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Article

Theoretical Prediction of Structural, Mechanical, and Thermophysical Properties of the Precipitates in 2xxx Series Aluminum Alloy

by
Xuewei Fang
1,
Yefei Li
1,
Qiaoling Zheng
1,
Jianye Guo
1,
Yanmei Yang
1,
Weiyun Ding
2,*,
Chunhui Ma
2,
Ke He
2,
Ningning Su
2,
Jingyi Jiang
2,
Xiaoxue Chen
2 and
Haoran Wang
2
1
National Innovation Institute of Additive Manufacturing, Xi’an 710065, China
2
China National Heavy Duty Truck Group Co., Ltd., Jinan 250000, China
*
Author to whom correspondence should be addressed.
Metals 2022, 12(12), 2178; https://doi.org/10.3390/met12122178
Submission received: 20 November 2022 / Revised: 14 December 2022 / Accepted: 14 December 2022 / Published: 17 December 2022
(This article belongs to the Special Issue Additive Manufacturing in Alloy Design and Development)
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Versions Notes

Abstract

:
We presented a theoretical study for the structural, mechanical, and thermophysical properties of the precipitates in 2xxx series aluminum alloy by applying the widely used density functional theory of Perdew-Burke-Ernzerhof (PBE). The results indicated that the most thermodynamically stable structure refers to the Al3Zr phase in regardless of its different polymorphs, while the formation enthalpy of Al5Cu2Mg8Si6 is only -0.02 eV (close to zero) indicating its metastable nature. The universal anisotropy index of AU follows the trend of: Al2Cu > Al2CuMg ≈ Al3Zr_D022 ≈ Al20Cu2Mn3 > Al3Fe ≈ Al6Mn > Al3Zr_D023 ≈ Al3Zr_L12 > Al7Cu2Fe > Al3Fe2Si. The thermal expansion coefficients (TECs) were calculated based on Quasi harmonic approximation (QHA); Al2CuMg shows the highest linear thermal expansion coefficient (LTEC), followed by Al3Fe, Al2Cu, Al3Zr_L12 and others, while Al3Zr_D022 is the lowest one. The calculated data of three Al3Zr polymorphs follow the order of L12 > D023 > D022, all of them show much lower LTEC than Al substance. For multi-phase aluminum alloys, when the expansion coefficient of the precipitates is quite different from the matrix, it may cause a relatively large internal stress, or even produce cracks under actual service conditions. Therefore, it is necessary to discuss the heat misfit degree during the material design. The discrepancy between a-Al and Al2CuMg is the smallest, which may decrease the heat misfit degree between them and improve the thermal shock resistant behaviors.

1. Introduction

2xxx series aluminum alloys are widely used in spacecraft parts, engine pistons, aircraft structures, missile components, propellers, aircraft skeletons, etc. The 2xxx series aluminum alloys mainly include Al-Cu, Al-Cu-Mg, Al-Cu-Mg-Mn-Zr alloys, etc., which could be strengthened by heat treatment, and thereby the hardness and strength will be further improved after solutionizing treatment and the following artificial aging treatment [1,2,3]. The 2xxx series aluminum alloy, in particular, keeps good performance at high temperatures and is commonly used in weldable forgings and structural parts.
During artificial aging treatment of 2xxx series aluminum alloys, etc., a succession of precipitates is developed from the supersaturated solid solution of α-Al. θ-Al2Cu, is obtained from the supersaturated solid solution, and then goes through the GP zone (Guinier-Preston zone), then the metastable θ’’, θ’ and finally the θ-Al2Cu phase; S-Al2CuMg undergoes a similar process to θ-Al2Cu [2]. Zr and Mn are also frequently used to improve the strength of the alloys by forming intermetallics such as Al3Zr, Al20Cu2Mn3, and Al6Mn. Particularly, Si and Fe elements are impurities in aluminum alloys, which exist as Al3Fe, Al3Fe2Si, Al5Cu2Mg8Si6, and Al7Cu2Fe.
For θ-Al2Cu and S-Al2CuMg phases, a lot of experimental work has been conducted to reveal their microstructures, mechanical and physical properties. Kairy et al. [4] claimed the hardness of the 2xxx series alloy can be increased with Sc and Zr additions. The fine spherical Al3(ScZr) effectively retards the recrystallization process, which benefits the high-temperature mechanical property of aluminum alloys. Actually, the ground state of Al3Zr has a tetragonal D023 structure, while the L12 structure can be stabilized by Cu or Mn [5]. Al20Cu2Mn3 is a kind of common dispersoid in 2xxx series aluminum alloys, which is formed during the homogenization [6].
The high temperature mechanical strength of aluminum alloys can be improved by fine and uniformly distributed Al20Cu2Mn3 particles [7]. Al20Cu2Mn3 has an orthorhombic structure with the space group of Bbmm [8]. Shen et al. [9] reported the atomic arrangement of the Al20Cu2Mn3 structure; and the optimized lattice parameters were estimated as a = 23.98 Å, b = 12.54 Å, and c = 7.66 Å. Huang et al. [10] reported Al6Mn is less steady than Al3Fe or Al3Fe2Si from the energetic point of view. Zhu et al. [11] found a high-strength die-cast aluminum alloy by optimizing the synergistic strengthening of Q-Al5Cu2Mg8Si6 and θ-Al2Cu phases, the yield strength of 225 MPa and elongation of 4.3% were obtained. For the Al7Cu2Fe phase, Tian et al. [12] reported the bulk, shear and young’s moduli are 107.8, 74.5, and 181.7 GPa, respectively.
Although many investigations associated with precipitates have been performed in previous experimental and theoretical work, most of them are mainly focused on the structures and mechanical strength. To date, little information is available about the thermodynamical stability, anisotropic mechanical property, temperature dependence thermal expansion, and thermal capacity due to the difficulties of experiments. Based on the importance of precipitates in 2xxx series aluminum alloys, the properties mentioned above should be investigated and discussed to reveal the intrinsic behavior of the 2xxx series aluminum alloys. In this paper, we will perform a comprehensive study on these properties of the precipitates using first-principles calculations based on the density functional theory (DFT).

2. Model and Computational Method

The crystal structures of precipitates considered in current paper are shown in Figure 1, which include Al2Cu, Al2CuMg, Al3Fe, Al3Fe2Si, Al5Cu2Mg8Si6, Al7Cu2Fe, Al6Mn, Al20Cu2Mn3, and Al3Zr. All these phases are built by their conventional crystal state, and the crystal structures and lattice parameters are shown in Table 1. This work was carried out by using the Cambridge Sequential Total Energy Package (CASTEP) code based on density functional theory (DFT) [13,14]. The criteria for convergence were 10−8 eV/atom for total energy, and 10−4 eV/Å for Hellmann-Feynman forces, respectively. The Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm is applied to optimize the crystal structure including lattice parameters and atomic fractional coordinates. The ultrasoft pseudo–potentials (USPPs) were used to represent the interactions between the ionic core and valence electrons. A plane–wave basis set with Ecut of 500 eV was used. The exchange and correlation relationship of Perdew-Burke-Ernzerhof (PBE) was applied for calculations [15]. For k-space, summation the 0.3 Å−1 for all phases with Monkhorst–Pack scheme in the first irreducible Brillouin zone [16] has been used. The valence electrons were considered as 3s23p1, 3d104s1, 2p63s2, 3s23p2, 3d54s2, and 4s24p64d25s2 for Al, Cu, Mg, Si, Mn, and Zr, respectively. In order to study the mechanical response of the crystals to external stress, the elastic properties are determined using the stress–strain relationship by deforming the unit cell using lagrangian strain modes based on Hooker’s law [17].
In order to calculate the thermal expansion coefficient (TEC), the Helmholtz free energy was given by F (V, T) = Egs(V) + Fvib (V, T) + Fele (V, T) [18]. The Egs refers to total energy with ground-state obtained directly by DFT calculation at 0 K. The vibrational free energy (Fvib) was calculated by means of the quasi-harmonic approximation (QHA) based on the empirical Debye model [19,20]. Fele is the electron thermal excitations at finite temperature and can be calculated by Mermin statistics Fele = Eele − TSele. Using isothermal curves (F (V, T) − V), the equilibrium volumes (V) at different temperatures can be obtained from the Birch-Murnaghan equation of state (EOS) [21]. Finally, the volumetric TEC λ (T) can be determined by λ = 1 V 0 ( d V d T ) .
E ( V ) = B o V o B 0 ′ [ 1 B 0 ′ − 1 ( V V o ) ( 1 − B 0 ′ ) + V V o + B 0 ′ 1 − B 0 ′ ] + E o
where E(V) is the total energy, B0, V0, and E0 refer to the equilibrium bulk modulus, volume and energy, and B0′ is the pressure derivation.
Table
Table 1. Theoretically calculated crystal structure, equilibrium lattice parameters (V in Å3, a, b, c in Å), formation enthalpies (eV/atom) and cohesive energy (eV/atom) of second phases of 2xxx series aluminum alloys.
Table 1. Theoretically calculated crystal structure, equilibrium lattice parameters (V in Å3, a, b, c in Å), formation enthalpies (eV/atom) and cohesive energy (eV/atom) of second phases of 2xxx series aluminum alloys.
PhaseCrystal StructureSpace GroupVCrystal ParametersΔHEcoh DF
abc
Al2CuTetragonalI4/MCM179.8 (179.8 a)6.04 (6.039 a, 6.063 b, 5.99 c)6.04 (6.039 a, 6.063 b, 5.99 c)4.928 (4.93 a, 4.872 b, 4.81 c)−0.151 (−0.164 a, −0.203 b, −0.203 c)−4.0 (−3.89 a, −3.99 b, −3.99 c)0.19 (0.19 a)
Al2CuMgOrthorhombicCMCM268.2 (268.1 a, 270.8 d)4.027 (4.026 a, 4.01 b, 3.89 c, 4.05 d, 4.01 e)9.319
(9.326 a, 9.25 b, 9.20 c, 9.28 d, 9.27 e)		7.147
(7.142 a, 7.15 b, 7.16 c, 7.21 d, 7.12 e)		−0.17 (−0.186 a, −0.25 b, −0.25 c)−3.5 (−3.35 a, −3.46 b, −3.46 c)0.16 (0.13 a)
Al3FeTriclinicP63/MMC108.2 (108.2 f)5.36 (5.357 f)5.36 (5.357 f)4.354 (4.354 f)−0.16 (−0.154 f)−5.30.33
Al3Fe2SiCubicFD-3M1224.9 (1225 g)10.70110.70110.701−0.423 (−0.402 f)−6.40.33
Al5Cu2Mg8Si6MonoclinicPM385.810.479 (10.423 h)4.016 (4.033 h)10.529−0.02 (−0.12 h)−3.50.13
Al7Cu2FeTetragonalP4/MNC590.56.324 (6.336 i, 6.338 j)6.324 (6.336 i, 6.338 j)14.763 (14.87 i, 14.83 j)−0.266 (−0.298 j)−4.60.14
Al6MnOrthorhombicCMCM431.46.468 (6.499 k)7.544 (7.555 k)8.841 (8.872 k)−0.195−4.70.14
Al3Zr_D023TetragonalI4/MMM280.14.020 (4.018 a, 4.015 b)4.020 (4.018 a, 4.015 b)17.332 (17.348 a, 17.454 b)−0.489 (−0.517 a, −0.459 b)−5.1 (5.14 a, 4.57 b)0.15 (0.15 a)
Al3Zr_D022TetragonalI4/MMM141.83.9633.9639.032−0.464 (−0.463 l)−5.10.2
Al3Zr_L12CubicPM-3M69.44.109 (4.111 m, 4.05 n, 4.09 o)4.109 (4.111 m)4.109 (4.111 m)−0.461 (−0.463 l, −0.487 m, −0.47 p)−5.10.18
Al20Cu2Mn3OrthorhombicBBMM2337.324.089 (23.98 q)12.601 (12.54 q)7.700 (7.66 q)−0.181 (−0.156 q)−4.640.19
a, c, d, f, h, j, m, p, and q are the theoretical data from Refs. [22,23,24,25,26,27,28,29], and [9], respectively. b, e, i, l, n, and o are the experimental data from Refs. [30,31,32,33,34,35], respectively. g The Inorganic Crystal Structure Database (ICSD ID: 422341). k Exp. data from Refs. [36,37].

3. Results and Discussion

3.1. Mechanical Properties

Table 1 listed the equilibrium lattice parameters, formation enthalpies and cohesive energy of precipitates in 2xxx series aluminum alloys. The optimized lattice parameters are in good agreement with the references. The thermodynamic stability usually requires the formation enthalpy and cohesive energy to be negative. This work provides the calculated formation enthalpies of −0.151, −0.17, −0.16, −0.463, −0.02, −0.266, −0.195, −0.489, −0.464, −0.461, and −0.181 eV/atom for Al2Cu, Al2CuMg, Al3Fe, Al3Fe2Si, Al5Cu2Mg8Si6, Al7Cu2Fe, Al6Mn, Al3Zr_D023, Al3Zr_D022, Al3Zr_L12, and Al20Cu2Mn3, respectively. Therefore, the most thermodynamically stable structure refers to Al3Zr phase regardless of its polymorphs, while Al5Cu2Mg8Si6 behaves the value of −0.02 eV (close to zero) indicating its metastable nature, which may decompose after high-temperature artificial aging.
The second order elastic constants were calculated and listed in Table 2, and the corresponding polycrystalline bulk and shear moduli are given in Table 3 calculated by Reuss and Voigt methods [38], respectively; the real polycrystalline values are estimated by Hill’s average [39]. For Al2Cu and Al2CuMg, C11, C22, and C33 are over 100 GPa, which are much higher than other elastic constants, indicating their strong anti-compressibility along principal axes. Especially, Al2Cu shows the same stiffness along the a- and b- axes, both of which are weaker than that along the c- axis, while Al2CuMg is stiffer along the b- axis compared with the a- or c- axis. The tabulated elastic modulus (C44) of both Al2Cu and Al2CuMg is significantly small. This corresponds to the shear mode and indicates their relatively small shear modulus (~43 GPa as shown in Table 3). Al3Fe, Al3Fe2Si, and Al7Cu2Fe show much higher elastic constants of C11, C22, and C33 (over 200 GPa), implying these phases are very hard to be deformed. Al5Cu2Mg8Si6 is an exception; all elastic constants are much smaller than others, implying its soft nature. Actually, Al5Cu2Mg8Si6 is a mechanically unstable phase since it disobeys the well-known Born-Huang stability criterion [40,41]. For Mn-based phases Al6Mn and Al20Cu2Mn3, Al6Mn shows better anti-compressibility along the principal axis than Al20Cu2Mn3, while the shear or complex mode corresponding to C44 or C12 and C23 of Al6Mn are much smaller than that of Al20Cu2Mn3. Al3Zr usually has three kinds of polymorphs; the calculated results of both D022 and L12 phases are very similar to other references as listed in Table 1, while for the D023 phase, the reference calculation results are dispersed from 201 to 284 GPa by taking C11 as an example [42]. To our knowledge, most precipitates of the 2xxx series aluminum alloys have not any experimental values, our study may provide valuable data especially for Al3Fe2Si, Al6Mn, etc.
Based on the values of the Voigt-Reuss-Hill approximation shown in Table 2, the Young’s modulus and Poisson’s ratio can be calculated by E = 9BG/(3B + G) and σ = (3B − 2G)/(6B + 2G), respectively. Generally, the mechanical moduli of precipitates of 2xxx series aluminum alloys are similar to previously reported values. As shown in Figure 2, the variation trend of bulk, shear, and Young’s muduli shares a similar tendency, in which Al3Fe2Si behaves the highest values. Bulk modulus, for instance, which is derived from 63.5 to 155.9 GPa, has the trend of: Al3Fe2Si > Al3Fe > Al7Cu2Fe > Al6Mn ≈ Al3Zr ≈ Al20Cu2Mn3 > Al2Cu > Al2CuMg > Al5Cu2Mg8Si6. All iron-based compounds of Al3Fe2Si, Al3Fe, and Al7Cu2Fe have a large bulk modulus, which is more or less smaller than BCC-iron (~174 GPa [47]). Young’s modulus usually reflects the plastic deformation ability of bulk materials; this work proves the ultrahigh anti-deformation nature of Al7Cu2Fe. For three polymorphs of Al3Zr, the Young’s modulus of D023 and D022 phases are similar (~196 GPa), which are much higher than the L12 phase (~156 GPa), indicating their outstanding mechanical behavior. Typically, for covalent and ionic materials, the value of Poisson’s ratio is 0.1 and 0.25, respectively, whereas for metallic materials, the value is 0.3 [48]. Most of second phases have the Poisson’s ratio of ~0.3, such as Al2Cu, Al3Fe, Al5Cu2Mg8Si6, and Al20Cu2Mn3, indicating their advanced metallic nature. For Al2CuMg, Al3Fe2Si, Al6Mn, and Al3Zr, they are dominated by a mainly covalent bond. The brittle index of B/G is applied to analyze the ductility of phases. The higher the value of B/G, the better the ductility of the materials. The present work indicates that Al2Cu, Al3Fe, Al5Cu2Mg8Si6, and Al20Cu2Mn3 are ductile phases, which are in agreement with their advanced metallic nature. For the three Al3Zr polymorphs, the L12 phase shows the best ductility. Furthermore, in the present work, a semi-empirical model proposed by Chen et al. [49] was employed to evaluate the Vicker’s hardness. As shown in Table 3, the hardness of Al3Zr_ D022 and D023 phases is very high (~18GPa), which is comparable to the value of Al7Cu2Fe; whereas Al5Cu2Mg8Si6 is quite soft since its hardness is only 2.7 GPa. The common precipitates of Al2Cu and Al2CuMg show a moderate hardness of 4.3 GPa and 6.8 GPa, which are much softer than Al6Mn or Al3Zr phases.
Table
Table 3. Theoretically calculated elastic properties including bulk modulus (B in GPa) and its pressure derivative (B′), shear modulus (G in GPa), Young’s modulus (E in GPa), Poisson ratio (σ), and anisotropy factors (AB, AG and AU) for second phases of 2xxx series aluminum alloys.
Table 3. Theoretically calculated elastic properties including bulk modulus (B in GPa) and its pressure derivative (B′), shear modulus (G in GPa), Young’s modulus (E in GPa), Poisson ratio (σ), and anisotropy factors (AB, AG and AU) for second phases of 2xxx series aluminum alloys.
PhaseBB’GBH/GHEσHvABAGAU
BVBRBHGVGRGH
Al2Cu90.5 (99.4 a, 87.7 b)90.5 (99.4 a, 87.6 b)90.5 (99.4 a, 87.7 b, 108.6 c)4.7146.1 (38.3 a, 52.1 b)38.2 (35.9 a, 42.3 b)42.1 (37.1 a, 47.2 b, 39 c)2.1109.5 (99 a, 120 b, 104.5 c)0.298 (0.33 a, 0.272 b, 0.34 c)4.300.091.03
Al2CuMg73.4 (80.9 b, 76.06 d)72.9 (80.7 b, 74.36 d)73.2 (80.8 b, 75.21 d, 79.48 e)4.63945.6 (63.3 b, 51.02 d)41.1 (57.9 b, 45.13 d)43.3 (60.6 b, 48.08 d, 46.8 e)1.7 (1.564 d)108.5 (145.5 b, 118.9 d, 117.3 e)0.253 (0.2 b, 0.237 d, 0.254 e)6.80.0030.0520.554 (0.675 d, 0.349 e)
Al3Fe129.6 (125 f)129.5 (125 f)129.6 (125 f)4.15654.9 (55 f)52.0 (51 f)53.4 (53 f)2.4140.90.319 (0.31 f)4.30.00030.0270.281 (0.35 f)
Al3Fe2Si155.9155.9155.94.49895.695.595.61.6238.10.24513.300.00050.0052
Al5Cu2Mg8Si663.6 (70.5 g)63.4 (69.7 g)63.5 (70.1 g)4.4395.8 (44.5 g)51.8 (42.8 g)28.8 (43.6 g)2.275.0 (108.4 g)0.303 (0.242 g)2.7///
Al7Cu2Fe109.1109.0109.1 (107.8 h)4.4291.390.891.0 (74.5 h)1.2213.6 (181.7 h)0.173 (0.219 h)19.60.00050.00270.0285
Al6Mn102.1101.9102.0 (102.6 i)4.02267.664.165.91.5126.60.23410.90.0010.02660.275
Al3Zr_D023101.6 (123.9 j)101.4 (117.4 j)101.5 (120.6 j, 100.2 k)3.97784.1 (88.4 j)82.8 (85.8 j)83.4 (87.1 j, 77.1 k)1.2196.5 (210.6 j, 184.1 k)0.177 (0.195 k)18.10.0010.0080.081
Al3Zr_D022101.4 (101.9 l)101.2 (101.6 l)101.3 (101.8 l)3.1187.4 (87.9 l)79.3 (80.6 l)83.3 (84.2 l)1.2196.2 (198 l)0.177 (0.176 l)18.10.0010.04860.513
Al3Zr_L12101.5101.5101.5 (104.4 m, 103 n)4.13463.162.262.7 (65.3 m, 64 n)1.6156.0 (162.2 m, 159.1 n)0.244 (0.241 m, 0.243 n)9.800.00720.0723
Al20Cu2Mn3101.5100.8101.14.74142.538.240.42.5106.90.3243.00.00350.05330.569
a, b, c, d, e, f, g, h, i, j, k, l, m, and n are the theoretical data form Refs. [22,23,24,25,26,27,28,43,44,50,51,52], and [45], respectively.
The mechanical anisotropy of crystals is also characterized by using 3-D mechanical modulus and anisotropy indexes. Firstly, the directional dependence of the Young’s modulus for different crystals can be evaluated by [53]:
For tetragonal (Al2Cu, Al7Cu2Fe, Al3Zr_D023 and D022):
E = [ ( l 1 4 + l 2 4 ) s 11 + l 3 4 s 33 + l 1 2 l 2 2 ( 2 s 12 + s 66 ) + l 3 2 ( 1 − l 3 2 ) ( 2 S 13 + S 44 ) ] − 1
For orthorhombic (Al2CuMg, Al6Mn, Al20Cu2Mn3):
E = [ l 1 4 s 11 + l 2 4 s 22 + l 3 4 s 33 + 2 l 1 2 l 2 2 s 12 + 2 l 1 2 l 3 2 s 13 + 2 l 2 2 l 3 2 s 23 + l 1 2 l 2 2 s 66 + l 1 2 l 3 2 s 55 + l 2 2 l 3 2 s 44 ] − 1
For triclinic (Al3Fe):
E = [ l 1 4 s 11 + 2 l 1 2 l 2 2 s 12 + 2 l 1 2 l 3 2 s 13 + 2 l 1 2 l 2 l 3 s 14 + 2 l 1 3 l 3 s 15 + 2 l 1 3 l 2 s 16 + l 2 4 s 22 + 2 l 2 2 l 3 2 s 23 + 2 l 2 3 l 3 s 24 + 2 l 1 l 2 2 l 3 s 25 + 2 l 1 l 2 3 s 26 + l 3 4 s 33 + 2 l 2 l 3 3 s 34 + 2 l 1 l 3 3 s 35 + 2 l 1 l 2 l 3 2 s 36 + l 2 2 l 3 2 s 44 + 2 l 1 l 2 l 3 2 s 45 + 2 l 1 l 2 2 l 3 s 46 + l 1 2 l 3 2 s 55 + 2 l 1 2 l 2 l 3 s 56 + l 1 2 l 2 2 s 66 ] − 1
For cubic (Al3Fe2Si, Al3Zr_L12):
E = [ s 11 − 2 ( s 11 − s 12 − 1 2 s 44 ) ( l 1 2 l 2 2 + l 2 2 l 3 2 + l 3 2 l 1 2 ) ] − 1
For monoclinic (Al5Cu2Mg8Si6):
E = [ l 1 4 s 11 + 2 l 1 2 l 2 2 s 12 + 2 l 1 2 l 3 2 s 13 + 2 l 1 3 l 3 s 15 + l 2 4 s 22 + 2 l 2 2 l 3 2 s 23 + 2 l 1 l 2 2 l 3 s 25 + l 3 4 s 33 + 2 l 1 l 3 3 s 35 + l 2 2 l 3 2 s 44 + 2 l 1 l 2 2 l 3 s 46 + l 1 2 l 3 2 s 55 + l 1 2 l 2 2 s 66 ] − 1
In the above equations, l 1 = sin θ cos φ , l 2 = sin θ sin φ , and l 3 = cos θ . The results are shown in Figure 3. Al2Cu and Al5Cu2Mg8Si6 show significantly stronger anisotropy than other phases, since the surface contours show a large deviation from the perfect spherical shape, while the iron-based compounds Al3Fe2Si and Al7Cu2Fe are very isotropy. As for the three Al3Zr phases, the L12 structure shows the best isotropic nature. More direct information can be seen from the planar projections on the (001) and (110) crystal planes in Figure 4. On the (001) plane, Al2CrMg shows the strongest anisotropy character, while on the (110) plane, Al5Cu2M8Si6 has the extremely strong anisotropy as can be seen from the green line in Figure 4b.
Three commonly used anisotropy indexes (AB, AG and AU) were calculated and shown in Table 3, where AB = (BV − BR)/(BV + BR), AG = (GV − GR)/(GV + GR), and AU = 5(GV/GR) + (BV/BR) − 6. For Al2Cu and Al2CuMg [54], the calculated AB of Al2Cu is zero, which indicates the perfect isotropy of Al2Cu in compression, while Al2CuMg has a high value of AB indicating its anisotropic nature. As for shear mode, Al2Cu has a high value of AG, implying its anisotropic character. The universal anisotropy index of AU follows the trend of: Al2Cu > Al2CuMg ≈ Al3Zr_D022 ≈ Al20Cu2Mn3 > Al3Fe ≈ Al6Mn > Al3Zr_D023 ≈ Al3Zr_ L12 > Al7Cu2Fe > Al3Fe2Si.

3.2. Thermophysical Properties

The precipitation influences thermophysical properties such as thermoelectric power, the thermal expansion coefficient, and thermal conductivity. In this part, the sound velocity and the Debye temperature were calculated. The longitudinal, transverse, and average wave velocities are given by [55]:
v m = [ 1 3 ( 2 v t 3 + 1 v l 3 ) ] − 1 / 3
v l = ( B + 4 3 G ) 1 ρ
Then, the Debye temperature is given by [56]:
Θ D = h k B [ 3 n 4 π ( N A ρ M ) ] 1 / 3 v m
where Θ D is the Debye temperature, and h, kB and NA are the Planck, Boltzmann and Avogadro constants, respectively; n is atomic number; M is molecular weight, and ρ is volumetric density; vl, vt and vm are longitudinal, transverse, and average acoustic velocities, respectively. The results are shown in Table 4, the sound velocity of Al3Fe2Si is highest among all considered phases, since it behaves the greatest mechanical moduli and small density, and therefore, the Debye temperature of Al3Fe2Si is as high as 622 K. The Debye temperatures obey the trend of: Al3Fe2Si > Al3Zr_D023 > Al6Mn > Al3Zr_D022 > Al7Cu2Fe > Al3Fe > Al2CuMg > Al20Cu2Mn3 > Al2Cu > Al3Zr_L12.
The thermal expansion is an important thermophysical character, which characterizes the anharmonicity of crystals. At different temperatures, the Helmholtz free energy of Al2Cu is shown in Figure 5a, based on which the volume expansion (ΔV/V0) and bulk modulus can be fitted by the well-known Birch-Murnaghan’s equation of states (EOS) [57]. Figure 5b shows the volume’ expansions (ΔV/V0) versus temperature curve by taking Al2Cu as an example. Then, the linear thermal expansion coefficient (LTEC) is obtained and plotted in Figure 6. For the Al2Cu phase, the LTEC along the a or c axis is also calculated, which implies that the value of αc is much higher than that of αa. The calculated average LTEC is ~16.2 ppm K−1 from 300 to 800 K, which is similar to the experimental data (~17.2 ppm K−1) [33]. For Al3Zr polymorphs, the results are plotted in Figure 6b, and our results indicate that the L12 phase has the highest LTEC value, followed by D023, and the lowest one refers to D022. From 300 K to 800 K, our calculated values are in extreme agreement with the calculated data given by Saha et al. [58], although both of our curves are slightly higher than experimental curve [59], as shown in Figure 6b. The calculated LTEC of three Al3Zr polymorphs follow the order of L12 > D023 > D022; all of them show a much lower value than pure Al substance. Overall, all calculated average LTECs are given in Figure 6c; in this figure the pure Al substance is also plotted. Al2CuMg shows the highest LTEC, followed by Al3Fe, Al2Cu, Al3Zr_L12 and others, while Al3Zr_D022 is the lowest one; the discrepancy between a-Al and Al2CuMg is the smallest, which may decrease the heat misfit degree between them and improve the thermal shock resistant property, and thereby delay the initiation and propagation of thermal crack at the interface.
The specific heats at constant pressure (CP) and constant volume (CV) are two fundamental parameters, which can be related by [62]:
CP−CV = λ2V(T)TB
where λ refers to the volumetric TEC, V(T) and B are the volume and bulk modulus at temperature T. The results are plotted in Figure 7; Al3Cu2Fe has much higher heat capacity than other phases. At very low temperatures, all curves increase rapidly due to the crystalline lattice vibration, and then the increasing rate reduces slowly; for CV, the curves tend to a well-known limit of Dulong-Petit, while for CP the curves keep increasing due to the work done by lattice expansion.
Based on the curves of (F (V, T) − V), the bulk modulus and its pressure derivative at different temperatures can be fitted based on Birch-Murnaghan EOS. The anti-compressibility of the second phases of 2xxx series aluminum alloys are given in Figure 8. At 0 K, Al3Fe, Al3Fe2Si, and Al7Cu2Fe are hard and anti-compressive, while Al2CuMg is the softest phase; meanwhile, three Al3Zr phases locate between them, which show moderate anti-compressibility. At temperatures of 300 or 700 K, the overall sequences of second phases are similar; Al2CuMg shows the most compressive character. However, with the increase of temperature, Al2Cu becomes more and more soft, and its compressibility tends to be similar to the Al2CuMg phase at 700 K.

4. Conclusions

Using first-principles calculations based on DFT, the structural stabilities, anisotropic mechanical, and thermophysical properties of precipitates of the 2xxx series aluminum alloys were investigated.
  • The calculated formation enthalpies are −0.151, −0.17, −0.16, −0.463, −0.02, −0.266, −0.195, −0.489, −0.464, −0.461, and −0.181 eV/atom for Al2Cu, Al2CuMg, Al3Fe, Al3Fe2Si, Al5Cu2Mg8Si6, Al7Cu2Fe, Al6Mn, Al3Zr_D023, Al3Zr_D022, Al3Zr_L12, and Al20Cu2Mn3, respectively. Al5Cu2Mg8Si6 shows thermodynamic and mechanical unstable.
  • The bulk modulus of all precipitates are derived from 63.5 to 155.9 GPa, which has the trend of: Al3Fe2Si > Al3Fe > Al7Cu2Fe > Al6Mn ≈ Al3Zr ≈ Al20Cu2Mn3 > Al2Cu > Al2CuMg > Al5Cu2Mg8Si6. The results of B/G imply that Al2Cu and Al20Cu2Mn3 are ductile precipitates. The hardness of Al3Zr_ D022 and D023 phases is very high (~18 GPa); whereas Al5Cu2Mg8Si6 shows low hardness value. The common precipitates of Al2Cu and Al2CuMg show a moderate hardness of 4.3 GPa and 6.8 GPa.
  • The thermal expansion characters are also calculated based on QHA; Al2CuMg shows the highest LTEC, followed by Al3Fe, Al2Cu, Al3Zr_L12 and others, while Al3Zr_D022 is the lowest one; the discrepancy between a-Al and Al2CuMg is the smallest.
  • The results of compressibility indicate Al3Fe, Al3Fe2Si and Al7Cu2Fe are hard and anti-compressive, while Al2CuMg is the softest one.

Author Contributions

Conceptualization, X.F., Y.L. and W.D.; formal analysis, X.F. and Y.L.; investigation, X.F., Q.Z. and J.G., writing-original draft preparation, X.F. and Y.L., supervision, X.F. and W.D., Methodology, Q.Z. and J.G., data curation, Y.Y., C.M., K.H., N.S., J.J., X.C. and H.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was funded by the National Key Research and Development Program of China (2021YFB3701204).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors wish to thank Y. Cheng from Xi’an Jiaotong University for providing CASTEP code support of this work.

Conflicts of Interest

The authors declare no conflict of interest.

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Metals 12 02178 g001 550
Figure 1. The unit crystal cells of second phases in the 2xxx series aluminum alloys.
Figure 1. The unit crystal cells of second phases in the 2xxx series aluminum alloys.
Metals 12 02178 g001
Metals 12 02178 g002 550
Figure 2. Mechanical properties of second phases in the 2xxx series aluminum alloys: (a) bulk, shear, and Young’s moduli, (b) Vicker’s hardness, and (c) Poisson’s ratio, and brittle index; the horizontal lines refer to the typical Poisson’s ratio value (~0.3) for pure metal, and criterion of ductility/brittleness (B/G = 1.75).
Figure 2. Mechanical properties of second phases in the 2xxx series aluminum alloys: (a) bulk, shear, and Young’s moduli, (b) Vicker’s hardness, and (c) Poisson’s ratio, and brittle index; the horizontal lines refer to the typical Poisson’s ratio value (~0.3) for pure metal, and criterion of ductility/brittleness (B/G = 1.75).
Metals 12 02178 g002
Metals 12 02178 g003 550
Figure 3. The surface constructions of the Young’s modulus of second phases in the 2xxx series aluminum alloys.
Figure 3. The surface constructions of the Young’s modulus of second phases in the 2xxx series aluminum alloys.
Metals 12 02178 g003
Metals 12 02178 g004 550
Figure 4. The planar projections of the Young’s modulus of second phases in the 2xxx series aluminum alloys: (a) (001) crystal plane, and (b) (110) crystal plane.
Figure 4. The planar projections of the Young’s modulus of second phases in the 2xxx series aluminum alloys: (a) (001) crystal plane, and (b) (110) crystal plane.
Metals 12 02178 g004
Metals 12 02178 g005 550
Figure 5. Dependence of the Helmholtz free energy, F (V, T), on the crystal volume under different temperatures (a) by taking Al2Cu for example, based on which the volume’ expansions (ΔV/V0) are given in (b).
Figure 5. Dependence of the Helmholtz free energy, F (V, T), on the crystal volume under different temperatures (a) by taking Al2Cu for example, based on which the volume’ expansions (ΔV/V0) are given in (b).
Metals 12 02178 g005
Metals 12 02178 g006a 550Metals 12 02178 g006b 550
Figure 6. Linear thermal expansion coefficients (LTECs) of second phases in the 2xxx aluminum alloys: (a) LTEC along a or c principal axis, and its average by taking Al2Cu for example, the references [33,60,61] are also ploted in this sub-figure; (b) average LTEC of Al3Zr by considering different polymorphs with references [58,59]; (c) LTEC of all second phases versus temperature, and calculated results of pure Al from reference [58].
Figure 6. Linear thermal expansion coefficients (LTECs) of second phases in the 2xxx aluminum alloys: (a) LTEC along a or c principal axis, and its average by taking Al2Cu for example, the references [33,60,61] are also ploted in this sub-figure; (b) average LTEC of Al3Zr by considering different polymorphs with references [58,59]; (c) LTEC of all second phases versus temperature, and calculated results of pure Al from reference [58].
Metals 12 02178 g006aMetals 12 02178 g006b
Metals 12 02178 g007 550
Figure 7. The specific heats at constant volume (a) or pressure (b) of second phases in the 2xxx aluminum alloys.
Figure 7. The specific heats at constant volume (a) or pressure (b) of second phases in the 2xxx aluminum alloys.
Metals 12 02178 g007
Metals 12 02178 g008a 550Metals 12 02178 g008b 550
Figure 8. The anti-compressibility of the second phases in the 2xxx aluminum alloys: (a) at 0 K, (b) at 300 K, and (c) at 700 K.
Figure 8. The anti-compressibility of the second phases in the 2xxx aluminum alloys: (a) at 0 K, (b) at 300 K, and (c) at 700 K.
Metals 12 02178 g008aMetals 12 02178 g008b
Table
Table 2. Theoretically calculated elastic constants (Cij in GPa) for second phases of 2xxx series aluminum alloys.
Table 2. Theoretically calculated elastic constants (Cij in GPa) for second phases of 2xxx series aluminum alloys.
PhaseC11C12C13C22C23C33C44C66
Al2Cu150.2 (150.3 a, 179.7 b, 163.8 c)97.4 (86.1 a, 72.7 b, 78.2 c)59.1 (62.6 a, 75.7 b, 14.7 c)--211.3 (171.7 a, 170.2 b, 246.7 c)34.5 (29.4 a, 28.0 b, 33.8 c)41.3 (45.5 a, 44.7 b, 37.3 c)
Al2CuMg124.7 (156.4 c, 115.9 d, 133.6 e)22.5 (33.4 c, 35.3 d, 42.1 e)66.5 (62.6 c, 46.8 d, 49.9 e)150.5 (175.9 c, 174.1 d, 138.8 e)40.4 (17.7 c, 38.7 d, 58.0 e)126.6 (168.8 c, 153.1 d, 145.2 e)41.2 (43.7 c, 50.9 d, 39.0 e)32.6 (50.7 c, 26.6 d, 37.7 e)
Al3Fe213.2 (211 f)100.0 (93 f)76.7 (73 f)--228.8 (228 f)41.3 (39 f)57.4 (59 f)
Al3Fe2Si288.289.8 ---93.2-
Al5Cu2Mg8Si6126.8 (146.3 g)39.7 (42.8 g)30.8 (33.3 g)--126.3 (123.3 g)<0<0
Al7Cu2Fe225.0 (206 h)50.7 (50.6 h)52.6 (65.7 h)--217.8 (194 h)99.5 (80.9 h)86.5 (71.1 h)
Al6Mn200.336.071.3229.751.6171.448.167.4
Al3Zr_D023203.4 (284.3 i, 206.7 j, 201.3 k)65.6 (67.8 i, 52.3 j, 70.5 k)43.1 (58.8 i, 50.7 j, 49.1 k)--204.0 (175.9 i, 182.6 j, 196.7 k)83.0 (79.2 i, 81.4 j, 80.8 k)101.4 (97.2 i, 75.9 j)
Al3Zr_D022183.2 (185.96 l)87.4 (85.34 l)42.0 (43.13 l)--203.7 (202.08 l)89.0 (90 l)126.0 (125.22 l)
Al3Zr_L12173.6 (182.8 m, 179 n)65.4 (65.2 m, 66 n)----69.2 (70.1 m, 69 n)-
Al20Cu2Mn3138.069.681.0177.573.0150.424.646.8
a, c, d, e, f, g, h, i, j, k, l, m, and n are the theoretical data from Refs. [22,23,24,25,26,27,28,42,43,44], and [45], respectively. b Exp. Data from Ref. [46].
Table
Table 4. Calculated sound velocities (km/s) of second phases of 2xxx series aluminum alloys; the Debye temperatures (K) are also shown below.
Table 4. Calculated sound velocities (km/s) of second phases of 2xxx series aluminum alloys; the Debye temperatures (K) are also shown below.
PhaseνlνtνmΘD
Al2Cu5.8123.1143.478420
Al2CuMg6.1053.5123.9438
Al3Fe6.9923.6224.055484
Al3Fe2Si7.74.4994.989622
Al7Cu2Fe6.8634.024.458532
Al6Mn7.6374.5535.041557
Al3Zr_D0237.1844.4844.94584
Al3Zr_D0227.0174.2584.705554
Al3Zr_L126.1283.1053.48387
Al20Cu2Mn36.4723.3033.701424
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MDPI and ACS Style

Fang, X.; Li, Y.; Zheng, Q.; Guo, J.; Yang, Y.; Ding, W.; Ma, C.; He, K.; Su, N.; Jiang, J.; et al. Theoretical Prediction of Structural, Mechanical, and Thermophysical Properties of the Precipitates in 2xxx Series Aluminum Alloy. Metals 2022, 12, 2178. https://doi.org/10.3390/met12122178

AMA Style

Fang X, Li Y, Zheng Q, Guo J, Yang Y, Ding W, Ma C, He K, Su N, Jiang J, et al. Theoretical Prediction of Structural, Mechanical, and Thermophysical Properties of the Precipitates in 2xxx Series Aluminum Alloy. Metals. 2022; 12(12):2178. https://doi.org/10.3390/met12122178

Chicago/Turabian Style

Fang, Xuewei, Yefei Li, Qiaoling Zheng, Jianye Guo, Yanmei Yang, Weiyun Ding, Chunhui Ma, Ke He, Ningning Su, Jingyi Jiang, and et al. 2022. "Theoretical Prediction of Structural, Mechanical, and Thermophysical Properties of the Precipitates in 2xxx Series Aluminum Alloy" Metals 12, no. 12: 2178. https://doi.org/10.3390/met12122178

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