Temperature-Dependent Generalized Planar Fault Energy and Twinnability of Mg Microalloyed with Er , Ho , Dy , Tb , and Gd : First-Principles Study

The generalized planar fault energies, Rice criterion ductility, and twinnability of pure Mg and Mg-RE (RE = Er, Ho, Dy, Tb, and Gd) alloys at different temperature have been investigated using density functional theory. It is shown that all the fault energies and twinnability in the same materials decrease with increasing temperature. However, the ductility has the opposite change trend. On the other hand, alloying rare earth elements will generally decrease the fault energies and increase the ductility and twinnability of Mg at different temperature. It is interesting to note that alloying larger atomic radius will enhance the ductility of Mg more easily and alloying smaller radius will make twinning tendency of Mgmore easily. Finally, the electron structure further reveals the underlying mechanisms for the reduction of fault energies with the addition of rare earth elements.


Introduction
Magnesium alloys have become increasingly attractive in the fields of microelectronics, aerospace, and automotive industries during the past two decades due to light weight and high specific stiffness [1].However, the potential use of magnesium in wrought forms, such as sheets and extrusions, is quite limited because in hcp magnesium a sufficient number of independent slip systems are in general not readily available.As a result, magnesium's poor ductility is most commonly rationalized in terms of its possession of only one close-packed plane called basal plane.Recently, introducing nanoscale twins has been proved to be useful for enhancing the ductility and fracture toughness of metals.Therefore, activation of deformation twinning plays an important role in the plastic deformation of Mg alloys [2,3].Deformation twinning can be considered to be a result of the fact that partial dislocations glide on successive slip planes.The intrinsic stacking fault energy has been traditionally used to describe the ease with which a metal plastically deforms by twinning in competing with dislocation-mediated slip.
However, the deformation mechanism in metals cannot be explained by means of the absolute value of intrinsic stacking fault energy as described by the molecular dynamics (MD) simulations [4][5][6][7].Furthermore, MD simulations [4,5] and mechanical models [8] reveal that generalized planar fault energy (GPFE) curve plays a critical role in the relevant deformation mechanism, especially the competition between twinning partial nucleation and trailing partial nucleation.
Tadmor and Bernstein have shown that the possibilities of mechanical twinning and dislocation-mediated slip in facecentered-cubic (fcc) metals depend significantly on the three typical energies: intrinsic stacking fault energy ( sf ), unstable stacking fault energy ( us ), and unstable twinning energy ( ut ) [6,7,[9][10][11].In general, the three typical energies can be modified by addition of alloying atoms because alloying atoms could disturb the local environment of matrix atoms such as strain field and charge density [12][13][14][15][16].For example, Han et al. [17] have examined alloying effects due to Li and Al in terms of generalized stacking fault energies associated with basal planes of Mg; their results show that Li alloying can facilitate dislocation-mediated processes while Al alloying 2 Advances in Materials Science and Engineering shows an opposite trend.Muzyk et al. [18] calculated the generalized stacking fault energies of Mg-based systems alloyed with Ag, Al, Cu, Fe, Li, Mn, Ni, Pb, Sn, Ti, Y, Zn, and Zr; they indicate that alloying with Y is expected to increase the tendency for mechanical twinning.Kwasniak et al. [19] investigated the deformation tendency of pure -Ti and -Ti alloys with C, H, O, and N additions; their results suggest that H addition decreases and C increases twin generation comparing with pure -Ti.Shang et al. [20,21] have studied the twinnability of dilute Mg-based alloys Mg 95 X (X = Al, Ca, Cu, La, Li, Mn, Sc, Si, Sn, Sr, Ti, Y, Zn, and Zr) in terms of firstprinciples calculations; they find that alloying elements Sr and La increase greatly the twinning propensity of Mg, while Mn, Ti, and Zr show opposite trends.
Obviously, all the above theoretical investigations about the effects of solute atoms on twinnability of Mg alloys are confined at 0 K.There is lack of investigation of temperature effects on the twinnability and other mechanical properties of pure Mg and Mg alloys [22,23].In practice, useful improvements in ductility are seen with elevated temperature, solute additions, and finer sizes [24].In this paper, the temperature effects on the GPFEs and twinnability of pure Mg have been studied systematically using a first-principles quasiharmonic approach.Recently, the rare earth metals (RE) and transition metals as the alloying elements are widely used in the Mg alloys.The rare earth elements especially are a class of special elements which are known to be of major significance for improving the mechanical properties such as ductility, creep resistance, and casting characteristics [25][26][27][28][29][30].For example, Gao et al. [26] investigated the effects of rare earth elements Gd and Y on the solid solution strengthening of Mg alloys by using both the hardness and the tensile tests; they found that solid solution strengthening by Gd and Y is much higher than the effect of Al and Zn.Zhang et al. [30] studied systematically the effects of RE (RE = Pr, Nd, Gd, Tb, and Dy) solute atoms on the stacking faults of Mg solid solutions using density functional theory; they found that the ductility of Mg is improved by the addition of RE atoms.Zhang et al. [31] employed a series of supercells to calculate the stable fault energies of basal stacking faults I 1 in various categories of Mg-based binary alloys; their results showed that the rare earth elements Er, Ho, Dy, Tb, and Gd decrease the stacking fault energy and are of great significance and effectiveness for improvement of the mechanical behaviors of Mg.Therefore, the GPFEs of Mg microalloyed with Er, Ho, Dy, Tb, and Gd are also investigated.

Computational Method
2.1.Details of First-Principles Calculations.The ab initio calculations are based on density functional theory (DFT) as implemented in the Vienna ab initio simulation package (VASP) code developed at the Institut für Materialphysik of Universität Wien [33][34][35].The ion-electron interaction is described by the projector augmented wave (PAW) method [36,37] and the exchange-correlation functional is described by the Perdew-Burke-Ernzerhof (PBE) [38,39] generalized gradient approximation (GGA).The cut-off energy of plane wave is chosen to be 500 eV for all the calculations.The recommended electron configuration by VASP is employed for each element in the present work, such as 14 electrons (4f 12 6s 2 ) used for the valence of Er, 13 (4f 11 6s 2 ) for Ho, 12 (4f 10 6s 2 ) for Dy, 11 (4f 9 6s 2 ) for Tb, 10 (4f 7 5d 1 6s 2 ) for Gd, and 8 (2p 6 3s 2 ) for Mg.Note that semicore p states are treated as valence electrons for Mg.The -centered Monkhorst-Pack scheme [40] is performed to generate a uniform grid of points along the three axes in reciprocal space.A 21 × 21 × 21 sampling grid is used to optimize the lattice constants of the unit cell of Mg.The optimized lattice parameters / = 1.625 with  = 3.195 Å without considering zero-point vibrational energy in this work are in excellent agreement with experimental values (/ = 1.624 with  = 3.209 Å) [41].In calculation of generalized planar energies of Mg-RE alloys at different temperature, periodic supercell of 3 × 3 unit cells with 12 layers is used.An ∼15 Å vacuum region is included along the cell's long -axis to avoid interactions with periodic images [42].For this periodic supercell, Brillouin zone sampling is performed using 9 × 9 × 2 special points mesh.The total energy calculation is performed using linear tetrahedron method with Blöchl et al. correction [43].The energy convergence criterion of the electronic self-consistency is chosen as 10 −6 eV, while the Hellmann-Feynman force on all atomic sites is less than 10 −4 eV/ Å for all calculations.All calculations are performed with the PREC = accurate in INCAR.In order to investigate the effect of spin polarization on the generalized planar energies of Mg-RE alloys, we have also compared the results obtained from the spin-polarized  sf and the nonpolarized one for fixed temperature.For example, we find that  sf at 0 K are 17.82 and 17.80 mJ/m 2 for magnetic and nonmagnetic MgEr, respectively; that is to say, the difference between the spin-polarized and the nonpolarized calculations is negligible.Therefore, the temperature-dependent generalized planar energies of pure Mg and Mg-RE (RE = Er, Ho, Dy, Tb, and Gd) alloys are performed within the nonpolarized calculations in this work.

Temperature-Dependent Properties.
In order to calculate the temperature-dependent generalized planar energies of Mg and Mg-RE alloys, we need the equilibrium volume  at the given .The Helmholtz free energy (, ) at temperature  and constant volume  based on the quasiharmonic approach can be expressed as where  static () is the static energy obtained from firstprinciples calculations directly and  ph (, ) is the phonon free energy arising from the lattice vibrations.Within the quasiharmonic approach, where ℏ is the reduced Planck constant,   is the Boltzmann constant, and   represents an individual phonon frequency.
Here, the sum is overall the phonon frequency in the first Brillouin zone.Phonon frequency calculations are performed by the supercell approach with the force constants calculated Frequency (THz) in the density functional perturbation theory implemented in the VASP code [44].Here, a 4 × 4 × 3 supercell with 96 atoms is employed based on our tests.The forces resulting from displacements of certain atoms in this supercell are calculated by VASP with 5×5×4 -points grid meshes.The other settings of VASP calculations are the same as described above.The phonon frequency is obtained by using the PHONOPY [45][46][47] package which can support VASP interface to calculate force constants matrix directly.The phonon dispersion curves for Mg along high-symmetry directions in the Brillouin zone computed with theoretical equilibrium lattice parameters are plotted in Figure 1.Obviously, the spectra are in agreement with those from the inelastic neutron scattering measurement depicted by red solid circles [32].
Based on (1), the equilibrium volume versus temperature relation () is straightforward.We have calculated the static energy  static () and the phonon free energy  ph (, ) at 15 volume points near the theoretical equilibrium volume for Mg.The equilibrium volume  at temperature  is obtained by minimizing Helmholtz free energy with respect to  from fitting to the integral form of the Vinet equation of state (EOS) [48].Figure 2(a) displays the values of Helmholtz free energy as a function of unit cell volume of Mg at every 100 K between 0 and 600 K, and the value depicted by the red circle of every fitted curve is the equilibrium volume at corresponding temperature.It is assumed that the ratio of / is 1.625 at different temperature.The thermal expansion is obvious as the equilibrium volume increases (see Figure 2(b)).Because the unit cell of Mg-RE contains 108 atoms, obtaining the phonon dispersion of typical Mg-RE needs a considerable amount of calculation.And the doping concentration is very low, so we estimate the fact that the five Mg-based alloys have the same equilibrium volumes at different temperatures with Mg.

Results and Discussion
The radius of Mg is 1.72 Å and the radii of rare earth metals Er, Ho, Dy, Tb, and Gd are 2.45, 2.47, 2.49, 2.51, and 2.54 Å, respectively.Obviously, all the rare earth metal atomic radii are larger than that of Mg.To study the effect of the rare earth metals on the twinnability of Mg-based alloys, the accurate positions of the rare earth metals to the 3 × 3 slab geometry with 12 (0001) atomic planes, corresponding to ∼0.9 at.% concentration of solute (∼11.1 at.% monolayer coverage at the doped plane), are firstly determined with the minimum energy for all the most possible occupation sites.Figure 3(a) displays the perfect atomic configuration of the supercell arranged into a ABAB sequence with 15 Å vacuum in the direction perpendicular to the slip plane.For the Mg-RE alloys, one Mg atom is substituted by a rare earth metal in the 4th, 5th, 6th, 7th, 8th, 9th, and 10th layers, respectively.All the calculated total energies are listed in Table 1.Obviously, we can conclude that the optimum position of the solute atom is in the 7th layer.
The calculated temperature-dependent generalized planar fault energy curves for Mg are shown in Figure 4.The first half of the generalized planar fault energy curves is evaluated by displacing the upper layers numbered 1-6 with respect to the atomic layers numbered 7-12 over a displacement / √ 3 and the stacking fault ABABAB(C)ACACA is created (see Figure 3(b)).The second half is then evaluated by successively sliding layers 8-12 in similar fashion and the deformation twin ABABAB(C)BABAB is generated (see Figure 3(c)).It is well known that deformation twinning in metals and alloys is initiated by preexisting dislocation configurations which dissociate into multilayered stacking fault structures to nucleate a twin.The first and the second maxima in Figure 4 are unstable stacking fault energy  us and unstable twinning energy  ut , respectively, and the first minimum is the stable stacking fault energy  sf .While  sf can be measured experimentally,  us and  ut are difficult to be obtained from experimental measurements and can be determined only through atomic simulation, which represent the minimum energy barriers for partial dislocation nucleation [49] and microtwin nucleation, respectively.All the calculated fault energies ( sf ,  us , and  ut ) at different temperature for pure Mg and Mg-based alloys are summarized in Table 2.The present stable stacking fault energy (33.5 mJ/m 2 ) of Mg at 0 K agrees well with other first-principles results (∼21-44 mJ/m 2 ) [50][51][52][53][54][55][56][57].However, the result (31.1 mJ/m 2 ) at 300 K is lower than the scattered experimental values measured at room temperature (<50 mJ/m 2 [58], 74, 102, 125, and 280 mJ/m 2 [59], and >90 mJ/m 2 [60]).It is worth mentioning that the existing experimental values of stable stacking fault energy in this paper are not reliable since the experimental measurements are often complex and elusive due to the smaller stable stacking fault energy.
The present results of other generalized planar fault energies of pure Mg including unstable stacking fault energy (95.8 mJ/m 2 ) and unstable twinning energy (114.5 mJ/m 2 ) at 0 K are in general agreement with previous theoretical values [18,21,51].Therefore, the calculation of this work should be reasonable and reliable.On the other hand the fault energies at finite temperature of pure Mg and Mg-based alloys are absent both experimentally and theoretically, which are worth verification in the future.From Table 2, it can be seen that all the fault energies of Mg and Mg-based alloys decrease slightly with increasing temperature, since thermal expansion may soften the stacking fault configuration at high temperature.Note that the decrease of fault energies of fcc metals with increasing temperature is predicted previously [20].Besides, it is interesting to find that doping rare earth metals at this concentration will generally decrease the fault energies at different temperature.Moreover, the reduction of  sf is more obvious than that of  us and  ut .For example, the  sf value of Mg at 0 K is 33.5 mJ/m 2 , while in the alloys  sf ranges from 14.4 to 17.8 mJ/m 2 .The size of  sf is decreased by nearly half with the alloying atoms at different temperature.
On the other hand, with the increase of the atomic radius of rare earth metals, the reduction of fault energies for Mg-RE alloys becomes strong.
To examine how alloying and temperature alter the ductility of pure Mg we employ Rice criterion [49].Rice has shown that the conditions for both brittle fracture and the nucleation and motion of dislocations can be simply measured from the ductility parameter  = 0.3  / us [49], where   is the (0001) surface energy of pure Mg or the appropriate Mg alloy, including the effects of surface segregation.If   is small relative to  us then under applied stress the metal will tend to form surface, that is, crack, rather than shearing by dislocation-mediated slip.The   value of pure Mg ranges from 546.17 mJ/m 2 to 548.47 mJ/m 2 in the temperature range 0-600 K and the increment is about 0.4%.Since the temperature and the rare earth metals have little influence on   value of pure Mg, we employ the   value (546.17mJ/m 2 ) at 0 K to calculate the ductility parameters for pure Mg and Mg-based alloys at different temperature.All the ductility parameters  for pure Mg and Mg-based alloys at different temperature are summarized in Table 2.Note that all the values are larger than 1, so Mg and Mg-based alloys are considered to be ductile at different temperature.The smaller  us of Mg-based alloys corresponds to the larger value of ductility parameter, which demonstrates that the crystals tend to nucleate dislocations rather than shear so as to form surfaces, so the ductility would be improved by the rare earth metals.It should be noted that the rare earth elements is of great significance and effectiveness for improvement of the mechanical properties such as ductility, creep resistance, and casting characteristics [25][26][27][28][29][30].Furthermore, the ductility parameter increases with increasing the atomic radius of rare earth metals, which demonstrates that alloying larger atomic radius of rare earth metals will enhance the ductility of Mg more easily.Besides, the ductility of pure Mg and Mg-based alloys increases with increasing temperature, and it results from the fact that  us decrease and   almost unchange with increasing temperature in the same materials.The fault energies ( sf ,  us , and  ut ) play a critical role in the activation of the extended partials and deformation twinning.The deformation mechanism and twinning tendency of the material are expected to depend on the value of  sf / us and  ut / us [4,6,19].Values of these parameters close to 1.0 imply that deformation by full dislocation is preferred.Decrease of these parameters favors deformation by partial dislocation and twinning.All the values of these parameters are listed in Table 2.For all additions  sf / us and  ut / us are smaller than that for pure Mg at different temperature.The results obtained in this work indicate that alloying rare earth metals at this concentration will make deformation by partial dislocations and twinning more easily.In fact, sorting the interstitials investigated here according to their decreasing  sf / us context, we obtain the ranking at different temperature: Er < Ho < Dy < Tb < Gd, which demonstrates that alloying larger atomic radius of rare earth metals will make deformation of partial dislocation more easily.The same elements can be ordered in terms of their decreasing  ut / us at different temperature in the sequence Gd < Tb < Dy < Ho < Er, which indicate that alloying smaller radius of rare earth metals will make twinning tendency more easily.In a word, the deformation mechanism and twinning tendency have the opposite change trend with the radius of rare earth metals at different temperature.On the other hand, temperature is beneficial to promote the dissociation of dislocation into partials more easily since  sf / us of Mg and Mg-based alloys decrease with increasing temperature.However, it can be seen from Table 2 that the  ut / us values of Mg and Mgbased alloys increase slowly with increasing temperature, so it can be concluded that the twinnability decreases slowly with increasing temperature, which is in accordance with the experimental phenomenon that low temperature can promote deformation twinning [61].
In general, variations of the fault energies of Mg and Mgbased alloys can be analyzed qualitatively in terms of the charge density distributions.Owing to the similar influence of rare earth elements on the fault energies, only Dy is taken as an example to be described in detail.Figure 5 illustrates the charge density distribution on (1010) plane of Mg and Mg-Dy alloy for four rigid shear configurations.From Figure 5(a), it can be seen that a high charge density area marked by red boat-like area is adjacent to each Mg atom at the nonfaulted structure (P).During shearing, Mg expels surrounding electrons away from itself and the charge density around it will thus be decreased.Since more electrons have to be redistributed during the formation process of the unstable stacking fault structure (USF), the total energy is increased.With continuous shear, the high charge density area around Mg is further reduced and tries to reach a new stable distribution.Finally, a new and relatively stable stacking fault configuration (SF) is formed.Therefore, the stable stacking fault energy is lower than unstable stacking fault energy.At the unstable twinning (UT) configuration, the high density area adjacent to Mg atom on the near fault plane (L8) becomes smaller and the charge redistribution is most serious.As a result, the unstable twinning energy is higher than the unstable stacking fault energy and the stable stacking fault energy.Figure 5(b) shows the similar situation of charge density distribution of Mg-Dy alloy during the formation process of twinning fault configuration.There is a progressive increase in the valence charge in the vicinity of Dy atom relative to Mg, showing that substitutional rare earth element increases charge at the fault plane.Owing to the enhanced charge density around Dy element, the interaction between atomic layers near the fault plane is stronger during the entire shear process, so all the fault energies are lower in comparison with the case of pure Mg.It should be noted that the charge transfer process is in accordance with Zhang et al. results [30].

Conclusions
In summary, we have employed first-principles approach to calculate the generalized planar fault energies for Mg and Mg-RE (RE = Er, Ho, Dy, Tb, and Gd) alloys as a function of temperature.The results indicate that all the fault energies decrease with increasing temperature and alloying rare earth elements will generally lower the fault energies of Mg at different temperature.Based on the calculated fault energies, the brittle-ductile behavior and the deformation twinning of Mg and Mg-based alloys have been investigated.It is found that the ductility increases and the twinnability decreases with increasing temperature for Mg and Mg-based alloys.Furthermore, alloying rare earth elements will generally increase the ductile and twinnability of Mg at different temperature.It is interesting to note that alloying larger atomic radius will enhance the ductility of Mg more easily and alloying smaller radius will make twinning tendency of Mg more easily.Finally, the electron structure further reveals the underlying mechanisms for the reduction of fault energies with the addition of rare earth elements.

Figure 1 :
Figure 1: Calculated phonon dispersion curves of bulk Mg along high-symmetry directions in the Brillouin zone.The red solid circles show the experimentally measured frequencies [32].

Figure 2 :Figure 3 :
Figure 2: (a) The values of Helmholtz free energy as a function of unit cell volume of pure Mg between 0 and 600 K at every 100 K.The minimum value depicted by red circle of every fitted curve is the equilibrium volume at corresponding temperature.(b) The equilibrium volume of pure Mg as a function of temperature.

Figure 4 :
Figure 4: The generalized planar fault energy curves of Mg at different temperature.The discrete points denote the values of firstprinciples calculations and the solid lines indicate the results from polynomial fitting.

Figure 5 :
Figure 5: The charge densities contour plots of perfect (P) configuration, unstable stacking fault (USF) configuration, stacking fault (SF) configuration, and unstable twinning (UT) configuration for pure Mg (a) and Mg-Dy solid solution (b) on the (1010) plane.The dots mark the fault layer (L7).

Table 2 :
Calculated fault energies (  , mJ m −2 ), ductility parameter ( = 0.3  / us ), and relative barrier heights ( sf / us ,  ut / us ) of Mg and Mg-based alloys in the temperature range 0-600 K by using the supercell of a 3 × 3 slab geometry with 12 atomic planes.