The Influence of Substitutional Defects of Transition Metal Elements on the Stability and Thermal Properties of Al at Finite Temperatures: A First-Principles Study

Abstract

Based on first-principles calculations, the effects of substitutional defects of the 3d–5d transition metal elements TM_Al on the stability and thermal conductivity of the aluminum matrix were investigated. The results show that with an increase in the atomic number of TM, the defect-forming energy E_f of TM_Al exhibits a periodic change feature, which depends on the valence electron configuration of the TM elements. The thermodynamic property parameters calculated with the Debye theory show that the addition of TM atoms does not change the stability of an Al system and can effectively reduce the thermal expansion coefficient of the material. But the equilibrium lattice constant a₀ of Al-TM_Al supercells changes very little. As the temperature increases, the relaxation time τ decreases, and both the electronic thermal conductivity κ_e and the total thermal conductivity κ decrease at the temperature range of 100–200 K, followed by a small increase or decrease. Because the lattice thermal conductivity κ_l is very small in the whole temperature range, the changes in electronic thermal conductivity and total thermal conductivity are basically the same. Moreover, when 1 at.% TM was added at both 300 K and 600 K, it was found that the influence of TM solute atoms on the thermal conductivity κ of Al was much greater than that of the second-phase particles. For solid solution atoms, Pd and Pt atoms have the greatest influence on the thermal conductivity of pure Al. This work is helpful for designing high-performance, heat-resistant Al-based alloys.

1. Introduction

For a long time, aluminum (Al) alloys have been widely used in vehicle and aerospace engineering industries due to their excellent physical and chemical properties, such as strong corrosion resistance, light weight, and high specific strength [1,2,3,4,5,6]. Nowadays, with the continuous development of technology, research related to high-performance Al alloys has ushered in a new era. In order to adapt to the progress of power electronics technology and the rapid development trend of 5G communication-network-related equipment, higher requirements are put forward for heat dissipation, creep resistance, and thermal conductivity. Moreover, these improved properties also make the Al-based alloys possess potential applications in computer heat sinks, automobile engine blocks, cylinder heads, communication base station radiators, overhead transmission wires, and other equipment [7,8,9]. However, the limited creep resistance and thermal conductivity of Al alloys greatly hinders their application at high temperatures. Fortunately, in the past few decades, related researchers have found that by adding appropriate transition elements (TMs) and rare earth elements (REs) to Al alloys, the L1₂ ordered binary precipitated phases can be produced with the appropriate heat treatment, namely, Al₃TM and Al₃RE intermetallic compounds. These binary intermetallics have excellent mechanical properties such as high specific strength, good elastic properties, and plasticity [10,11,12], as well as excellent physical properties such as a high melting point, good oxidation resistance, good thermal stability, and low mass density [13,14,15,16,17,18,19,20,21,22,23,24,25], thereby improving the creep resistance and thermal conductivity of Al alloys even at high temperatures.

The addition of RE elements will form a complex second phase with Fe, Si, and other elements in the alloy and promote precipitation in the form of Al₃RE, so that the element will change into the precipitation state, which is conducive to the improvement of electrical conductivity. At the same time, the formed second-phase particle Al₃RE can effectively restrict the dislocation and sub-grain boundary and hinder the grain growth, and thus, increase the recrystallization temperature to improve the heat resistance of the Al-based materials. Al₃Sc and Al₃Y are of great significance in providing superplastic ductility for Al-based alloys [26,27,28]. Jahnátek [29] et al. have shown that the presence of Al₃Sc precipitation increases the creep resistance of coarse-grained Al-Mg-Sc ternary alloys. It has also been shown that the uniform distribution of Al₃Sc effectively refines the grain boundaries and is beneficial to the thermal stability of the refined grain size [30,31]. Compared with Al₃Li, Al₃Sc has higher structural stability and stronger reinforcement [32].

On the other hand, related studies have shown that Al₃Ti has excellent properties of high melting point (1610 K), relatively low density (3.4 g/cm³) [33], high Young’s modulus (216 GPa) [34], and strong oxidation resistance [35]. Compared with D0₂₂ Al₃Ti, the compressive strain of L1₂ type Al₃Ti at high temperature has a significant promoting effect [36,37]. At the same time, Al₃Zr particles with an L1₂ structure have good thermal stability, which can effectively prevent dislocation and sub-grain boundary migration and increase the recrystallization temperature and thus improve the heat resistance of the Al-based material. Guan et al. [38] found that Al₃(Zr, Sc) complex precipitates could further improve the mechanical properties and the electrical conductivity of Al-based alloy cables. The appropriate addition of Zr in Al can improve the heat resistance of Al alloys; proper addition of Er can not only improve the heat resistance of the Al-based alloy, but also react with Si and Fe to purify the alloy [39]. When 0.1 wt % Er and 0.04 wt % Zr were added to Al alloys, it was found that the electrical conductivity and heat resistance of the Al alloys were good [40,41]. Relevant studies have confirmed that the introduction of the TM element Nb successfully tunes the microstructure and properties of Al alloys and enables the formation of Al₃Nb to make the matrix have good oxidation resistance [21,42,43,44,45,46]. Tzeng et al. [47] found that metastable L1₂-Al₃Sc can improve the strength of Al alloys by inhibiting recrystallization and refining grains [48]. There have been many studies on the precipitation of the L1₂-Al₃TM phase of transition elements, especially Sc elements, but the high cost of Sc hinders the wide application of the Al₃Sc phase in the strengthening process of Al alloys. Because the Al₃RE phase with the L1₂ structure has been comprehensively studied, this study turns its attention to the L1₂-Al₃TM phase, in the hope of exploring feasible candidates for cheaper TM elements that can replace the Sc element.

A large number of studies have shown that the improvement of Al alloy properties is mainly due to the action of the second-phase particles. Taking Al₃Sc as an example, the second-phase particle can act as the heterogeneous nucleation center of grains during the solidification of the alloy liquid and play a role in weakening or even eliminating dendrites. At the same time, the recrystallization temperature of the alloy can be increased by inhibiting the nucleation and growth of recrystallization grains during the process of thermal deformation and heat treatment, and the dislocation movement and sub-grain boundary migration and merger can be prevented to improve the alloy strength. It can be seen that the improvement of the second phases on performance of the Al alloy is crucial.

The crystal structure and thermophysical properties can be further improved by doping TM elements [49,50,51,52]. However, due to the dissolution of TM in Al, forming TM_Al, it acts as the center of electron scattering, resulting in a great reduction in thermal conductivity. Vandersluis et al. reported that the thermal conductivity of the cylinder head T7 sample was improved by the dissipation of solute atoms in the Al matrix that acted as electron-scattering centers [53]. Therefore, point defects, especially substitutional impurity defects, are also important factors affecting the mechanical and thermodynamic properties of Al alloys at high temperatures, which is a difficult problem to be solved by the majority of scientific researchers. At present, although a large number of experimental and theoretical studies have been carried out on Al₃TM phases, the effects of point defects in TM_Al on the thermal conductivity of Al alloys are rarely considered, and a systematic database is lacking. In recent years, with the unprecedented development of computer technology, computational materials science has formed a research upsurge, which not only has laid the foundation data for the experimental research, but also provides a new way of thinking for material research. Its theoretical research methods have gradually matured and can provide a theoretical basis for the optimal design of material properties and basic data support for production and application. In this paper, based on the first-principles calculations, the defect formation energy of substitutional impurities TM_Al in Al is calculated and the influence of TM solute atoms on the thermal conductivity of an Al matrix at various temperatures and concentrations is further analyzed. It is hoped that the research on high thermal conductivity of Al alloys can provide further basis and guidance.

2. Computational Detail

The first-principles calculation software Vienna ab initio simulation package (VASP) [54] code, based on density function theory [55], was adopted. The Perdew–Burke–Ernzerhof (PBE) version of the generalized gradient approximation (GGA) [56] was used to deal with the exchange-correlation functional, which can more accurately describe the energy of the structure and improve the convergence speed. In this work, the effects of substitution point defects TM_Al (including 3d (Sc-Zn), 4d (Y-Cd), and 5d (Hf-Hg)) on the stability and thermal conductivity of Al were investigated. A 3 × 3 × 3 supercell model with 108 atoms was employed for the study, and the PAW_GGA pseudopotentials of Al, Sc-Zn, Y-Nb_sv, and Mo-Hg were used during the relaxation calculations. The cutoff energy of 400 eV for plane wave was set to ensure high accuracy of calculation, and the Brillouin zone sampling was considered using the gamma-centered Monkhorst–Pack method [57] with 4 × 4 × 4 k-point meshes. The electronic self-consistence convergency criterion for electron iteration was 10⁻⁶ eV/atom; the ground-state geometries were relaxed via conjugate gradient (CG) minimization until the Hellman–Feynman forces [58] on each atom were less than 0.01 eV/Å. To calculate the influence of substitutional point defects TM_Al on the electronic transport of Al alloys, the electronic Boltzmann transport equation (BTE) was solved through performing in the BoltzTraP2 code [59], and a larger k-mesh with 6 × 6 × 6 and cutoff energy of plane wave basis of 500 eV were used. Thermodynamic properties were calculated using the Debye theory [60], and lattice thermal conductivity (LTC) was calculated using the Slack equation based on the Debye model [61,62,63].

3. Result and Discussion

3.1. Defective Formation Energy

In order to study the stability of defects TM_Al in the Al matrix, we first calculated the lattice constant a₀ of Al-TM_Al and the defect formation energy E_f of TM_Al. Generally speaking, there are two kinds of defects in the Al matrix that have a great influence on its performance, namely, vacancy and substitutional impurity defects. The calculation formula of defect formation energy is as follows [64,65]:

$$\mathsf{\Delta }{E}_{\mathrm{f}}=E_{\mathrm{d}\mathrm{e}\mathrm{f}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-E_{\mathrm{p}\mathrm{e}\mathrm{r}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-{\sum }_{\mathrm{i}}{n}_{\mathrm{i}}{\mu }_{\mathrm{i}}$$

(1)

where $E_{\mathrm{d}\mathrm{e}\mathrm{f}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ and $E_{\mathrm{p}\mathrm{e}\mathrm{r}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ are the total energies of defective and perfective supercells, respectively; n_i is the coefficient of increase (n_i > 0) or decrease (n_i < 0) in the number of atoms caused by the defect; and µ_i is the chemical potential corresponding to a specific type, which is derived from the energy of a single atom in the most stable system, from the database calculated with the VASP package [54]. ${\sum }_{\mathrm{i}}{n}_{\mathrm{i}}{\mu }_{\mathrm{i}}$ is the total variation chemical energy as a result of the defects.

Following the rule of thermal laws, it should be ensured that the chemical potential µ_i of the solute atom defects TM_Al is less than that of the bulk chemical potential ${\mu }_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}$ of pure TM to avoid an accumulation of TM, as follows:

$${\mu }_{\mathrm{i}}-{\mu }_{\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{k}}\le 0$$

(2)

Figure 1a illustrates the computational model of the Al-TM_Al solid solution, in which one TM element is placed in the center of the crystal, corresponding to a concentration of ~1 at.%, and the calculated results of the lattice constant a₀ of the Al-TM_Al supercell and the defect formation energy of TM_Al in Al are shown in

Table 1. The calculated equilibrium lattice constants a₀ (Å) of Al-TM_Al solid solution and defective formation energy ΔE_f (eV) of TM_Al in Al.

Systems	Ef	a0
Al	--	4.039; 4.039 [66]; 3.983 [66]; 4.032 [67]
VAl	0.68; 0.67 [66]; 0.73 [66]; 0.67 [68] 0.54 [69]; 0.70 [69]	4.035
ScAl	−1.16	4.045
TiAl	−1.17	4.038
VAl	−0.67	4.035
CrAl	−0.21	4.032
MnAl	−0.41	4.032
FeAl	−0.41	4.032
CoAl	−0.65	4.031
NiAl	−0.76	4.033
CuAl	−0.10	4.035
ZnAl	0.11	4.038
YAl	−0.62	4.052
ZrAl	−1.25	4.044
NbAl	−1.06	4.039
MoAl	−0.94	4.036
TcAl	−1.26	4.033
RuAl	−1.50	4.032
RhAl	−1.92	4.033
PdAl	−1.51	4.036
AgAl	0.10	4.039
CdAl	0.63	4.045
HfAl	−1.12	4.043
TaAl	−0.75	4.039
WAl	−0.40	4.035
ReAl	−0.85	4.033
OsAl	−1.16	4.032
IrAl	−1.81	4.033
PtAl	−1.87	4.034
AuAl	−0.55	4.039
HgAl	0.74	4.045

. Taking Al-W as an example, the lattice constant and atomic position information of the optimized structure have been placed in Table S6. It can be seen that the equilibrium lattice constant a₀ of the supermonomer Al-TM_Al is in good agreement with the reported theoretical predictions and experimental values, indicating that the calculation method has high accuracy. The equilibrium lattice constant a₀ changes very little, implying that the TM elements can hardly change the volume of Al alloys. The calculated defect formation energy ΔE_f of TM_Al varies with the atomic number. To clearly present the variations feature, we have further plotted the ΔE_f as a function of the atomic number of the TM element in Figure 1b. It is shown that the ΔE_f of TM_Al in the Al matrix shows a similar changing trend for 3d–5d cycles, and they all present a “W” shaped periodic changing feature.

The positive values of ΔE_f indicate the difficulty of the dissolution of atoms in the Al matrix. From this point of view, the elements Zn, Cd, Ag, and Hg are hard to form Al-TM binary solid solutions. Furthermore, it is interesting to note that the formation energy differences ΔE_f of the replacement defects TM_Al of the later elements (3d (Ni-Zn), 4d (Rh-Cd), and 5d (Pt-Hg)) are significantly higher than those of the preceding TM elements, which means that these elements are also more difficult to dissolve in the Al matrix. From the above results, it is reasonable to speculate that the valence electrons of elements determine the defect formation energy ΔE_f and equilibrium lattice constant a₀ of Al-TM_Al binary solid solutions. However, it should be noted that the defect formation enthalpy values of group IB and group IIB are inclined to or higher than 0, indicating that the substitutional defects formed by these two subgroup elements in the aluminum matrix are difficult.

3.2. The Thermodynamic Properties

Based on the quasi-harmonic Debye model [70,71,72] in the Gibbs2 program, the thermodynamic properties of the materials were calculated at 0 Pa and the temperature range was 0–1000 K [73]. According to the Debye theory [60], a crystal quasi-harmonic vibration model was obtained, and the Debye function ${\Theta }_D$ was calculated to analyze the Debye cutoff frequency ${\omega }_D$ [61,62,63,74,75]:

$${\Theta }_D=\frac{h}{2\pi {\kappa }_B}{\omega }_D$$

(3)

where h is Planck’s constant (6.63 × 10⁻³⁴ J/s) and ${\kappa }_B$ is Boltzmann’s constant (1.38 × 10⁻²³ J/K). ${\omega }_D$ can be expressed using Debye’s sound velocity ${\nu }_D$ as:

$$\frac{{\omega }_D}{{\nu }_D}={\left(\frac{{6\pi }^2}{V}\right)}^{1/3}$$

(4)

where V is the volume of the solid in the steady state. But the speed of sound has three different types and is anisotropic. For isotropic media, Anderson [76] expressed the mean crystal sound velocity at low temperature as:

$$\frac{1}{{\nu }_D^3}=\frac{1}{3}\left(\frac{1}{{\nu }_L^3}+\frac{2}{{\nu }_S^3}\right)$$

(5)

and

$${\nu }_L=\sqrt{\frac{L}{\rho }}, {\nu }_S=\sqrt{\frac{S}{\rho }}$$

(6)

where ${\nu }_L$ and ${\nu }_S$ are the longitudinal and transverse sound velocities, respectively, and $\rho$ is the density of the material. The Poisson’s ratio $\nu$ of the crystal can be expressed as:

$$\nu =\frac{3B-2G}{2(3B+G)}$$

(7)

where B and G are the bulk modulus and shear modulus of the crystal, respectively.

According to Formulas (3)~(7), a simpler method for calculating ${\nu }_D$ and ${\Theta }_D$ can be obtained by calculating the transverse modulus S and longitudinal modulus L. The calculation is as follows:

$$L=\frac{3\left(1-\nu \right)}{1+\nu }B, S=\frac{3(1-2\nu )}{2\left(1+\nu \right)}B$$

(8)

Then, the ${\nu }_D$ can be obtained:

$${\nu }_D=f\left(\nu \right)\sqrt{\frac{B}{\rho }}$$

(9)

$$f_{\nu }={\left\{\frac{1}{3}\left[{\left[\frac{1+\nu }{3\left(1-\nu \right)}\right]}^{\frac{3}{2}}\right]+2\left[{\left[\frac{2\left(1+\nu \right)}{3\left(1-2\nu \right)}\right]}^{\frac{3}{2}}\right]\right\}}^{-\frac{1}{3}}$$

(10)

This method has been proved to be suitable for obtaining a reasonable Debye sound velocity. Therefore, ${\Theta }_D$ can be obtained by combining the above formula:

$${\Theta }_D=f_{\nu }\frac{h}{2\pi {\kappa }_B}{\left(48{\pi }^5\right)}^{\frac{1}{6}}\sqrt{\frac{r_{0}B}{M}}$$

(11)

where the M is the atomic weight and r₀ is the equilibrium Wigner–Seitz radius.

The thermodynamic parameters of Gibbs free energy G, entropy S, coefficient of thermal expansion $\alpha$, and the relevant formulas for calculating the constant volume specific heat capacity $C_V$ can be further deduced from Formula (12):

$$F\left(V;T\right)=E\left(V\right)-{\kappa }_{B}T\left\{D\left(\frac{{\Theta }_D}{T}\right)-3\ln \left[1-e^{\left(\frac{-{\Theta }_D}{T}\right)}\right]+\frac{9}{8}{\kappa }_B{\Theta }_D\right\}$$

(12)

$$S=3{\kappa }_B\left\{\left(\frac{{4D\Theta }_D}{3T}-In\left[1-exp\left(\frac{{-\Theta }_D}{T}\right)\right]\right)\right\}$$

(13)

$$C_V=3{\kappa }_B[4D\left(\frac{{\Theta }_D}{T}\right)-\frac{3\left(\frac{{\Theta }_D}{T}\right)}{e^{\frac{{\Theta }_D}{T}}-1}]$$

(14)

$${\gamma }_D=-\frac{\ln \partial {\Theta }_D}{\ln \partial InV}; a_T=\frac{\gamma C_V}{B_{0}V}$$

(15)

The calculated results are shown as Figure 2. As we all know, the Gibbs free energy G can easily determine the direction and limit of the reaction, and the reduction in G in the reaction process is the maximum limit for the system to perform non-volume work. Both the Gibbs free energy G of all Al-TM solid solutions and Al matrices decrease with the increase in temperature, as seen in Figure 2a; the decrease in G in the reaction process is ~0.4 eV/atom, and the change trend is all the same. However, the Gibbs free energy G of Al-TM solid solutions (−3.7~−4.2 eV/atom) is slightly higher than that of Al (−3.7~−4.1 eV/atom), and the addition of TM keeps the system stable. In an isolated system, there is no energy exchange between the system and the environment, and the system always changes spontaneously in the direction of increasing the degree of disorder; that is, the system changes from an ordered state to a disordered state. It can be seen from Figure 2b that both the entropy S of the Al-TM solid solution and Al increase with the increase in temperature due to the more intense molecular motion caused by the increase in temperature, which further deviates from the stable state. The expansion $\alpha$ is one of the key properties of the thermodynamic properties of materials. It is commonly used to characterize the degree of thermal expansion of solid materials, that is, the change in material volume with temperature. Generally speaking, a rising temperature will increase the volume of materials, and materials with low expansion coefficients are expected to be applied in the field of high-precision instrument engineering. In Figure 2c, the coefficient of thermal expansion α at 1000 K was 5.6 × 10⁻⁵. In addition to Al vacancy, the addition of Zn, Cd, and Hg solid solution atoms will slightly increase the degree of thermal expansion α, which may be due to the thermodynamic instability of Zn metal, while Cd metal is heated violently and Hg can rapidly swing at high temperatures, resulting in the addition of these elements to increase the coefficient of thermal expansion α of Al. However, the addition of other solvable atoms reduces the expansion coefficient of Al, indicating that the addition of TM elements is expected to make the Al alloy develop into high-temperature precision materials. Under the condition of constant volume, the heat absorbed by Al and Al-TM solid solutions at 0–1000 K for every 100 K increase was observed, and the results are shown in Figure 2d. According to the images, at 0–300 K, the constant volume specific heat capacity $C_V$ increases exponentially with the increase in temperature, and the increment at this stage is about 23 Jmol⁻¹K⁻¹. At the stage of 300–1000 K, the specific heat capacity at constant volume increases linearly with the increase in temperature, and the increment of this stage is only 2 Jmol⁻¹K⁻¹. This shows that at the stage of 0–300 K, the material has completed most of the heat absorption.

All in all, according to the thermodynamic property parameters, the study shows that the addition of TM atoms does not change the system stability of Al and can effectively reduce the thermal expansion coefficient of the material, which provides support for the application of Al in the field of high-temperature precision instruments.

3.3. Thermal Conductivity

Understanding the heat conduction of materials can better measure their adiabatic and thermal conductivity properties to adapt to the rapid development of modern technologies such as electronics, artificial intelligence, and thermoelectricity. The importance of developing high-thermoelectric materials has become increasingly prominent, which puts forward higher requirements on the thermal conductivity of materials to improve the heat dissipation demand, thereby extending their service life and improving their operating efficiency [77,78]. Generally speaking, in metal systems, the interaction between electrons has a decisive influence on the thermal and electrical transport of materials, while the thermoelectric transport of phonons at medium and high temperatures is limited by electron-phonon scattering [79,80]. Therefore, the thermal conductivity block κ can be approximated by the composition of lattice thermal conductivity (LTC) κ_l and electron thermal conductivity (ETC) κ_e. That is to say, the total thermal conductivity (TC) formula is κ = κ_{e +} κ_l.

In this paper, the lattice thermal conductivity κ_l is obtained by combining VASP with BoltzTraP and using Slack’s model [81,82]. The calculated Debye temperature θ_α and Grüneisen parameter γ are both put into the Slack equation to obtain the lattice thermal conductivity κ_l [83]:

$${\kappa }_l\left(\mathrm{T}\right)=\frac{0.849\times 3\sqrt[3]{\left(4\right)}}{20{\mathsf{\pi }}^3\left(1-0.154{\gamma }^{-1}+0.228{\gamma }^{-2}\right)}\times {\left(\frac{{\mathsf{\kappa }}_{\mathrm{B}}{\theta }_{\mathsf{\alpha }}}{\mathrm{\hslash }}\right)}^2\frac{{\mathsf{\kappa }}_{\mathrm{B}}{M}_{\mathrm{a}\mathrm{v}}{V}^{\frac{1}{3}}}{\mathrm{\hslash }{\gamma }^2}\times \frac{{\theta }_{\mathsf{\alpha }}}{\mathrm{T}}$$

(16)

where V, M_av, ${\kappa }_{\mathrm{B}}$, and $\mathrm{\hslash }$ are the primitive cell volume, average atomic mass, Boltzmann constant, and reduced Planck constant, respectively.

Further, by using the Boltzmann transport theory [84,85,86,87], the expression of conductivity σ as a function of temperature can be given by the following formula:

$${\sigma }_{\mathsf{\alpha }\mathsf{\beta }}(T,\mu )=\frac{1}{V}\int {\mathsf{\sigma }}_{\mathsf{\alpha }\mathsf{\beta }}\left(\epsilon \right)\left[-\frac{\partial {\mathrm{ƒ}}_0(T,\epsilon , \mu )}{\partial \epsilon }\right]\mathrm{d}\epsilon$$

(17)

where e, ε, ${\mathrm{ƒ}}_0$, and μ are the electronic charge, band energy, chemical potential, and the Fermi–Dirac distribution function, respectively.

The group velocity $v_{\alpha }(i,k)$ can be obtained from band structure calculations as

$$v_{\alpha }\left(i,k\right)=\frac{1}{\hslash } \frac{\partial {\epsilon }_{i,k}}{\partial k_{\alpha }}$$

(18)

From the combination of Equations (4) and (5), the projected energy transport distribution tensor σ_αβ can be expressed as

$${\sigma }_{\mathsf{\alpha }\mathsf{\beta }}(\epsilon )=\frac{{\mathrm{e}}^2}{\mathrm{N}}{\sum }_{i,k}{\tau }_{i,k}{\nu }_{\mathsf{\alpha }}(i,k){\mathsf{\nu }}_{\mathsf{\beta }}(i,k)\frac{\mathsf{\delta }(\epsilon -{\epsilon }_{i,k})}{\mathrm{d}\epsilon }$$

(19)

where N, i, k, and ν_α,β (i, k) are the number of the k-points, the band index, the wave vector, and the group velocity of the acoustic wave, respectively. Equation (19) is solved under constant relaxation time approximation (RTA) [84,85], and, for simplicity, we adopted a relaxation time of 10⁻¹³ as obtained by the experiment [88]. Near the Fermi level of the system, the chemical potential has an energy range of 0.09 Ry [85].

Then, we calculated the κ_e according to the Wiedemann–Franz law [89],

κ_e = LσT,

(20)

where

$$L=\frac{{\pi }^2{\kappa }_B^2}{3e^2}$$

(21)

where L is the Lorentz number, σ is the electrical conductivity, and T is the temperature in Kelvins.

The relaxation time τ in this study is calculated according to the formula τ = A × T^-r, where A is a proportional constant and r is the scaling factor [90,91,92]. Their fitted results are used in the present investigation for both pure Al and Al-TM_Al solid solutions, which is shown in Figure 1c. With the increase in temperature, the relaxation time decreases. Based on this basic assumption, the calculated values of κ_l, κ_e, and κ obtained from the above formula are shown in

Table 2. The calculated κ_l, κ_e, and κ (units in Wm⁻¹K⁻¹) of TM_Al containing a perfect supercell at room temperature of 300 K.

	κl	κe	κ
Al	7.38 4.83 [93]	223.61	230.99 190.90 [93]; 240.00 [89]
VAl	7.48	178.95	186.43
ScAl	7.71	176.13	183.85
TiAl	8.14	181.60	189.75
VAl	8.22	177.43	185.65
CrAl	8.20	178.41	186.61
MnAl	8.15	176.13	184.28
FeAl	8.06	178.67	186.73
CoAl	7.91	177.65	185.57
NiAl	7.69	183.41	191.10
CuAl	7.44	184.12	191.55
ZnAl	7.28	205.54	212.82
YAl	7.53	161.69	169.21
ZrAl	8.04	158.06	166.09
NbAl	8.24	158.30	166.54
MoAl	8.30	162.37	170.67
TcAl	8.25	153.81	162.06
RuAl	8.12	150.51	158.62
RhAl	7.92	153.73	161.65
PdAl	7.65	140.95	148.60
AgAl	7.33	144.98	152.30
CdAl	7.21	194.94	202.15
HfAl	7.97	175.14	183.10
TaAl	8.21	174.88	183.09
WAl	8.28	162.70	170.98
ReAl	8.20	158.10	166.30
OsAl	8.11	151.66	159.76
IrAl	7.92	142.88	150.80
PtAl	7.67	115.51	123.19
AuAl	7.34	127.02	134.36
HgAl	7.09	191.10	198.19

. The thermal conductivity of Al and Al-TM_Al solid solutions as a function of temperature and atomic number are shown in Figure 3a–f, respectively.

In order to explicitly exhibit the variation feature of LTC as a function of both temperature and atomic number, Figure 3a,c,e shows the calculated κ_l, κ_e, κ as a function of atomic number and Figure 3b,d,f presents the corresponding results as a function of temperature, respectively. The detail values are summarized in Supplementary Tables S1–S5. LTC κ_l changes with the atomic number at both 300 K and 600 K are illustrated in Figure 3a. It is seen that with the increase in atomic number, for 3d–5d TM elements, the variation trend of the LTC at 300 K and 600 K is the same, a trend of slow rising and then slow decline. But it is obvious that the LTC κ_l at 300 K is significantly higher than that at 600 K, in the range of ~5.16 Wm⁻¹K⁻¹. As is shown in Figure 3b, the LTC κ_l is roughly inversely proportional to temperature, consistent with the functional relationship in the literature. As is well known, the LTC mainly comes from the vibration of the atoms in the crystal lattice, and the larger the noise caused by the thermal movement of the atoms, the worse transportation ability the phonon of the metal material is. Compared with the values of one another, it is found that the numerical differences of the LTC κ_l decreases with increasing temperature. It can be found that the LTC κ_l of both defects containing Al-TM_Al and perfect Al supercells decreases sharply with the increase in temperature (0~600 K), and then the deceasing velocity slows down, but the variation range is only about 26.64 Wm⁻¹K⁻¹, far smaller the that of the ETC.

Comparing Figure 3c,e, it can be found that the relationship between the ETC κ_e and atomic numbers of TM are related to the position of the atoms in the periodic table. The ETC κ_e of all Al-TM_Al at 300 K and 600 K is nearly the same. For 3d transition elements, the ETC κ_e increases slowly from Sc to Cu, and then jumps to Zn by about 20 Wm⁻¹K⁻¹. Looking at the 4d and 5d TM elements, it is found that the ETC κ_e of Al-TM_Al solid solution is the smallest for Pd and Pt in the group VIII, and thus, with Pd and Pt as nodes, it first slowly decreases and then rapidly increases. In the same way, the total TC κ as a function of temperature and atomic number tends to be basically the same as the function of the ETC κ_e. According to Figure 3d, the calculated value of the ETC κ_e of the Al matrix increases fast from 203.34 Wm⁻¹K⁻¹ at 100 K to 217.81 Wm⁻¹K⁻¹ at 200 K, and then the increasing velocity decreases. On the contrary, the Al-TM_Al solid solutions show a decreasing trend in the temperature range of 100–200 K at first, and then they all show a small variation. Among them, the change in the ETC κ_e and the TC κ of Al-Zn_Al is the closest to that of Al, while the change in Al-Pt_Al is quite different from that of Al. At the same time, due to the small influence of LTC κ_l, the functional relationship between the TC κ and temperature, as is shown in the Figure 3f, is nearly the same as that of ETC κ_e, as illustrated in Figure 3d.

3.4. The Effect of 1 at.% TM Added on Thermal Conductivity

To quantitatively analyze the effects of second-phase particles L1₂-Al₃TM and TM solute on the TC κ of Al, 1 at.% TM defects as above two states added to the Al matrix were further investigated to quantitatively exhibit the trends of different elements on the thermal conductivity of the Al matrix at finite temperatures. The relationships between thermal conductivity κ and the atomic number of TM for Al-L1₂-Al₃TM phase and Al-TM_Al systems when 1 at.% TM is added at 300 K and 600 K are shown in Figure 4a,b, respectively. Meanwhile, the thermal conductivity κ of the Al matrix at 300 K and 600 K is provided for comparing. Specifically, Figure 4a,b shows that the thermal conductivity κ of TM solute atoms at 300 K and 600 K as a function of atomic number is consistent with Figure 3f. It is found that when 1 at.% TM is added, the thermal conductivity κ of the second-phase particles remains almost at a horizontal level for all three 3d–5d periodic TM elements. In 3d (Sc-Zn), the thermal conductivity κ of Al-TM_Al increases with the increase in atomic number, the change is relatively stable, and the maximum thermal conductivity reduces in the Al matrix by no more than 5%. In 4d (Y-Cd) and 5d (Hf-Hg), the thermal conductivity κ decreases with the increase in atomic number, from about 30% at the beginning to about 50% at the lowest, and finally rises to 15%. Moreover, the lowest values appear in Pd and Pt for 4d and 5d, respectively, and then the values increase.

In summary, the effect of TM solute atoms on the thermal conductivity of Al is much greater than that of the second-phase particles, and the effect of Pd and Pt atoms is the most significant. The reason may be that the thermal conductivity of the second-phase particles is closer to the Al matrix and can retain the high thermal conductivity of the Al matrix to a greater extent.

4. Conclusions

In summary, based on first-principles calculations, the effect of substitution point defects TM_Al on the stability and thermal conductivity of aluminum was investigated. The main results show as follows:

• The equilibrium lattice constant a₀ of Al-TM_Al supercells changes very little, while the defect formation energy of substitution defects TM_Al on the Al matrix exhibits a similar “W”-shaped periodic change with the increase in atomic number for 3d–5d TM elements. The E_f of TM_Al for both groups IB and IIB is greater than 0, indicating that these two subgroup elements are more difficult to form substitution defects in Al with. It is speculated that the valence electron structures of the elements play a crucial role.
• The thermodynamic property parameters calculated with the Debye theory show that the addition of TM atoms can effectively reduce the thermal expansion coefficient of the material without changing the stability of the Al system, which provides support for the application of Al in the field of high-temperature precision instruments.
• The ETC κ_e and the TC κ decrease at the temperature range of 100–200 K and then show a small variation. Moreover, because the lattice thermal conductivity κ_l has little effect, the electron TC κ_e and total TC of Al-TM_Al κ have the same linear increase trends with temperature. But all are slightly lower than that of a pure aluminum matrix.
• At 300 K and 600 K, the total TC κ and the ETC κ_e for 3d TM increase slowly from Al-Sc_Al to Al-Cu_Al and then show a rapid increase to Al-Zn_Al with the increase in atomic number. The total TC κ in 4d (Y-Cd) and 5d (Hf-Hg) first drops slowly to Pd and Pt and then rises sharply after reaching the lowest point.
• Finally, we chose to add 1 at.% TM to an Al matrix as L1₂ s phases and solute atoms at 300 K and 600 K. It was obviously found that the TC of the second-phase-containing Al matrix is slightly lower than that of pure aluminum, while the solute atoms TM_Al significantly reduce the thermal conductivity of the Al material. Therefore, the second phase should be favorable and must avoid solid solution atoms in the matrix from the point of view of thermal conductivity.