles prediction of electronic transport in fabricated semiconductor heterostructures via physics-aware machine learning. NPJ Comput. Mater. 2021, 7, 93. [Google Scholar] [CrossRef]
  22. Liu, B.; Zhao, J.; Liu, Y.; Xi, J.; Li, Q.; Xiang, H.; Zhou, Y. Application of high-throughput first-principles calculations in ceramic innovation. J. Mater. Sci. Technol. 2021, 88, 143–157. [Google Scholar] [CrossRef]
  23. Liu, L.; Wang, M.; Hu, L.; Wen, Y.; Jiang, Y. Structural, elastic, and electronic properties of MgB2 C2 under pressure from first-principles calculations. Int. J. Quantum Chem. 2021, 121, e26442. [Google Scholar] [CrossRef]
  24. Wan, H.; Yao, W.; Zeng, D.; Zhou, J.; Ruan, W.; Liu, L.; Wen, Y. Structural, Elastic, Electronic, and Magnetic Properties of a New Full-Heusler Alloy Mn2MgGe: First-Principles Calculations. J. Supercond. Nov. Magn. 2019, 32, 3001–3008. [Google Scholar] [CrossRef]
  25. Wen, Y.; Zeng, X.; Ye, Y.; Gou, Q.; Liu, B.; Lai, Z.; Jiang, D.; Sun, X.; Wu, M. Theoretical Study on the Structural, Elastic, Electronic and Thermodynamic Properties of Long-Period Superstructures h- and r-Al2Ti under High Pressure. Materials 2022, 15, 4236. [Google Scholar] [CrossRef]
  26. Ciftci, Y.O.; Coban, C.; Evecen, M.; Durukan, İ.K. Pressure effects on structural, electronic and anisotopic elastic properties of Si doped RuGe compound with different concentrations by first-principles calculations. Mater. Chem. Phys. 2022, 291, 126695. [Google Scholar] [CrossRef]
  27. Chen, C.; Liu, L.; Wen, Y.; Jiang, Y.; Chen, L. Elastic properties of orthorhombic YBa2Cu3O7 under pressure. Crystals 2019, 9, 497. [Google Scholar] [CrossRef]
  28. Wen, Y.; Yu, X.; Zeng, X.; Ye, Y.; Wu, D.; Gou, Q. Ab initio calculations of the mechanical and acoustic properties of Ti 2-based Heusler alloys under pressures. Eur. Phys. J. B 2018, 91, 1–10. [Google Scholar] [CrossRef]
  29. Pebdani, Z.H.; Janisch, R.; Pyczak, F. Effect of V-VIB group ternary elements on the properties of Ti2AlM-type O-phases: A first-principles study. Comput. Condens. Matter 2024, 40, e00945. [Google Scholar] [CrossRef]
  30. Fan, S.; Hou, H.; Zhang, S. Understanding of the crystal structure, mechanical, vibrational properties of α-MgSO4 under pressure. Vacuum 2021, 190, 110280. [Google Scholar] [CrossRef]
  31. Liu, Z.J.; Sun, X.W.; Zhang, C.R.; Zhang, S.J.; Zhang, Z.R.; Jin, N.Z. First-principles calculations of high-pressure physical properties anisotropy for magnesite. Sci. Rep. 2022, 12, 3691. [Google Scholar] [CrossRef]
  32. Evecen, M.; Ciftci, Y. Theoretical investigation of the electronic structure, elastic, dynamic properties of intermetallic compound NiBe under pressure. Eur. Phys. J. B 2021, 94, 19. [Google Scholar] [CrossRef]
  33. Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. 1964, 136, B864. [Google Scholar] [CrossRef]
  34. Clark, S.J.; Segall, M.D.; Pickard, C.J.; Hasnip, P.J.; Probert, M.I.; Refson, K.; Payne, M.C. First principles methods using CASTEP. Z. Krist.-Cryst. Mater. 2005, 220, 567–570. [Google Scholar] [CrossRef]
  35. Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77, 3865. [Google Scholar] [CrossRef] [PubMed]
  36. Mull, T.; Baumeister, B.; Menges, M.; Freund, H.J.; Weide, D.; Fischer, C.; Andresen, P. Bimodal velocity distributions after ultraviolet-laser-induced desorption of NO from oxide surfaces. Experiments and results of model calculations. J. Chem. Phys. 1992, 96, 7108–7116. [Google Scholar] [CrossRef]
  37. Hamann, D. Optimized norm-conserving Vanderbilt pseudopotentials. Phys. Rev. B—Condens. Matter Mater. Phys. 2013, 88, 085117. [Google Scholar] [CrossRef]
  38. Bradley, A.; Taylor, A. XCIX. The crystal structures of Ni2Al3 and NiAl3. Lond. Edinb. Dublin Philos. Mag. J. Sci. 1937, 23, 1049–1067. [Google Scholar] [CrossRef]
  39. Le Page, Y.; Saxe, P. Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress. Phys. Rev. B 2002, 65, 104104. [Google Scholar] [CrossRef]
  40. Wu, X.; Vanderbilt, D.; Hamann, D. Systematic treatment of displacements, strains, and electric fields in density-functional perturbation theory. Phys. Rev. B—Condens. Matter Mater. Phys. 2005, 72, 035105. [Google Scholar] [CrossRef]
  41. Gunda, N.H.; Michi, R.A.; Chisholm, M.F.; Shyam, A.; Shin, D. First-principles study of Al/Al3Ni interfaces. Comput. Mater. Sci. 2023, 217, 111896. [Google Scholar] [CrossRef]
  42. Shi, D.; Wen, B.; Melnik, R.; Yao, S.; Li, T. First-principles studies of Al–Ni intermetallic compounds. J. Solid State Chem. 2009, 182, 2664–2669. [Google Scholar] [CrossRef]
  43. Cheng, H.C.; Yu, C.F.; Chen, W.H. First-principles density functional calculation of mechanical, thermodynamic and electronic properties of CuIn and Cu2In crystals. J. Alloys Compd. 2013, 546, 286–295. [Google Scholar] [CrossRef]
  44. Pugh, S. XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. Lond. Edinb. Dublin Philos. Mag. J. Sci. 1954, 45, 823–843. [Google Scholar] [CrossRef]
  45. Frantsevich, I.N. Elastic Constants and Elastic Moduli of Metals and Insulators; Reference book; Naukova Dumka: Kyiv, Ukraine, 1982. [Google Scholar]
  46. Pettifor, D. Theoretical predictions of structure and related properties of intermetallics. Mater. Sci. Technol. 1992, 8, 345–349. [Google Scholar] [CrossRef]
  47. Mao, P.; Yu, B.; Liu, Z.; Wang, F.; Ju, Y. First-principles investigation on mechanical, electronic, and thermodynamic properties of Mg2Sr under high pressure. J. Appl. Phys. 2015, 117, 115903. [Google Scholar] [CrossRef]
  48. Kube, C.M. Elastic anisotropy of crystals. AIP Adv. 2016, 6, 095209. [Google Scholar] [CrossRef]
  49. Brugger, K. Determination of third-order elastic coefficients in crystals. J. Appl. Phys. 1965, 36, 768–773. [Google Scholar] [CrossRef]
  50. Zeng, X.; Peng, R.; Yu, Y.; Hu, Z.; Wen, Y.; Song, L. First-Principles Calculations on Structural Property and Anisotropic Elasticity of γ 1-Ti4Nb3Al9 under Pressure. Materials 2018, 11, 2025. [Google Scholar] [CrossRef]
Figure 1. Cell structure model of Al3Ni.
Figure 1. Cell structure model of Al3Ni.
Crystals 15 00003 g001
Figure 2. The structural parameter ratios of Al3Ni as a function of pressure.
Figure 2. The structural parameter ratios of Al3Ni as a function of pressure.
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Figure 3. Bulk modulus (B), shear modulus (G), and Young’s modulus (E) of Al3Ni at different pressures.
Figure 3. Bulk modulus (B), shear modulus (G), and Young’s modulus (E) of Al3Ni at different pressures.
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Figure 4. (a) The curves of the modulus ratio G / B and (b) Poisson’s ratio ν versus pressure for Al3Ni.
Figure 4. (a) The curves of the modulus ratio G / B and (b) Poisson’s ratio ν versus pressure for Al3Ni.
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Figure 5. The Cauchy pressure of Al3Ni alloy under different pressures.
Figure 5. The Cauchy pressure of Al3Ni alloy under different pressures.
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Figure 6. Hardness of Al3Ni at different pressures.
Figure 6. Hardness of Al3Ni at different pressures.
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Figure 7. Anisotropy factors A { 100 } , A { 010 } , and A { 001 } (a); compressive anisotropy percentage A B and shear anisotropy percentage A G (b); and broad-spectrum anisotropy indices A U and logarithmic Euclidean anisotropy indices A L (c) for Al3Ni under different pressures.
Figure 7. Anisotropy factors A { 100 } , A { 010 } , and A { 001 } (a); compressive anisotropy percentage A B and shear anisotropy percentage A G (b); and broad-spectrum anisotropy indices A U and logarithmic Euclidean anisotropy indices A L (c) for Al3Ni under different pressures.
Crystals 15 00003 g007
Figure 8. (a) The sound longitudinal and transverse velocities v l , v t 1 , and v t 2 in the [100] direction; (b) the sound longitudinal and transverse velocities v l , v t 1 , and v t 2 in the [010] direction; and (c) the sound longitudinal and transverse velocities v l , v t 1 , and v t 2 in the [001] direction for Al3Ni at different pressures. (d) The longitudinal sound velocity ( V L ), transverse sound velocity ( V T ), and mean sound velocity V M of the Al3Ni crystal at different pressures.
Figure 8. (a) The sound longitudinal and transverse velocities v l , v t 1 , and v t 2 in the [100] direction; (b) the sound longitudinal and transverse velocities v l , v t 1 , and v t 2 in the [010] direction; and (c) the sound longitudinal and transverse velocities v l , v t 1 , and v t 2 in the [001] direction for Al3Ni at different pressures. (d) The longitudinal sound velocity ( V L ), transverse sound velocity ( V T ), and mean sound velocity V M of the Al3Ni crystal at different pressures.
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Figure 9. Debye temperature of Al3Ni at different pressures.
Figure 9. Debye temperature of Al3Ni at different pressures.
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Table 1. The lattice constants a, b, and c (in Å) of Al3Ni at zero pressure were calculated.
Table 1. The lattice constants a, b, and c (in Å) of Al3Ni at zero pressure were calculated.
Methodabc
Present6.6697.3564.850
Exp. [38]6.5897.3524.80
Theo. [41]6.6207.3904.81
Theo. [16]6.6987.3524.801
Theo. [42]6.5657.2574.75
Table 2. The elastic constants of Al3Ni at varying pressures C i j (in GPa).
Table 2. The elastic constants of Al3Ni at varying pressures C i j (in GPa).
Pressure C 11 C 12 C 13 C 22 C 23 C 33 C 44 C 55 C 66
0194.18899.82592.714188.24985.107159.50782.48770.18650.690
5191.43793.15185.004207.70486.025169.72291.69278.45361.053
10247.767124.152119.637233.570108.005204.37597.67585.12770.922
15299.935118.541131.479292.731115.682260.29485.19879.37079.709
20296.481120.618126.124326.432116.447264.94993.45484.88789.939
25308.355127.146141.222344.547123.846287.48699.16891.47197.117
30333.877156.503159.168370.778153.446369.997120.139115.574111.860
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Xiao, C.; Yang, B.; Lai, Z.; Chen, Z.; Yang, H.; Wang, H.; Zhou, Y.; Zeng, X. First-Principles-Based Structural and Mechanical Properties of Al3Ni Under High Pressure. Crystals 2025, 15, 3. https://doi.org/10.3390/cryst15010003

AMA Style

Xiao C, Yang B, Lai Z, Chen Z, Yang H, Wang H, Zhou Y, Zeng X. First-Principles-Based Structural and Mechanical Properties of Al3Ni Under High Pressure. Crystals. 2025; 15(1):3. https://doi.org/10.3390/cryst15010003

Chicago/Turabian Style

Xiao, Chuncai, Baiyuan Yang, Zhangli Lai, Zhiquan Chen, Huaiyang Yang, Hui Wang, Yunzhi Zhou, and Xianshi Zeng. 2025. "First-Principles-Based Structural and Mechanical Properties of Al3Ni Under High Pressure" Crystals 15, no. 1: 3. https://doi.org/10.3390/cryst15010003

APA Style

Xiao, C., Yang, B., Lai, Z., Chen, Z., Yang, H., Wang, H., Zhou, Y., & Zeng, X. (2025). First-Principles-Based Structural and Mechanical Properties of Al3Ni Under High Pressure. Crystals, 15(1), 3. https://doi.org/10.3390/cryst15010003

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