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The thermodynamic stabilities, electronic structures, and mechanical properties of the Pd-based superalloys are studied by first principles calculations. In this work, we discuss the effect of Pd-based superalloys made from Al, Si, Sc, Ti, V, Cr, Mn, Fe, Cu, Zn, Y, Zr, Nb, Mo, Tc, Hf, Ta, W, Re, Os, Ir and Pt, and we also calculate a face centered cubic (fcc) structure 2×2×2 superalloy including 31 Pd atoms and one alloying element TM (Pd31TM). The mixing energies of these Pd-Based superalloys are negative, indicating that all Pd-based superalloys are thermodynamically stable. The Pd31Mn has the lowest mixing energy with a value of −0.97 eV/atom. The electronic structures of the Pd-based superalloys are also studied, the densities of states, elastic constants and moduli of the mechanical properties of the Pd-based superalloys are determined by the stress-strain method and Voigt–Reuss–Hill approximation. It is found that Pd31TM is mechanically stable, and Pd31Tc has the largest C11, with a value 279.7 GPa. The Pd31Cr has the highest bulk modulus with a value of 299.8 GPa. The Pd31Fe has the largest shear modulus and Young’s modulus with the values of 73.8 GPa and 195.2 GPa, respectively. By using the anisotropic index, the anisotropic mechanical properties of the Pd31TM are discussed, and three-dimensional (3D) surface contours and the planar projections on (001) and (110) planes are also investigated by the Young modulus.
Due to the fact that platinum-group-metals (PGMs) have excellent physical and mechanical properties, they have received a great deal of attention from high-temperature applications.[1–7] The PGMs have the same face centered cubic (fcc) structures as Ni-Based superalloys.[8] Pan has studied the structure, electronic and elastic properties of RuAl2 of typical TiSi2-type structure.[9] Among the platinum-group-metals, the melting-point of Palladium is 1837 K, which is higher than those of Ni-based superalloys. With one or more alloy element added, the mechanical properties of Pd-based superalloys have been obviously improved.[10,11] Ning et al.ʼs study showed that the rare earth elements not only improve the room temperature strength and elasticity of Pd, but also increases the high temperature instantaneous strength, creep life and activation energy of Pd.[12] Zhou has studied the microstructure and mechanical properties of Pd–Ag–Cu alloy.[13] Wang et al. showed that the addition of Rh significantly improves the mechanical properties of Pd–Rh–V.[10] To investigate the development of the Pd-based superalloys, it is desirable to gain the effects of alloying elements on Pd-based superalloys. In this work, we investigate the thermodynamic stability, electronic structures, and mechanical properties of the Pd-based superalloys. To understand the thermodynamic stability of the Pd-based superalloys, we calculate the mixing of their energies. We also calculate the total density of states, the partial density of states via first principles calculation, to study the electronic structures of the Pd-based superalloys. The modulus of elasticity measures the ability of the material to recover shape after deformation and provide information of the deformation behavior of the material. The single crystals are anisotropic. The occurrence of micro cracks in materials is always related to anisotropy; therefore, it is necessary to describe the parameter of the anisotropy. In this work, the following 21 alloying elements in Pd-based superalloys are determined: Al, Si, Sc, Ti, V, Cr, Mn, Fe, Cu, Zn, Y, Zr, Nb, Mo, Tc, Hf, Ta, W, Re, Os, Ir, and Pt. We calculate the structural stability, electronic structure and mechanical properties of Pd31TM via first principles calculations.
In this work, we calculate a face centered cubic (fcc) structure 2×2×2 superalloy including 31 Pd atoms and one alloying element TM. The first principles calculation is based on density functional theory (DFT), using the Cambridge Serial Total Energy Package (CASTEP) code.[14,15] The relation between valence electrons and ionic solid was described by the generalized gradient approximation (GGA) of the Perdew–Burke–Ernzerhof for surfaces for solids (PBEsol) approach.[16] The plane wave cutoff energy was 500 eV. In the first irreducible Brillouin zone, the k-point method used the ultrasoft pseudopotential. The exchange correlation energy was described by the GGA combined with the Monkhorsr-Pack method,[17] and a Monkhorst-Pack k-point mesh was 1×1×1 of the Pd31TM. In the geometric optimization process, the total energy change ultimately converges to
In this paper, the dilute solution of the Pd31TM is shown in Fig.
![]() | Fig. 1. (color online) Dilute solutions of Pd31TM (![]() |
The energy parameters can be defined by the following expressions:
For the dilute solution of thermodynamic stability, the mixing energy of Pd31TM is negative, and the lower the negative value, the more stable the dilute solution is. In this work, the density, volume, and mixing energy of the Pd31TM superalloy are listed Table
![]() | Table 1.
Values of denisty, volume, mixing energy ( |
The change of the mixing energy of Pd31TM with the atomic radius increases is shown in Fig.
In this paper, in order to study the chemical bond properties and electronic structures of Pd31TM, the total density of state (TDOS), partial density of state (PDOS), electron density distribution and the electron density difference map of Pd31TM superalloys are calculated.
Figure
In this paper, the elastic constants of Pd31TM are obtained by the stress-strain method based on the generalized Hooke law, then all elastic modulus (
In Table
![]() | Table 2.
Calculated values of elastic constant ( |
The results of elastic constants and moduli of Pd-Based superalloy from calculations and experiments of Pd are listed in Table
In order to further understand the effects of the elements on the elasticity of Pd31TM superalloy, the elastic moduli varying with the density of Pd31TM are plotted in Fig.
In this work, we calculate the mechanical anisotropy of Pd31TM. The anisotropy is related to the micro cracks in a material; therefore, it is necessary to describe the anisotropy of Pd31TM. In order to understand anisotropy, we calculate the universal anisotropic index
![]() | Table 3.
Calculated values of universal anisotropic index |
To better understand the characteristic of elastic anisotropy, we plot the three-dimensional (3D) surface of Pd31TM. The Young modulus is plotted in spherical coordinates. The 3D representation of Young’s modulus for cubic crystal is given by the following equations:[26]
Figure
The thermodynamic stabilities, electronic structures, and mechanical properties of the Pd-based superalloys are studied by first principles calculations. The mixing energies of these Pd-based superalloys are negative, so the Pd31TM (TM = Al, Si, Sc, Ti, V, Cr, Mn, Fe, Cu, Zn, Y, Zr, Nb, Mo, Tc, Hf, Ta, W, Re, Os, Ir, and Pt) are thermodynamically stable. The electronic structures of the Pd-based superalloys are also studied. The bonding characteristics of the Pd31TM superalloys are dominated by metallic bonds. The elastic constants and moduli of the mechanical properties of the Pd-based superalloys are determined by the stress-strain method and Voigt–Reuss–Hill approximation. It is found that Pd31TM are mechanically stable, and Pd31Tc has the largest C11, with a value of 279.7 GPa. The Pd31Cr has the highest bulk modulus, with a value of 299.8 GPa. The Pd31Fe has the largest shear modulus and Young’s modulus, whose values are 73.8 GPa and 195.2 GPa, respectively. In this work, we summarize the systematical theoretical researches of Pd31TM superalloys, and the fundamental physical picture of the properties of the compound can be established, which will be useful for the experimental research and applications of Pd-based superalloys in the future.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] |