Advances in Materials Science and Engineering

Volume 2016 (2016), Article ID 7365906, 9 pages

http://dx.doi.org/10.1155/2016/7365906

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

^{1}Department of Physics, Chongqing Three Gorges University, Chongqing 404100, China^{2}Institute for Structure and Function, Chongqing University, Chongqing 401331, China^{3}College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China

Received 22 June 2016; Revised 30 September 2016; Accepted 11 October 2016

Academic Editor: Jinghuai Zhang

Copyright © 2016 Lili Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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 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.

#### 1. 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 [4–7]. 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 face-centered-cubic (fcc) metals depend significantly on the three typical energies: intrinsic stacking fault energy (), unstable stacking fault energy (), and unstable twinning energy () [6, 7, 9–11]. 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 [12–16]. 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 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 (X = Al, Ca, Cu, La, Li, Mn, Sc, Si, Sn, Sr, Ti, Y, Zn, and Zr) in terms of first-principles 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 [25–30]. 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 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.

#### 2. 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 [33–35]. 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 (46) used for the valence of Er, 13 (46) for Ho, 12 (46) for Dy, 11 (46) for Tb, 10 (456) for Gd, and 8 (23) 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 sampling grid is used to optimize the lattice constants of the unit cell of Mg. The optimized lattice parameters with Å without considering zero-point vibrational energy in this work are in excellent agreement with experimental values ( with Å) [41]. In calculation of generalized planar energies of Mg-RE alloys at different temperature, periodic supercell of 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 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 eV, while the Hellmann-Feynman force on all atomic sites is less than 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 and the nonpolarized one for fixed temperature. For example, we find that at 0 K are 17.82 and 17.80 mJ/ 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.

##### 2.2. 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 aswhere is the static energy obtained from first-principles calculations directly and is the phonon free energy arising from the lattice vibrations. Within the quasiharmonic approach,where 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 in the density functional perturbation theory implemented in the VASP code [44]. Here, a 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 -points grid meshes. The other settings of VASP calculations are the same as described above. The phonon frequency is obtained by using the PHONOPY [45–47] 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].