First-Principles Investigation of Point Defects on the Thermal Conductivity and Mechanical Properties of Aluminum at Room Temperature

Abstract

The effects of point defects on the mechanical and thermal conductivity of aluminum at room temperature have been investigated based on the first-principles calculations combined with the Boltzmann equation and the Debye model. The calculated results showed the equilibrium lattice constants a₀ of all RE_Al are larger than that of Al, and the defective formation energy E_f of all RE_Al is lower than that of V_Al. Both a₀ and E_f increase from Sc to La and then decrease linearly to Lu. The effects of solute atoms on the mechanical properties of the Al matrix were further calculated, and compared with Al, it is found that the RE_Al defects decrease the elastic constant C_ij, Cauchy pressure C₁₂–C₄₄, bulk modulus B, shear modulus G, Young’s modulus E, B/G and Poisson’s ratio ν of Al, except for C₄₄ of RE_Al (RE = La-Nd). With the increase of atomic number, the C₁₁ and E of Al-containing RE_Al decrease from Sc to La and then slowly increase to Lu, whereas C₁₂, C_44, B, and G have little change. Meanwhile, the values of C₁₂–C₄₄ and B/G of Al-containing RE_Al increase from Sc to Ce, and it slightly change after Ce, while ν is nearly unchanged. All defects containing Al present nonuniform and ductility. Finally, the effects of rare earth (RE) atoms on the thermal conductivity (TC) of Al alloys have been investigated based on the first-principles calculations. The reduction of TC of Al alloys by RE solute atoms RE_Al is much greater than that by the L1₂ Al₃RE phase with the same concentration of RE, which is in good agreement with the experiments. With the RE atomic number increasing, the total TC κ of the Al-RE solid solution decreases from Sc to La firstly and then increases linearly to Lu. Moreover, the decrement of TC Δκ of the Al matrix by early RE_Al (RE = La-Sm) is larger than that by V_Al, while the later RE_Al (RE = Gd-Lu) shows the opposite influence.

1. Introduction

Al-based alloys are widely applied in automotive, aerospace, and electronic industries as they exhibit low density, high specific strength, and welding strength [1,2,3], and their applications as potential in automobile engines and brakes, gearbox housings, and electronic packaging are increasing, because of their good creep-resistance, high specific strength and high thermal conductivity (TC) [4,5,6,7,8,9,10]. In detail, automotive engine blocks and cylinder heads mainly come from hypo-eutectic Al-Si alloys with Mg and Cu, e.g., A356 alloy (Al-7 Si-0.3 Mg), 319 alloy (Al-7 Si-4 Cu), and A380 alloy (Al-9Si-3Cu) [11,12]. Although they are known to have a good tensile strength of the order of ~270~320 MPa, the thermal conductivity within 120 Wm⁻¹K⁻¹ is far less than pure Al of 240 Wm⁻¹K⁻¹ [13,14,15,16]. The reason mainly comes from the influence of solid solution and particles by adding various micro-alloy elements, especially point defects [15,17].

Point defects that include intrinsic and impurity defects are common in materials and play key roles in the mechanical and thermal performance of Al alloys [18,19,20]. It is reported that the TC of materials is significantly influenced by the impurities dissolved in [21,22,23]. Vandersluis et al. reported the reason for the increased TC of the T7 samples of cylinder heads, which was the cast of 319-type Al alloy by heat treatment [16]. They pointed out that the more complete precipitation of Al₂Cu during over-aging was suggested as predominant in improving the TC of the T7 samples by depleting the solute atoms in the Al matrix, which act as electron scattering centers. Very recently, a high-performance Al-Si alloy with TC of 169.34 Wm⁻¹K⁻¹ was reported by Weng et al. [24]. They explained that the prominent performance is due to the systematic optimization of micro-structure, e.g., reducing the solute atoms in the primary phase. The addition of rare-earth (RE) elements has become an effective measure to further improve the physical properties of Al-based alloys and thus are widely investigated [25,26,27,28,29,30,31,32,33,34,35,36]. Liu et al. studied the effect of RE elements on the resistivity of Al wires [37]. Their results showed that when RE elements were added to the Al melt, forming the transitional compounds with impurities Fe and Si, namely, REFe₂ and RESi₂, respectively, which reduced the amount of impurity elements in the Al solid solution matrix, and thus the TC of 6063 Al alloy has been largely improved.

However, up to now, due to a series of various and extensive experimental workloads, it’s difficult to accurately and extensively study the effect of specific factors on thermal conductivity in the experiment [38,39,40], and most of the research mainly studied the effects of the second phase particles [41,42,43,44,45,46,47,48], and the effects of solute atoms on the stability, mechanical properties, TC of Al matrix are rarely investigated. In addition, the relationship between stability, mechanical properties, TC, and RE atomic number has not been respectively obtained, and the effects of the second phases and RE solute atoms on the TC have not been systematically compared. Fortunately, in recent years, with the improvement of computing power, theoretical calculation, e.g., first-principles calculations, has been widely used in the analysis of various materials [49,50,51,52,53,54,55,56,57,58]. Herein, we have calculated the defective formation of impurities and then have analyzed the effects of RE solute atoms and vacancy on TC of the Al matrix; finally, the effects of dissolution of 1% RE, as solute atoms and L1₂ Al₃RE second phases, respectively, on the TC of Al matrix has been discussed.

2. Computational Detail

In the theoretical framework of density functional theory [59], all calculations within the generalized gradient approximation (GGA) [60] were accomplished in the Vienna ab initio simulation package (VASP) of 5.4.4 version [61], and the projector augmented wave (PAW) potentials [62] were employed for treating the core-valence interactions. In the current calculation, a 3 × 3 × 3 supercell model containing 108 atoms expanded from a stable unit cell, which can be found from work conducted by Liu et al. [48], as shown in Figure 1, was employed for this study. During the execution, the PAW_GGA pseudopotentials of Sc, Y_sv, La, and (Ce-Lu)_3 were used; where _sv and _3 present the electrons with s shell as valence electron and the version of pseudopotentials. The wave energy cut-off of 350 eV was chosen, and the first Brillouin zone sampling was carried out using the Gamma centered Monkhorst–Pack method [63] with 3 × 3 × 3 k-point mesh for the warrant of high accuracy. The convergence criterion for total energy and Hellman-Feymann forces [64] was established until the values reached 10⁻⁶ eV and 0.01 eV/Å, respectively, to stop the calculation.

Based on the energy strain method, the elastic constant C_ij carried out in the VASPKIT code [65] is calculated with the same accuracy as above. Solving the Boltzmann transport equation (BTE), the electronic transport properties of defective compounds, including perfect supercells, were further studied, performing in the BoltzTraP2 code [66], and a larger k-mesh with 6 × 6 × 6 and cut-off energy of plane wave basis of 500 eV was used to calculate electronic transport properties.

3. Result and Discussion

3.1. Defective Formation Energy

In the system of Al, two types of point defects containing vacancies V_Al and impurity elements replacing the Al atom in the lattice X_Al are necessary to investigate, as they generally occur in Al. Here, an evaluation for the stability of these defects is considered, as the following expression [67,68]:

$$E_{\mathrm{f}}=E_{\mathrm{d}\mathrm{e}\mathrm{f}}-E_{\mathrm{p}\mathrm{e}\mathrm{r}}-\sum _{\mathrm{i}}{n}_{\mathrm{i}}{\mu }_{\mathrm{i}}$$

(1)

where E_def and E_per are the total energies of the defective-containing and perfect supercells, respectively. n_i is the number of atoms increased (n_i > 0) or deleted (n_i < 0), and µ_i is the corresponding chemical potential of specific species, respectively. Further, to avoid the occurrence of precipitate of solid, the chemical potential µ_i in defective phases should be less than the chemical potential of bulk µ_bulk.

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

(2)

To study the effects of point defects on the mechanical and thermal conductivity of aluminum, the lattice constant a₀ of Al bulk and E_f of 18 defectives are first calculated, and the results are summarized in

Table 1. The calculated equilibrium lattice constants a₀ (Å) and defective formation energy E_f (eV) in Al.

Def.	a0	Ef
Al	4.038; 4.039 [69]; 3.983 [69]; 4.032 [70]	-
VAl	4.034	0.626; 0.665 [69]; 0.729 [69]; 0.670 [71] 0.540 [72]; 0.700 [72]
ScAl	4.045	−1.062
YAl	4.052	−0.507
LaAl	4.061	0.316
CeAl	4.060	0.314
PrAl	4.059	0.137
NdAl	4.058	−0.004
PmAl	4.057	−0.140
SmAl	4.055	−0.219
EuAl	4.055	−0.316
GdAl	4.054	−0.402
TbAl	4.054	−0.463
DyAl	4.051	−0.513
HoAl	4.051	−0.551
ErAl	4.050	−0.584
TmAl	4.050	−0.604
YbAl	4.050	−0.633
LuAl	4.049	−0.641

. It can be seen that the lattice constant a₀ of Al bulk and defective formation energy E_f of V_Al are in good agreement with experiment values and other theoretical predictions [69,70,71,72], and the errors are not more than 1.36% and 16.45%, respectively. Furthermore, to illustrate the potential relation between the a₀ and E_f of rare elements substitution RE_Al and the atomic number, the a₀, and E_f as a function of atomic number of RE are depicted in Figure 2a,b. Clearly, it is seen that the equilibrium lattice constants a₀ of Al-RE_Al is larger than that of Al of 4.038 Å as RE solute atoms force lattice distortion, and it increases from 4.045 Å for Sc to 4.061 Å for La firstly and then decreases linearly to 4.049 Å for Lu. One can see that the E_f of all RE_Al is lower than that of V_Al, indicating that the impurity atoms are easier to replace the Al site than Al escaping the Al matrix. The results further show that with the increase of atomic number, the E_f of RE_Al increases from −1.062 eV for Sc to 0.316 eV for La at first and then decreases linearly to −0.641 eV for Lu, and the Sc_Al has the lowest E_f. The phenomenon shows the Sc element has larger solubility than other RE elements in the Al matrix, which is in agreement well with the results of the experiment and theoretical data of Zhang et al. [28,73,74].

To investigate the underlying physical mechanisms of RE_Al defects on the properties of the Al matrix, the electron structures are calculated, and the obtained density of states (DOSs) for Sc_Al and La_Al, presenting the best and the worst stability, respectively, are shown in Figure 2. As can be seen from Figure 2c,d, the addition of Sc results in higher peaks near the Fermi level compared to that of the addition of La, implying that the effect of a chemical bond between Sc and Al atoms is larger. This phenomenon shows the stronger Al-Sc bond, and the defect Sc_Al is more stable than La_Al.

3.2. Mechanical Properties

The bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio ν determined by [75] the elastic constants C_ij are important family members of mechanical parameters. The C_ij are calculated from the energy-strain relationship by adding small strains to the original optimization lattice ordering, and the energy-strain relation ΔE (V, {ε_i}) of material under the quadratic function is given by [65,76]:

$$\Delta E_{\left(V,\left\{{\epsilon }_{\mathrm{i}}\right\}\right)}=E_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}}\left(V,\left\{{\epsilon }_{\mathrm{i}}\right\}\right)-E_0\left(V_0,0\right)=\frac{V_0}{2}\sum _{\mathrm{i},\mathrm{j}=1}^6{C}_{\mathrm{i}\mathrm{j}}{\epsilon }_{\mathrm{i}}{\epsilon }_{\mathrm{j}}$$

(3)

where $E_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}}\left(V,\left\{{\epsilon }_{\mathrm{i}}\right\}\right)$ and $E_0\left(V_0,0\right)$ are the total energies of the strained-lattice V and equilibrium at the volume of V₀, respectively.

In Born stability criteria, determining the mechanical stability for a Cubic crystal system is as following formulas [48,77,78]:

$$C_{11}>0,C_{44}>0,C_{11}-C_{12}>0,C_{11}+{2C}_{12}>0$$

(4)

The bulk modulus B can be used to evaluate the ability to resist volume deformation and strength of the average atomic bond, and the shear modulus G can describe the resistant of the materials from shape-changing [79], and both are used to estimate the Vickers’ hardness of materials according to $H_{\mathrm{v}}={2\left(KG\right)}^{0.585}-3$ [80,81], where $K=G/B$. And the Young’s modulus E mainly demonstrates the detailed value of the tensile strength of materials. They can be calculated by the elastic constant matrix according to Voigt-Reuss-Hill’s theory, which can be obtained as follows [48,82,83,84,85,86]:

$$B_{\mathrm{v}}=B_{\mathrm{R}}=B=\frac{C_{11}+{2C}_{12}}{3}$$

(5)

$$G_{\mathrm{v}}=\frac{C_{11}-C_{12}+{3C}_{44}}{5}$$

(6)

$$G_{\mathrm{R}}=\frac{5\times (C_{11}-C_{12})\times C_{44}}{4\times C_{44}+3\times (C_{11}-C_{12})}$$

(7)

$$G=\frac{G_{\mathrm{V}}+G_{\mathrm{R}}}{2}$$

(8)

The Young’s modulus E and Poisson’s ratio ν can be given by refs. [87,88,89]:

$$E=\frac{9GB}{3B+G},\hspace{1em}\nu =\frac{3B-2G}{2\times (3B+G)}$$

(9)

where G_V, B_V and G_R, B_R are the shear and bulk moduli of Voigt and Reuss approximations, respectively.

The calculated values with the experimental data and previously computed results are presented in

Table 2. The calculated elastic constants (units in GPa), bulk modulus B, shear modulus G, Young’s modulus E, Poisson’s ratio ν, and B/G of RE_Al containing the perfect supercell and unit cell.

Systems	C11	C12	C44	C12–C44	B	G	E	B/G	ν
Al Al *	121.05 110.47 107.00 [a] 109.98 [b]	63.64 67.33 61.00 [a] 60.11 [b]	34.50 33.94 28.00 [a] 31.33 [b]	29.14 33.39 28.78 [b]	82.78 81.71 76.73 [b]	32.05 28.30 28.59 [b]	85.16 76.11 76.30 [b]	2.583 2.887 2.68 [b]	0.329 0.345 0.33 [b]
VAl	93.71	43.19	35.45	7.74	60.03	30.95	79.24	1.939	0.280
ScAl	96.02	42.33	35.06	7.27	60.23	31.51	80.48	1.912	0.277
YAl	93.18	43.26	34.93	8.34	59.90	30.53	78.28	1.962	0.282
LaAl	91.05	43.14	33.98	9.16	59.11	29.54	75.96	2.001	0.286
CeAl	91.18 s	43.29	33.99	9.31	59.26	29.54	75.99	2.006	0.286
PrAl	91.72	42.84	34.23	8.61	59.13	29.91	76.78	1.977	0.284
NdAl	91.45	43.29	34.25	9.04	59.34	29.74	76.45	1.995	0.285
PmAl	91.75	43.28	34.40	8.88	59.44	29.90	76.81	1.988	0.285
SmAl	92.23	43.30	34.59	8.71	59.61	30.11	77.31	1.980	0.284
EuAl	92.37	43.20	34.59	8.61	59.59	30.17	77.44	1.975	0.283
GdAl	92.01	43.60	34.68	8.92	59.74	30.03	77.15	1.990	0.285
TbAl	92.96	43.06	34.74	8.33	59.70	30.42	78.02	1.962	0.282
DyAl	93.10	43.37	34.98	8.39	59.95	30.51	78.26	1.965	0.282
HoAl	93.88	43.05	35.03	8.02	59.99	30.80	78.91	1.947	0.281
ErAl	94.14	42.99	35.08	7.91	60.04	30.91	79.14	1.943	0.280
TmAl	94.29	42.90	35.07	7.83	60.03	30.96	79.26	1.939	0.280
YbAl	94.48	42.80	35.06	7.74	60.03	31.03	79.40	1.935	0.280
LuAl	94.67	42.73	35.09	7.65	60.04	31.10	79.57	1.931	0.279

* Experimental values. ^[a] Ref. [90] and ^[b] Ref. [48] represent the results of the experiment at 175 K and GGA-PBE calculation at 0 K, respectively.

. A good coincidence can be clearly identified, and the difference between the computed values and the experiment data is within 10%. It can be found that compared with Al, the parameters involving C₁₁, C₁₂, C₁₂–C₄₄ B, G, E B/G, and ν of all defective supercell decrease while C₄₄ increases, except for RE_Al (RE = La-Nd). Furthermore, it can be seen that all situations meet the mechanical stability criteria according to Equation (4). To study the variation regulation of mechanical parameters, the C₁₁, C₁₂, C₄₄, C₁₂–C₄₄ B, G, E B/G, and ν of RE_Al-containing Al with the variation of the atomic number of RE are further illustrated in Figure 3. A clear result seen in Figure 3a shows that the C₁₁ of RE_Al decreases from 96.02 GPa for Sc to 91.05 GPa for La and slowly increases to 94.07 Ga for Lu, while there is a little variation for C₁₂ and C₄₄. The B, G, and E of RE_Al are described in Figure 3b. The same law as C₁₁ can be seen in that the E decreases from 80.48 GPa for Sc to 78.28 GPa for Y and slowly increases to 79.57 GPa for Lu with the increase of atomic number, while B and G have little change.

To study the influence of solute atoms on the characteristics of bonds, we have calculated the Cauchy pressure $C_{12}-C_{44}$, which is a critical index to describe the directional chemical bonding features of materials [91]. It is proved that the atomic bonds have metallic characteristic if $C_{12}-C_{44}>0$, while those satisfying $C_{12}-C_{44}<0$ has directional covalent bonds, and the larger absolute value shows the stronger bonding characteristic of materials for both cases [92,93]. Figure 3c shows the change of $C_{12}-C_{44}$ with the increased atomic number, and the dissolution of all RE elements leads to a significant decrease in Cauchy pressure with a minimum of 68.05%, indicating the dissolution of RE elements in the Al matrix reduces the metallic characteristic of atomic bonding. Meanwhile, it is found that the values of $C_{12}-C_{44}$ of defects increase from Sc to Ce, and it slightly change after Ce. The Poisson’s ratio ν is usually used to judge the heterogeneity of materials [94], and greater/smaller ν than the critical value of 0.26 is an implication of uniform/nonuniform strain response to exerted stress, respectively. The Pugh’s ratio B/G relates mainly to the plasticity of materials, and a higher/lower B/G than the critical value of 1.75 [48] leads to the ductile/brittle behavior of materials, respectively. The calculated ν and B/G of RE_Al with the increases of atomic number, as shown in Figure 3d. It is seen that the ν and B/G of defects range from ~0.286 to ~2.006, respectively. The B/G shows the same trend compared to C₁₂-C₄₄ with the increase of atomic number while ν is unchanged. Moreover, all defects containing perfect supercells present nonuniform and ductility.

3.3. Thermal Conductivity

The power system in modern electrical equipment needs materials possessing high thermal conductivity (TC) for dissipating heat faster so as to prolong its life and improve its operation efficiency [95,96]. The TC of an objective usually contains two aspects; the first one comes from the lattice TC (LTC) κ_l, and the other one originates from the electronic TC (ETC) κ_e. That is to say, the total thermal conductivity κ = κ_e + κ_l. For pure metal and their compounds, heat transfer mainly involves the heat exchange between electrons. Then, we have firstly calculated the κ_e by using the Wiedemann–Franz law [97], κ_e = LσT, where $L=\frac{{\pi }^2{\kappa }_B^2}{3e^2}$ denotes the Lorentz number, σ and T represent the electrical conductivity and temperature in Kelvins, respectively. Here, the σ of materials are further obtained by using Boltzmann transport theory [98,99,100,101], and it’s given by the following formula:

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

(10)

where e, μ, and ε are the electronic charge, the band-energy, and the chemical potential, respectively. And the core of Equation (10) is the tensor σ_αβ, which mainly indicates the energy projected transport distribution function and contains the system-dependent information, which is computed as follows:

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

(11)

where N, i, K, and ν_α,β(i, K) are the number of k-points, the band index, the wave vector, and the group velocity of the acoustic wave, respectively. Equation (10) is settled from constant relaxation time approximation (RTA) [98,99]. For convenience, in the present work, we used a common relaxation time for metal Al of 10⁻¹³ s. The energy limit of the system is set to be 0.30 Ry for the chemical potential around the Fermi level [99].

We have further calculated the κ_l, which is solved by two key parameters: Debye temperature θ_α and Grüneisen parameter γ, using the Slack equation:

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

(12)

and

$${\kappa }_{\mathrm{l}}\left(\mathrm{T}\right)={\kappa }_{\mathrm{l}}\left({\theta }_{\mathsf{\alpha }}\right)\frac{{\theta }_{\mathsf{\alpha }}}{\mathrm{T}}$$

(13)

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.

The ETC κ_e of all defects-containing Al supercell at 300 K is calculated first, and the obtained results as a function of the atomic number of RE are drawn in Figure 4a. The κ_e value of 208 Wm⁻¹K⁻¹ for pure Al is in good agreement with the calculation value of 186.10 Wm⁻¹K⁻¹ at a room temperature of 300 K by Wen et al. [102]. A clear law can be seen that the κ_e of Al with RE_Al varies with the atomic number of RE, it decreases from 153.23 Wm⁻¹K⁻¹ for Sc to 128.34 Wm⁻¹K⁻¹ for La at first and then increases linearly to 153.73 Wm⁻¹K⁻¹ for Lu, and the κ_e of Al with RE_Al is greatly reduced as compared to that of pure Al, the decrements percent ranges from 24% to 38%. Moreover, the κ_e of Al with early RE (La-Sm) solute atom is lower than that with V_Al, while later RE (Gd-Lu) replacing Al shows the opposite influence. The LTC κ_l of Al with RE_Al as a function of atomic number at 300 K has been further computed, and the results are elucidated in Figure 4b. It can be seen that the calculated LTC κ_l of 9.82 Wm⁻¹K⁻¹ for Al in this work is in good agreement with the calculated value of 4.80 Wm⁻¹K⁻¹ by Wen et al. [102]. And the LTC κ_l of Al with RE_Al decreases slightly from Sc to Y and then nearly unchanged. The maximum loss of κ_l of the Al matrix by Sm solute is only 0.53 Wm⁻¹K⁻¹. For the V_Al, the loss of κ_l in the Al matrix is also small, ~0.33 Wm⁻¹K⁻¹.

Based on the calculation of the above two parts, we have computed the total TC κ of defective supercells, and the obtained values as a function of the atomic number of RE are depicted in Figure 4c; a very consistent can be clearly ascertained as compared with both theoretical and experimental values [102,103]. It is found that the total TC κ of Al is greatly reduced by the RE solute atom; with the increase of the atomic number of RE, it decreases from 163.01 Wm⁻¹K⁻¹ for Sc to 137.33 Wm⁻¹K⁻¹ for La firstly and then increases linearly to 168.29 Wm⁻¹K⁻¹ for Lu. The total TC κ of the Al matrix with early stage RE_Al (RE = La-Sm) is lower than that with V_Al, which is denoted by an orange horizontal dot-dot-dashed line, while the total TC κ of the Al matrix with Sc_Al, Y_Al, and later stage RE_Al (RE = Gd-Lu) possesses higher value.

To quantitatively analyze the different effects between second phase particles (L1₂ Al₃RE phases) and RE solute atom on the total TC κ of the Al matrix, we have calculated the effects of adding 1% RE at 300 K in the Al matrix for both cases, and the results are shown in Figure 5. It can be seen that for the effect of solute atoms, the reduction of total TC κ of the Al matrix increases from 27.17% for Sc to 39.91% for La firstly and then decreases linearly to 24.57% for Lu. Whereas the reduction of the total TC κ of the Al matrix by L1₂ Al₃RE second phases increases from 1.50% for Sc to 3.43% for La and then slightly decreases to 2.21% for Lu. In short, the influence of RE solute atoms on the TC of Al is much greater than that of L1₂ Al₃RE phase particles.

4. Conclusions

In summary, the effects of point defects on the thermal conductivity of aluminum are studied based on the first principles calculations. The main obtained results are as follows:

(1)

The equilibrium lattice constant a₀ of RE_Al is larger than that of Al, and it increases from Sc to La and linearly decreases to Lu with the increase of atomic number.

(2)

The defective formation energy E_f of all RE_Al is lower than that of V_Al, and with the increase of the atomic number of RE, they increase from Sc to La at first and then decrease linearly to Lu.

(3)

Compared with Al, the elastic constant C_ij, Cauchy pressure C₁₂-C₄₄, bulk modulus B, shear modulus G, Young’s modulus E, B/G, and Poisson’s ratio ν of all RE_Al decrease except for C₄₄ of RE_Al (RE = La-Nd); With the increase of atomic number, the C₁₁ and E of RE_Al decreases from Sc to La and slowly increases to Lu while C₁₂, C_44, B and G have little change.

(4)

The values of $C_{12}-C_{44}$ and B/G of defects increase Sc to Ce with the increase of atomic number, and it slightly change after Ce, while ν is unchanged. All defects containing perfect supercells present nonuniform and ductility.

(5)

The total TC κ is greatly reduced by the RE solute atom in the Al matrix, and it decreases Sc to La firstly and then increases linearly to Lu. The total TC κ of the Al matrix with early stage RE_Al (RE = La-Sm) is lower than that with V_Al, while the total TC κ of the Al matrix with Sc_Al, Y_Al, and later stage RE_Al (RE = Gd-Lu) possesses a higher value.

(6)

When 1% RE atoms are added to the Al matrix, the reduction of TC by RE solute atoms is much greater than that by the formation of the L1₂ Al₃RE second phase.