EFFECT OF ALLOYING ELEMENTS ON THE STACKING FAULT ENERGIES OF DILUTE AL-BASED ALLOYS

A systematic study of the stacking fault energy (γ SF ) for the dilute Al-based alloys (Al 23 X, Al 47 X and Al 71 X, where X = Al, Ag, Be, Ca, Cd, Co, Cu, Cr, Fe, Ga, Ge, Hf, In, K, La, Li, Mn, Mg, Ni, Na, Pb, Sc, Sn, Sr, Si, Ti, V, Zn, and Zr) has been performed by means of first-principles calculations. Alias shear deformation is adopted in the present investigations. The presently calculated γ SF for Al is in favorable accordance with experimental and other theoretical data. For the targeted elements, the calculations indicate that Na, Si, K, Ca, Sc, Ga, Ge, Sr, Zr, In, Sn, La, Hf, and Pb, in any concentration we considered, decrease the γ SF of Al, while Ag, Be, Cd, Co, Cu, Cr, Fe, Li, Mn, Mg, Ni, Ti, V, and Zn increase the γ SF of Al, when the concentration of alloying elements is 1.39 at. % in the system. With increasing concentration of alloying elements, Li, Mg, V, Ti, and Cd change from increasing the γ SF of Al to decreasing it, based on present investigations. Among the alloying elements, which decrease the γ SF of Al, La decreases the γ SF most significantly. It is also found that the γ SF of Al-X generally decreases with the increase of equilibrium volume. The results obtained in the present work provide an insight into the design of Al based alloys.


Introduction
Intrinsic stacking fault is common and important in materials, because it usually can affect the deformation mechanism and mechanical properties of metallic materials，such as plastic deformation, crystal growth, phase transition, strength, and ductility [1][2][3]. The strength of materials is controlled by the formation and propagation of dislocations, while the ductility or brittleness of a material is related to the strength [4]. It is observed that the kinetic process of partial dislocation movements, such as cross-slip or climb, can be retarded by a lower stacking fault energy (γ SF ) [3]. Meanwhile, the steadystate creep rate can be reduced by a lower γ SF [5][6][7] as well, which is usually expressed approximately as below [8][9], (1) (2) In Eq. (1), is the normalized stacking fault energy and can be evaluated by Eq. (2), where n is an empirical parameter around 3 to 4 [10]. G and b in Eq.
(2) are the shear modulus and the Burgers vector, respectively.
In a perfect fcc (face centered cubic) crystal, the stacking sequence of atomic layers along <111> direction is …ABCABC…, where A, B, and C represent nonequivalent three kinds of atomic layers, as shown in Fig. 1. When one atomic layer is removed or glided a distance along the < > direction on the {111} layer, the stacking sequence will turn to be ABCABABC…. As a result, an intrinsic stacking fault occurs [11]. Herein, a fcc is the lattice parameter of perfect fcc crystal. γ SF can be calculated by the energy difference between the crystal with a stacking fault and the original perfect crystal, being normalized via the stacking fault area.
Unlike Cu and Ag, the γ SF of Al is large, resulting from the fact that electrons cannot readapt readily in Al due to the directionality of the bonds when intrinsic stacking fault occurs [2]. Many researchers have performed calculations and experiments to deduce the γ SF of Al [12][13][14][15][16][17][18][19][20][21]. However, the reported results vary widely, as seen in Table 1. The measured γ SF of Al are 280±50 [15], 135±20 [19], 166 [18], and 150±40 mJm -2 [17]. The large uncertainty of these measurements is possibly due to the different methods, experimental conditions, sample purity, the inaccuracies in measuring small differences for the separation distance between partial dislocations [2], and assumptions in the analysis. In the aspect of measuring method, more advanced experimental methods including TEM (transmission electron microscopy) [19] and HRTEM (high-resolution transmission electron microscopy method) [17] must lead to more accurate results than the earlier measured value 280±50 [15]. Computationally, the values of γ SF in Al vary from 95.4 [14] to 162 mJm -2 [22]. The difference of 66.6 mJm -2 can be regarded as the uncertainty of calculations, which partially comes from the inaccuracies in modeling electronic interactions between atoms using an empirical potential (EP) [23], such as the glue potentials [14] and the phenomenological many-body potentials [24], and partially comes from the models used in the calculations, such as the slab model [22] and the alias shear model [4]. For Al-based alloys, the alloying elements, which include Mg, Ga, Zn, Si, and Cu targeted by Muzyk et al. [22] and Fe, Mn, Cr, Cu, Zn, Mg, Si, Ga, Ti, and Ge by Qi et al. [2] have been investigated through first-principles calculations. In the work of Muzyk et al. [22], the concentration of alloying elements is 2.08% (atomic percent, the same below) in the adopted supercell and 25% at the sliding layer adjacent to stacking fault, while in the work of Qi et al. [2], the concentration of solute atom is as high as 3.3% for the whole system and 33% at the sliding layer adjacent to stacking fault. From Table 1, it can be seen that the results obtained by Muzyk et al. [22] are larger by 20~30 mJ/m 2 than those by Qi et al. [2]. Such a difference may be due to that different mole fractions of alloying elements at the sliding layer adjacent to stacking fault result in different γ SF of Al, which is also verified for Ni based alloys by the recent work of Shang et al. [25]. In addition, in the work of Qi et al. [2] it can be observed that the γ SF values of Al alloyed with Ge and Ti decrease mostly, while the higher γ SF values of Al are from adding Fe and Mn. However, Muzyk et al. [22] reported that alloying elements Fe and Ga lead to higher reduction in the γ SF of Al, while Al alloyed with Mn and Si result in higher γ SF , in which the common alloying element Fe acts on the contrary with that in the work of Qi et al. [2]. In addition, the effects of Mg, Cu, and Zn on the γ SF of Al have been investigated experimentally [26][27][28][29]. The reported concentrations of alloying elements in Al from experiments are 0.65% [26], 0.5% [27], 1.1% [27], and 3.25% [27] for Mg, 0.47% [27] and 0.86% [29] for Cu, and 10% [28] for Zn, respectively. For Al alloyed with Mg, the measured values of γ SF decrease with the increasing concentration of Mg. However, the calculated data deviate from the measured ones to some extent, even though the concentrations of Mg are similar. For instance, the data for 3.25% Mg and 3.33% Mg are far away from each other. For Al alloyed with Cu, the values of γ SF decrease with the increasing concentration of Cu, and there exist some divergences between the calculated results and the measured ones. For Al alloyed with Zn, only measured γ SF by Soliman and Mohamed [28] is 150, which agrees well with the result from the work of Muzyk et al. [22], but the concentrations of Zn in the work of Soliman and Mohamed [28] is as high as 10%. As summarized above, there are too many factors, which can affect the accuracy of results. The divergences in the previously reported data and the lacking of systematical study of γ SF for Al-based alloys motivate the present investigation through firstprinciples calculations. Consequently, the present work aims to investigate the alloying elements (Ag, Be, Ca, Cd, Co, Cu, Cr, Fe, Ga, Ge, Hf, In, K, La, Li, Mn, Mg, Ni, Na, Pb, Sc, Sn, Sr, Si, Ti, V, Zn, and Zr, shown in Table 2), which have been reported as additions in Al alloys [30][31]. More importantly, these results from a systematic study can serve as a guide for future experimental researches as well as computational investigations by using γ SF and corresponding properties.
The remainder of this paper is organized as follows. The method for evaluation of stacking fault energy and the details of first-principles calculations are presented in section 2. The predicted stacking fault energies for Al-based alloys are discussed in section 3. Conclusions obtained in the present work are summarized in section 4.

Prediction of stacking fault energy
Stacking fault energy of an fcc lattice can be evaluated roughly from the energy difference between the hcp and fcc phases [32], or between the hcp, dhcp, and fcc phases [33]. The models to calculate the γ SF have been developed in the literature [11,[34][35][36]. In the present work, the alias shear deformation is selected to study the stacking fault energies because the numbers of atoms employed in the supercell is only half for a similar study compared to other models [11]. For an alias shear deformation, there exist two kinds of deformation schemes: the simple shear and the pure shear [3][4]21], in which the simple alias shear scheme uses much less computational resource. More importantly, the value of γ SF from the simple alias shear is similar to the one from the pure alias shear [4]. Hence, the simple alias shear is adopted in the present work to study the effect of alloying elements on the stacking fault energies of Al-based alloys. < > direction on the {111} plane is the weak direction for shear deformation in fcc lattice, during which the intrinsic stacking fault occurs [11]. For the sake of simplifying the process for first-principles calculations, an orthorhombic cell containing six   Fig. 1. The lengths for a, b, and c of the orthorhombic cell are , and (a fcc is lattice parameter of the fcc cell), respectively. By applying the simple alias deformation along < > direction on {111} plane for fcc lattice, which is along <100> direction on {001} plane in the orthorhombic cell, the lattice vectors R are deformed to , where R is the initial lattice vector of the orthorhombic cell, D is the deformation matrix, and ε is the engineering shear strain, which is the ratio of displacement after strain ε with respect to the height of the orthorhombic cell.
In the present calculations, three kinds of supercell are employed: 1) a 24-atom (1×2×2) supercell (with respect the 6-atom orthorhombic cell) with 6 {111} layers and 4 atoms in each layer; 2) a 48-atom (2×2×2) supercell with 6 {111} layers and 8 atoms in each layer; and 3) a 72-atom (2×3×2) supercell with 6 {111} layers and 12 atoms in each layer. Alloying element X is placed on the top layer of each supercell, which is the sliding layer adjacent to stacking fault. Correspondingly, the alloy compositions are Al 23 X, Al 47 X and Al 71 X, respectively. The compositions of X in these three supercells are therefore 1/4, 1/8 and 1/12, respectively, within the sliding layer adjacent to stacking fault. The overall mole fraction y of alloying element X can be derived from the mole fraction within the sliding layer adjacent to stacking fault x, y = x/6 (5)

First-principles calculations details
All first-principles calculations in the present work are based on the density functional theory (DFT), as implemented in the Vienna ab initio simulation package (VASP) code [37]. The projectoraugmented-wave (PAW) method [38] is used to describe the electron-ion interaction and the generalized gradient approximation (GGA) by Perdew-Burke-Ernerholf (PBE) [39] is adopted to describe the exchange-correlations functional. The recently recommended electronic configurations of pure elements are employed in the present work, as seen in Table 2. According to the series of tests, a 400 eV cutoff energy, 10×8×3 k-points mesh for Al 23 [40] is adopted, based on which the force acts on atoms where they are relaxed to at least -0.01 eV/Å. The Methfessel-Paxton algorithm [41] and the linear tetrahedron method including Blöchl corrections [42] are adopted for structural relaxations and the final energy calculations, respectively. For magnetic elements (Ni, Mn, Fe, Co, and Cr), the spin-polarized approximation is adopted [3].

Results and discussions
In this section, we present and discuss the static properties for Al 71 X, Al 47 X, and Al 23 X at 0 K without considering zero point vibrational energy (ZPE). The targeted properties include the equilibrium volume (section 3.1) and the stacking fault energy (section 3.2). Table 3 summarizes the equilibrium volumes (V 0 ) at 0 K, the relative volumes ( ) for the initial and the stacking fault structures of Al 71 X, Al 47 X, and Al 23 X. It is worth mentioning that the presently predicted V 0 for fcc Al agrees well with the data in the literature [43][44][45][46][47][48]. For example, the presently computed V 0 of 16.497 Å 3 /atom is in excellent agreement with the available theoretical and the experimental value of 16.49 Å 3 /atom [43,46]. As shown in Table 3, in comparison with the initial structures, the structures with stacking fault possess slightly larger volumes with the exception of Al 23 Cr, Al 47 Fe, and Al 23 Zr. Addition of alloying elements usually affect the behavior of charge density redistribution, which may lead to the unusual changes of volumes [3].

Equilibrium volume
To observe the systematic change for V 0 of series of dilute Al alloys, as shown in Table 3, the equilibrium volumes of Al 71 X are plotted in Fig. 2. It can be observed from Fig. 2 that all of the fourth and the fifth period alloying elements (the fourth and fifth line in Table 2) considered in the present work increase the equilibrium volumes of Al 71 X. However, alloying elements K, Ca, Sc, Ga, and Ge of the third period (the third line in Table 2) can increase the equilibrium volumes of Al 71 X, whereas Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn decrease them. For the first and the second period alloying elements (the first and second line in Table 2), Na and Mg increase the equilibrium volumes of Al 71 X slightly, while Si, Li, and Be act inversely. The biggest increase of equilibrium volume comes from the fourth and fifth period alloying elements. With increasing atomic number of alloying elements in each row, the equilibrium volumes of Al 71 X decrease firstly and then increase, except for the first and second period alloying elements which are only a few for these two periods. The same tendency of V 0 can be found for Al 47 X and Al 23 X.
To facilitate analysis, calculated variation of volume (V) relative to alloying elements (X) for the dilute Al-X alloys is fitted according to the equation below, where k V is a fitting parameter, and x is the mole fraction of alloying element X. V 0 (Al) of 10.497 Å 3 /atom is the volume of Al. This value is selected from Al 71 Al, as shown in Table 3, the equilibrium volumes for Al 71 X, Al 47 X and Al 23 X are all employed in the linear fittings. The negative value of k V demonstrates that the equilibrium volume decrease with the increasing concentration of alloying element, while the positive value implies that the alloying element acts the opposite way. From Table 3, it is seen that the largest k V is caused by alloying element Sr, followed by K and La. Combined with Table 2, it can also be concluded that the radius of alloying element smaller than that of Al usually leads to the positive k V , while the radius larger than that usually leads to a negative value of k V , in spite of Ga, Ge, Li, and Ti, whose radius is adjacent to that of Al. The fitted errors, used to reflect the fitting degree between the fitting data and the real values, are also shown in Table 3. For this fitting error, the closer that this value is to 1, the better the linear fitting is. For pure fcc Al, the fitted kv should be zero, and the fitting error should be close to 1. The non-zero value of kv and the error deviated from 1 indicate the systematic errors of first-principles calculations.

Stacking fault energy
Predicted stacking fault energies (γ SF ) for Al 71 X, Al 47 X and Al 23 X at 0 K are listed in Table 4. The relative stacking fault energy (△γ SF ), which is the energy difference between Al 71 X and Al 71 Al, Al 47 X and Al 47 Al, or Al 23 X and Al 23 Al, are also given in Table 4. The stacking fault energies are estimated at the average volume of the initial structure and the one with stacking fault. The present stacking fault energies predicted using Al 23 Al, Al 47 Al and Al 71 Al are 131.2, 130.8 and 101.7 mJm -2 , respectively, as shown in Table 4. By comparing with the reliable measured data, 135±20 [19] , 166 [18] , 150±40 mJm -2 [17], and the predicted values 120 [49], 122 [13], 124 [13], 134 [13], 130 [20], 142 [16], 146 [12], 158 [21], and 162 mJm -2 [22] discussed above, the present results of fcc Al from Al 23 Al and Al 47 Al lie in the error range of the reliable experimental data, and also agree well with Q. Gao (10)) due to the effect of alloying element X are also listed, as well as the corresponding fitting errors (err) are also listed. All quantities for volume are in Å 3 /atom most of the theoretical data. The value of Al 71 Al is a little lower than most of the experimental and theoretical data, which is mainly due to the complexity of Al [2] and the systematic error of the 72-atom supercell which is supported by the previous discussion of V 0 (as shown in Table 3).
Considering the experimental data of Al alloyed with Mg, Cu, and Zn in reference [26][27][28][29], it is believable that the concentration of alloying elements impose some extent of impact on the γ SF of Al. The concentrations of Mg, Cu, and Zn in the present study are all different from the ones in available experiments. In addition, there are too many factors that can affect the accuracy of results as described in the introduction section, so there existing some divergences between the results from the present work Q. Gao [14]; b Woodward et al. [13]; c Kibey et al. [20]; d Hirth et al. [16]; e Brandl et al. [12]; f Ogata et al. [21]; g Dillamore et al. [15]; h Smallman et al. [19]; i Murr et al. [18]; j Mills et al. [17]; k Qi et al. [2]; l Muzyk et al. [22]; m Soliman et al. [27]; n Chaudhury et al. [29]; o Kritzinger et al. [26]; p Soliman et al. [28] and the experiments are reasonable. When compared with the previous calculated data by Muzyk et al. [22], the present results with 2.08% X (X = Mg, Ga, Zn, Si, Cu) are 15-25 mJm -2 lower. Although we adopted the same content of alloying elements (2.08%) as Muzyk et al. [22] in the present work, different models (alias shear in the present work and slab model in the work of Muzyk et al. [22]) and different content of alloying elements at the sliding layer adjacent to stacking fault (12.5% in the present work and 25% in the work of Muzyk et al. [22]) are adopted. Considering Al 23 X with the same concentration 25% X at the sliding layer adjacent to stacking fault as that in work of Muzky et al [22], the present predicted results are about 20 mJm -2 lower. This is mainly because of the different concentration of alloying elements in the whole system (4.17% in the present work and 2.08% in the work of Muzyk et al. [22]) and the different models. The situation between the present work and Qi et al. [2] is the same as that between the present work and experiments, which is due to the different concentration of alloying elements (1.39%, 2.08%, and 4.17% in the present work and 3.3% in the work of Qi et al. [2] for the whole systems, the corresponding concentrations of alloying elements at the sliding layer adjacent to stacking fault are 8.3%, 12.5%, and 25% in the present work and 33% in the work of Qi et al. [2]) are adopted. Figure 3 shows the comparisons between the present results of relative stacking fault energy for Al 23 X with respect to the results evaluated by Muzyk et al. [22]. In both works, the mole fraction of alloying element at the sliding layer adjacent to stacking fault is 25%. Figure 3 shows that the present relative stacking fault energy for Al 23 X agree well with those from Muzyk et al. [22].
Stacking fault energies of Al 71 X are illustrated as a function of atomic number of the alloying elements ( Fig. 4(a)) and equilibrium volume of Al 71 Al ( Fig.  4(b)). Fig. 4(a) also shows that the alloying elements Na, Si, K, Ca, Sc, Ga, Ge, Sr, Zr, In, Sn, La, Hf, and Pb decrease the γ SF of Al, while the other alloying elements act the opposite way. It is also shown that alloying elements at both ends of each period decrease the stacking fault energy of Al, while the elements in the middle of each period tend to increase it. For all the present alloying elements which decrease the stacking fault energy of Al, the further the element from Al on the periodic table, the larger decrease of the stacking fault energy of Al. Taking the far left alloying elements on each period as an example, the stacking fault energy of Al alloys follows the following trend: Al 71 La < Al 71 Sr < Al 71 K < Al 71 Na < Al 71 Li. The largest γ SF decreases of Al for the second, the third, the fourth, and the fifth period alloying elements are Si, K, Sr, and La, respectively. Fig. 4 Fig. 4, it also can be seen that the alloying elements which decrease the stacking fault energy of Al greatly are adjacent to each other, which is similar to the alloying elements which decrease slightly the stacking fault energy of Al. For a better understanding, the relative stacking fault energy values of Al 71 X, Al 47 X, and Al 23 X, as shown in Table  4, are plotted in Fig. 5. From Fig. 5, it is seen that with increasing concentration of alloying element, the effect of Li on the first period, Mg on the second period, V and Ti on the third period, and Cd on the fourth period transfer from increasing the stacking fault energy of Al to decreasing it. The results of first-principles computations described here verify the dependence of stacking fault energy on the presence of alloying elements. It is found that alloying elements of La, Sr, K, Pb, Ca, Sn, Zr, Hf, Ge, Sc, In, Si, Na, and Ga can reduce the stacking fault energy of Al with 1.39% alloying elements in the whole system. With increasing concentration of alloying elements, Li, Mg, V, Ti, and Cd turn to reduce the stacking fault energy of Al as well. High strength and good ductility are important properties for structural materials. This finding will have an implication for the design of Al alloys, which generally exhibit low ductility due to the lack of efficient strengthening mechanisms.

Conclusion
Stacking fault energy for the dilute Al-X alloys (Al 71 X, Al 47 X, and Al 23 X) has been calculated through first-principles calculations in the frame of the alias shear deformation. Twenty-nine alloying elements X (X = Al, Ag, Be, Ca, Cd, Co, Cu, Cr, Fe, Ga, Ge, Hf, In, K, La, Li, Mn, Mg, Ni, Na, Pb, Sc, Sn, Sr, Si, Ti, V, Zn, and Zr) are considered in the present work. The following conclusions are obtained.
Within the considered concentration, alloying elements Na, Si, K, Ca, Sc, Ga, Ge, Sr, Zr, In, Sn, La, Hf, and Pb decrease the stacking fault energy of fcc Al, while the other alloying elements increase it when the concentration of alloying elements is 1.39 at. % in the whole system. With increasing concentration of alloying elements, Li, Mg, V, Ti, and Cd turn to reduce the stacking fault energy of Al.
The alloying elements at both ends of each period (except for the first period which just contains two alloying elements) tend to decrease the stacking fault energy of Al, while the elements in the middle range tend to increase it. For instance, Na and Si at both ends of the second period alloying elements, Ga, Ca,  (see table 3 ) Sc, Ga, and Ge at both ends of the third period alloying elements, Sr, Zr, In, and Zn at both ends of the fourth period alloying elements, and La, Hf, and Pb at both ends of the fifth period alloying elements decrease the stacking fault energy of Al, while the other elements, such as Mg, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ag, and Cd in the middle part of each period increase it. For all the alloying elements which decrease the stacking fault energy of fcc Al, the further the element from Al on the periodic table is, the larger decrease of stacking fault energy.
It is also found that the stacking fault energy of Al-X generally decreases with the increase of equilibrium volume.