Generalized Stacking Fault Energies of Aluminum Alloys–Density Functional Theory Calculations

: Generalized stacking fault energies of aluminum alloys were calculated using density functional theory. Stacking fault energy of aluminum alloys was correlated with the d-electrons number of transition metal alloying elements. The tendency to twinning is also modiﬁed by the presence of the alloying element in the deformation plane. Our results suggest that Al alloys, with such elements as Zr, Nb, Y, Mo, Ta, and Hf, are expected to exhibit a strong work hardening rate due to emission of the partial dislocations.


Introduction
High strength and ductile low-density materials are required for lightweight design in the future.The high strength of alloys might be caused by the nanostructure of the materials [1].Nanocrystalline materials (NCMs) usually exhibit high tensile strength due to grain-size hardening, but this is achieved at the expense of ductility and toughness [2].Ductility of NCMs can be increased by partial dislocations emission and deformation twins [3].Karim poor et al. have demonstrated that for the nanocrystalline, Co, ductility increases due to mechanical twinning [4].Deformation twins are widely observed in low stacking fault energy metals, i.e., Cu and Cu-based alloys [2,5].High stacking fault energy metals are deformed by full dislocation emission, and ductility is low due to the extremely low rate of accumulation of dislocations in the nanosized grains [6].This is because, for a grain size below a certain critical value, dislocation accumulation is impossible [7].Aluminum has high stacking fault energy (SFE).As such, aluminum shows a tendency to cross-slip dislocations and has a relatively small work hardening rate [8].
Tadmor and Bernstein showed that the tendency to partial dislocation emission and twinning in a face-centered cubic (fcc) lattice depended on stacking fault energy (SFE), unstable stacking fault energy (USFE), and unstable twinning energy (UTE), which may be modified by alloying elements.However, experimental measurement of USFE and UTE is impossible [9].Practical realization of this idea requires atomic-scale calculations of the generalized stacking fault energies (GSFE) for the specified alloy.Such estimates can be currently obtained using computational methods [10,11].
In the present study, we used density functional theory calculations to analyze the possible effect of alloying elements on GSFE of aluminum alloys.This study presents the results of GSFE calculations for other metallic and non-metallic elements in the aluminum matrix.Arelationship between SFE and thenumber of d-electrons in transition metals was found.Recent progress in experimental studies on aluminum alloys shows that the deformation mode may be effectively managed by alloying elements [17,24].The elements, which significantly promote partial dislocation emission and deformations twins in Al alloys, were selected.GSFE calculations of aluminum alloys may be useful for material design.

Materials and Methods
Calculations were based on density functional theory (DFT) as implemented in the Vienna Ab-initio Simulation Package (VASP) [25,26].GSFE calculations were performed using projector augmented wave potentials and generalized gradient approximation, with the electron exchange-correlation described by Perdew-Burke-Ernzerhof (PBE) parameterization [27,28].Fermi smearing of electronic occupancy with 0.2 eV and plane-wave cut-off energy of 400 eV were used.The Monkhorst-Pack scheme was used to sample the Brillouin zone with an 8 × 8 × 1 k-point mesh.The ionic relaxation was terminated when energy convergence was greater than 10 −4 eV.GSF energies were computed by using a periodic supercell consisting of twelve (111) planes with a 2 × 2 slab geometry which gives 48 atoms in a system.Energies were calculated for configurations obtained by displacing one part of the crystal on the other along a ‹112› direction on a (111) plane.GSFE was calculated by displacing parts of the model in two operations.In the first process, one-half of the crystal was displaced (1-6 layers of a supercell).In the second step, the second part of the crystal was displaced in the opposite direction (8-12 layers of a supercell) (see Figure 1).Calculations of GSFE were carried out for Al and systems with one atom of Al in the slip plane being replaced by the alloying element.Taking into account the size of a supercell, the concentration of alloying elements was about 2% and about 25% in the slip plane.GSFEs may have been overestimated, in comparison to real materials, due to the high concentration of the alloying elements in the slip plane.More detailed calculations considering different impurity concentration (a supercell with 3 × 3 and a 4 × 4 slab geometry in the slip plane) are necessary in the future.The vacuum space (6 Å), and relaxation of atom positions perpendicular to free surface were applied.Cell shape and volume were kept constant during relaxation.

Results and Discussion
Obtained values of GSFE are plotted in Figure 2 as a function of displacement along the ‹112› direction.The first maximum on the energy plot determined USFE, and the next minimum is the intrinsic SFE at displacement equal to a0/6 1/2 .The second maximum on the GSFE curve is UTE, and the following minimum defined the twin formation energy (TE).It should be noted that SFE and TE could be determined experimentally while USFE and UTE can only be obtained by atomistic scale calculations.The unstable energies (USFE and UTE) are the maximum energies on the GSFE curve, which were required to shift one part of the crystal on to the remaining one.The curves indicated that the alloying Al, at the analyzed concentration, changed the GSFE.Results of computations are summarized in Table 1 which lists computed values of USFE, SFE, UTE, TE, ratios of SFE/USFE and UTE/USFE.Results for Mg, Ga, Zn, Si, and Cu alloys have been previously reported [13].The SFE of Al is calculated as 162 mJ/m 2 .SFE is reduced by a series of non-transition metals: Na, Mg, Li, Si, Ga, Ge, and Te.The lowest value of SFE, in alloyed Al by the analyzed non-transition metals, is 20 mJ/m 2 (Te).USFE and UTEdecreased by all analyzed non-transition metals, except Mg.
The main goal of this study was focused on fault formation in alloying Al by transition metals.The results of SFE calculations for transition metals are presented in Figure 3a where the black line indicates the SFE of pure Al.SFE was reduced by adding transition metal alloys: 3d (Sc, Ti, V, and Cu), 4d (Y, Zr, Nb, Mo, and Ag), and 5d (Hf, Ta, W, and Au).The lowest energies are for elements from the beginning of transition metals series (Ti, Zr, and Hf).The lowest value of SFE equal 25 mJ/m 2 is for Y. Liu et al. have found that SFE is lower by Y and Sc [20] The highest energies are for elements close to the middle of the series (Fe, Ru, and Os).SFE revealed a specific parabolic trend.For the addition of alloy with a small number of valence electrons, SFE was lower for the Al alloy than for pure Al.The lowest value is for Y with three valence electrons.SFE increased with the increasing number of valence electrons.For metals with six valence electrons, SFE is close to the value of Al.Maximum energy is observed for impurity additions with eight and nine valence electrons, and the trend reduced for metals with ten and eleven valence electrons.We also observed the specific trend on the other two analyzed values: USFE (Figure 3b) and UTE (Figure 3c).However, the maximum is shifted to the elements with alower number of valence electrons than in the case of SFE.The maximum of USFE is observed for the addition of alloys with seven valence electrons (Mn, Tc, and Re).The minimum of USFE is observed for the alloying Al with a low number of valence electrons.The trend of UTE is similar to the one for USFE.The maximum is observed for the addition of alloys with seven valence electrons.
As mentioned above, SFE exhibited a specific trend (Figure 3a).To explain the trend we performed additional calculations.As presented on Figure 1b, around the stacking fault there was a ...BCBC... atom arrangement, and that arrangement was anhexagonal-close packed (hcp)-likesequence.We compared the energy difference between Al and Al-transition metal alloys in thefcc and hcp structures.In both cases (fcc and hcp) the supercell with 48 atoms and 12 layers was used.In fcc ABCABC (see the structure on Figure 1a) and in hcp-BCBCBC atom sequences were used.In the sixth layer the impurity atom was substituted.Hcp-fcc energy difference in Al is about 29 meV/atom.If hcp-fcc energy difference for selected alloy is lower than 29 meV/atom, it means that the impurity stabilizes hcp structure.When comparing the Figures 3a and 4 we see that the trend is very similar.The impurity, which decrease the hcp-fcc energy difference (Y, Sc, Ti, Hf, Zr, Ta,Nb, V, W), also decrease the SFE.The accuracy of the relationshipis visible for metals with six valence electrons (Cr, Mo and W), in which energies are close to Al line (Figures 3a and 4).The SFE of the Al-Cr alloy is a little bit above the SFE of pure Al, as in the case of hcp-fcc stability of the Al-Cr system.However, for full filled d-shell atoms (Cu, Ag, and Au) the relationship is not exact.Energy difference is above the Al line, but SFE is lower for Al-X (X = Cu, Ag, Au) than for the Al.The deformation mechanism and twinning tendency expect to depend on the values of SFE/USFE and UTE/USFE [29,30], respectively.SFE/USFE value close to 1.0 imply that deformation by full dislocation emission is preferred.If this parametersis lower than 1.0, then partial dislocations are observed.Twins tendency increases with decreasing UTE/USFE ratio.The ratios of SFE/USFE are included in Table 1.The results obtained in this study are presented in Figure 5 in the context oftendency to the emission of partial dislocations.The value of SFE/USFE for Al is 0.862.Alloy addition, which decreased the value, promoted partial dislocation emissions of alloying Al.The overwhelming majority of analyzed elements decrease the SFE/USFE ratio, except Si, Pd, Ag, Pt, Au, and all analyzed post-transition metals: Cd, In, Sn, Sb, and Pb.The SFE/USFE ratio is presented in Figure 5.It can be seen that alloy additions are ordered in almost the same way as in the periodic table.The elements from the beginning of the period reduce the SFE/USFE ratio to very low values, i.e., Zr to 0.376, which is a very low value in comparison to Al (0.862).It can be noted that the fifth-period elements: Y, Zr, and Nb reduce the SFE/USFE ratio to the lowest values, thus preventing the cross-slip of dislocations.The obtained results showed that alloying with i.e.Sc, Y, Ti, Zr, Hf, V, Nb, and Ta is expected to bring a significant increase in the tendency to the formation of partial dislocations.As in the cases of SFE, USFE, and UTE, the SFE/USFE ratio are the lowest one for the alloying Al by transition metals with a low number of valence electrons.Thisis another specific trend observed in the Al alloys.The tendency to twinning is analyzed by the UTE/USFE ratio (see Table 1 and Figure 6).It can be seen that there is no noticeable trend for the UTE/USFE ratio.The lowest values were for Na and Mg.The twinning tendency of Al-Mg alloys has already been confirmed by experimental results.Velsco and Hodge have found that the fraction of twinned grains increases from 40% to 70% in magnetron sputtered thick films (>10 μm) of Al and Al-5.3%wt.Mg alloy, respectively [24].Lipinska at all.found that the mechanism of grain refinement in Al-0.57%wt.Mg-0.43%wt.Si alloy is distinctly different from that in pure aluminum, with the grain rotation being more prominent than the grain subdivision due to lower stacking fault energy and the reduced mobility of full dislocations in the alloy [29,31].Other experimental work has found that Mg and Si concentration in Al impact on the deformation mechanism and grain size in aluminum alloys [15,17,30,32].A recent study on Al/Ti composites has showed that the Al/Ti interphase boundaries significantly promoted the generation of dislocation during high-pressure torsion deformation [33].This agrees with our prediction that Ti is one of the most promising elements from the point of view of twin ability of Al alloys (Figure 6).

Conclusions
All analyzed transition metal alloy additions increased the UTE/USFE ratio.In our previous study of GSFE for Mg alloys, it has been found that the UTE/USFE ratio for Mg and Mg alloys was mainly in the range of 1.20 to 1.33 [14].In the case of Al alloys, the range is from 1.2 to 2.0.However, most of the elements increased the UTE/USFE ratio above the value of non-alloyed Al.Some specific trends were observed in alloying Al.The SFE, USFE, and UTE of alloying Al were correlated with the number of valence electrons of alloy addition.Furthermore, the SFE/USFE ratio was also correlated with the number of valence electrons.Our systematic calculations showed that transition

Figure 1 .
Figure 1.Schematic explanation of the model used in the computations: (a) Initial configuration separated from the rest by a "vacuum slab" 0.6 nm thick; (b) configuration imposed by the first

Figure 2 .
Figure 2. Generalized stacking fault energies(GSFEs) of selected aluminum-transition metal alloys.3d, 4d and 5d elements are plotted from the bottombox to the top box, respectively.GSFEs of thesystem without the alloying element (Al) are shown on each plot for comparison.

Figure 3 .
Figure 3. (a) Stacking fault energy (SFE), (b) unstable stacking fault energy (USFE), and (c) unstable twinning energy (UTE) of aluminum-transition metal alloys.The black line on each plot is a value of considered energy of pure Al.The circle, triangle and square denote 3d, 4d and 5d elements, respectively.All energies are in mJ/m 2 .

Figure 4 .
Figure 4. Stability of hexagonal-close packed hcp vs. face-centered cubic (fcc)aluminum-transition metals alloys.The supercell contained 47 Al atoms and one impurity atom.In the case of the fcc structure, the atoms were arranged as shown on Figure 1a.In the case of the hcp structure, a BCBCBC... atom arrangement was used.The supercell volume and shape was identical in hcp and fcc cases (48 atoms).The black line indicates energy difference between hcp and fcc structures of Al.Transition metals impact edon hcp-fcc stability in the Al alloys.Y, Sc, Ti, Hf, Zr, Ta, Nb, V, W, and Mo decreasing the stability of Al-fcc vs. hcp ones.The shape of the curve was very similar to stacking fault energies (Figure3a).The same transition metal impurity, which decreased the fcc stability, decreased the stacking fault energy of the Al alloys, except Cu, Ag and Au.

Figure 5 .
Figure 5. Stacking fault energy (SFE) to unstable stacking fault energy (USFE) ratio for analyzed aluminum alloys.The position of the analyzed alloy indicates SFE/USFE value.

Figure 6 .
Figure 6.Unstable twinning energy (UTE) to unstable stacking fault energy (USFE) ratio for analyzed aluminum alloys.The positionof the analyzed alloy indicates the UTE/USFE value.

Table 1 .
[12]ulated values of unstable stacking fault energy (USFE), stacking fault energy (SFE) and unstable twining energy (UTE) for Al and their alloys.Results for Al alloys with Mg, Ga, Zn, Si and Cu are from our previous calculations[12].The most reduced SFE/USFE and UTE/USFE values are underlined.All energies are in mJ/m 2 .