Pd-based system used in this work includes three alloys: Pd₂SnYb, PdSnYb and PdSn₂Yb. The symmetry group and international table number of Pd₂SnYb are and 216. PdSnYb and PdSn₂Yb are orthorhombic system. Their space groups are PNMA (No. 62) and CMCM (No. 63). In Pd₂SnYb, Pd possesses the 8c site (0.25, 0.25, 0.25), Sn perches the 4a site (0, 0, 0), Yb possesses the 4b site (0.5, 0.5, 0.5). In PdSnYb, atoms of Pd, Sn, and Yb respectively possess the 4c site (0.28675, 0.25, 0.3988), (0.17329, 0.25, 0.07563) and (0.0176, 0.25, 0.69161). In PdSn₂Yb, Pd and Yb perch the 4c site (0, 0.70228, 0.25), (0, 0.42899, 0.25), Sn possesses the 8f site (0, 0.14024, 0.04483). In order to obtain reliable structural optimization results, the lattice constants we employed are all from experiments.

CASTEP code [28] was used for this work, which grounded on the density functional theory [29] . The exchange correlation functional employed the PBE method in the generalized gradient approximation (GGA) [30] . Ultra soft pseudo potential (USPP) [31] was chosen for interaction potential between ionic potential and valence electrons. The atom orbits Pd 4d¹⁰, Sn 5s²5p², and Yb 4f¹⁴5s²5p⁶6s² were considered as valence electrons in the calculation of pseudo potential. The cut-off energy of 450 eV was set for plane waves in the wave-vector K space. For Brillouin regions k-point sampling, the Monkhors-Pack mesh was set as 4 × 4 × 4 [32] . The lattice parameters of Pd₂SnYb, PdSnYb and PdSn₂Yb were optimized successively by using the BFGS scheme [33] [34] [35] [36] . On this basis, the magnetic, alloy corrosion resistance, elastic, thermal conductivity and anisotropy are being computed.

The equilibrium lattice constants of Pd₂SnYb, PdSnYb, and PdSn₂Yb are obtained by geometry optimization with spin polarization. The paramagnetic (NM), ferromagnetic (FM) and anti-ferromagnetic (AFM) coupling between Yb atoms are taken into account in the calculations. Atomic initial magnetic order affects the convergence of ground state. Therefore, the different magnetic orders of Yb atoms are considered to ensure the convergence of ground state. In the condition of different magnetic orders, the curves of the relative energy are drawn out in Figure 1, whose minimum energy is set up to be the ground state (0 eV).

As the Figure 1 shown, the energy of AFM-2 (each layer of Yb atoms spin in the opposite manner along the crystal orientation [001]) in Pd₂SnYb is higher than other magnetic orders. And this proves spin polarization displaying in Pd₂SnYb. However, the energy of NM in PdSnYb and PdSn₂Yb are the highest. It demonstrates the ground state of these three materials, which is in accordance

with the experiments [37] [38] . The calculated magnetic moment value of each Yb is 2.4 μ_B for the alloy composition Pd₂SnYb. The total magnetic moment of per formula unit for Pd₂SnYb and atoms Yb, Pd, Sn are all 0 μ_B. As we can see, Pd₂SnYb is barely the magnetic one among the three Pd-based alloys. According to Mulliken’s bond population and length shown in Table 1, Pd₂SnYb contains the bond type Pd-Yb, which the other two alloys don’t. And the lack of Pd-Yb bonding maybe indicates the reason for appearing a transition from non-mag- netism (Pd₂SnYb, Pd₂SnYb) to anti-ferromagnetism (Pd₂SnYb). The bond population means the distribution of overlapping electron charge between two atoms. It is usually used to evaluate the ionicity or covalency of a bond. Compared with the positive values of Pd-Sn, the negative of Pd-Yb displays its iconicity, which is also connected with the magnetism in Pd₂SnYb.

Based on the calculation of magnetic ground state in 3.1 Magnetic properties, lattice constant, volume, density, total energy, cohesive energy, formation enthalpy and partial experiment values of Pd₂SnYb, PdSnYb and PdSn₂Yb are listed in Table 2. As we all know, GGA calculation usually overestimates lattice constants. On the contrary, the elastic constants are underestimated. Therefore, lattice parameters calculated by GGA are slightly larger. While the error is negligible, and the computed results still agree well with the experiment data.

For further details of the bonding properties in these alloys, the cohesive energy and formation enthalpy per atom of Pd, Sn and Yb atoms are defined

as the calculated Equation (1) and (2) [42] [43] .

Here, ΔH and ΔE_coh respectively represent the formation enthalpy and cohesive energy of Pd-based compounds. E_tot stands for the energy of a unit cell., , are the energy of each Pd, Sn and Yb atom in the bulk state, and, , show the total energy of insular Pd, Sn, Yb atom, respectively. x, y and z are the number of Pd, Sn, and Yb atom in unit cell.

It is clear that the calculated formation enthalpy and the cohesive energy given in Table 2 are negative: 0 > Pd₂SnYb > PdSnYb > PdSn₂Yb. The results show that Pd₂SnYb, PdSnYb and PdSn₂Yb are all thermally stable. Among them, PdSn₂Yb is the easiest to synthesis and the most stable alloy. And Pd₂SnYb, which has the poorest stability and reacts easily with Cl⁻ or H⁺ resulting in corrosion, is just on the contrary.

Fermi level (E_f) also can be known as the Fermi energy. If the electrons accumulation in semiconductor is regarded as a thermodynamic system, the statistic theory has been proved that Fermi energy is the electronic chemical potential of this system.

in which μ is the chemical potential, F is the free energy, N represents the total number of electrons, T is temperature.

The corrosion behavior on alloys is complicated. In the light of the electron theory, each fermion obeys Fermi-Dirac statistics. According to Pauli exclusion- principle, the minimal energy principle, and Hund rule, fermion occupies the quantum state respectively. On behalf of the top level of electron filling, Fermi energy loses electron in the first. And the higher Fermi level reaches, the easier outermost shells are to lose.

Fermi energy of Pd₂SnYb, PdSnYb, and PdSn₂Yb are shown in Figure 2. Due to the different types and structures of the system, Fermi level is different in the ground state. Corrosion potential is bound on with the Fermi level. So the higher Fermi level reaches, the smaller corrosion potential will be. As shown in Figure 2, the Fermi energy (E_f) values of these compounds with E_f (PdSnYb) > E_f (Pd₂SnYb) > E_f (PdSn₂Yb) indicate that PdSnYb is most likely to lose electrons, while PdSn₂Yb is difficult. Their corrosion potential and complexity of corroding are in the order of PdSn₂Yb > Pd₂SnYb > PdSnYb.

The reaction to external stress in the elastic limit of crystal lattice can be charac-

terized by elastic constants. It’s of important significant on the stability and stiffness of materials. Table 3 lists the elastic constants of these three alloys. The elastic constants of cubic and orthorhombic system need to satisfy the generalized stability criteria which can be expressed as [44] :

For cubic phase (Pd₂SnYb):

For orthorhombic phase (PdSnYb and PdSn₂Yb):

As mentioned in Table 3, the mentioned Pd-based alloys are stable in mechanics, due to the elastic constants satisfy the corresponding stability criterions.

It’s well known that the elastic constants C₁₁ and C₃₃ are depicted as the ability to resist linear compression along x and z-axis [45] . The present C₁₁ is equal to C₃₃ in Pd₂SnYb, indicating that the compression of x and z-axis is isotropy. The largest C₁₁ of PdSnYb implies that it is the most incompressible material along x-axes obviously. For PdSn₂Yb, the value of C₃₃ is slightly higher than the C₁₁, which indicates that the z-axis is less compressible than x-axis. The calculated elastic constants of Pd₂SnYb follow the order:, suggesting the anisotropies of shear moduli for Pd-based intermetallics are relatively weak. The PdSnYb is in the order of. Thus, the bonding strength of adjacent atoms are the highest in (1 0 0) plane. The present C₂₂ is higher than the C₁₁ and C₂₂ for PdSn₂Yb. Therefore, the bonding strength of (0 1 0) plane is higher than (1 0 0) and (0 0 1) planes. In conclusion, all the three compounds have the highest binding strength in (1 0 0) plane.

Additionally, C₄₄, which measures the ability to resist monoclinic shear strain

in (1 0 0) plane, is a vital parameter indirectly affecting the indentation hardness [46] . The highest C₄₄ for PdSnYb indicates that it has the strongest resistance to shear deformation in (1 0 0) plane. The equation (C₁₂-C₄₄) is a classical representation of Cauchy pressure. When the value of Cauchy pressure is positive, it reveals the material is ductile, whereas the negative value represents brittleness [47] . The computed Cauchy pressure for Pd-based intermetallics follows this order: PdSnYb (232 GPa) > PdSn₂Yb (93 GPa) > PdSn₂Yb (60 GPa) > 0. The largest value of Cauchy pressure for PdSnYb and the smallest one for PdSn₂Yb manifest PdSnYb is the most ductile structure and PdSn₂Yb is the least one.

For the polycrystalline system, elastic modulus can be got via independent elastic constants. In order to obtain the bulk modulus and shear modulus, we consult the Voigt and Reuss models. Ref. [44] sums up the expressions of bulk and shear modulus for different systems:

For cubic phase (Pd₂SnYb):

For orthorhombic phase (PdSnYb and PdSn₂Yb):

where, and, , which represent the bulk modulus and shear modulus respectively, are calculated by Voigt and Reuss approximation.

According to the extreme value principle, the Reuss’s and Voigt’s models have been proved to be the lower and upper limits of the elastic constant by Hill [48] . The formula called Voigt-Reuss-Hill (VRH) agrees well with the experiments:

where B and G represent the bulk and shear modulus.

The value of bulk modulus and shear modulus, Young’s modulus and Poisson’s ratio using Hill’s models are obtained:

Melting point, characterizing the thermodynamic stability of alloy, has always been considered as an important parameter. Deduced from Ref. [49] , the melting temperature of materials, which is closely related to elastic constants, is estimated as follows:

Table 4 lists the elastic modulus (GPa), bulk modulus (GPa), shear modulus (GPa), Poisson’s ratio ν, Pugh modules ratio and the melting temperature (˚C) of Pd₂SnYb, PdSnYb, and PdSn₂Yb.

Generally, the bulk modulus reflects the average values of bonding strength and the ability to resist volume change. Shear modulus measures the resistance to plastic deformation. As Table 4 shown, the bulk modulus is in the sequence of Pd₂SnYb (106 GPa) > PdSnYb (71 GPa) > PdSn₂Yb (45 GPa), indicating Pd₂SnYb is the least compressible material in all structures. However, the shear moduli of them are almost the same. Young’s modulus serves as a measure of the stiffness. The higher the Young’s modulus is, the stiffer the material will be.

Poisson’s ratio ν and Pugh modules ratio further confirm the brittleness and ductility of materials. Poisson’s ratio v reflects the elastic parameter of uniaxial deformation, especially in atom binding force. With 0.5 as the critical point, Poisson’s ratio v = 0.5 suggests the constancy of volume. When the variation of v is between 0.25 - 0.5, the atom binding force is central force. The v value

for Pd₂SnYb, PdSnYb and PdSn₂Yb is higher than 0.25, which shows the atomic forces are remarkably central forces. Pd₂SnYb presents the largest v, reflecting its resistance of shear strain is the weakest. In accordance with Pugh’s criterion [50] , material with is brittle; whereas material with shows ductility. In Table 4, the G/B ratios of Pd₂SnYb, PdSnYb, and PdSn₂Yb are less than 0.57. Thus, these three alloys are deemed to be ductility. The melting point of alloys is implied its thermodynamic stability. In Table 4 the melting temperature of Pd₂SnYb, PdSnYb, and PdSn₂Yb are 984˚C, 1230˚C, 747˚C, respectively, which further verifies the stability.

The thermal conductivity is a measure of material’s heat conduction ability. Therefore, the research on it of Pd-based alloys in this work is significant.

Owing to the lattice vibration influences the crystal macroscopic thermodynamic properties, the lattice vibration becomes important factors we want to know. And lattice vibration is determined by phonon system. Thus it has great significance to the materials’ thermal conductivity. The transverse acoustic wave velocity (v_t), longitudinal acoustic wave velocity (v_l), and wave velocity (v_m) are calculated [55] :

In the condition of high temperature, the value of thermal conductivity will decrease with increasing temperature [56] . Hence the minimum thermal conductivity value for materials in the applications of high temperature is extremely important. The minimum thermal conductivity of Pd₂SnYb, PdSnYb, and PdSn₂Yb is calculated on the basis of Clark’s model [56] and Cahill’s model [57] :

Clark’s Model: (28)

Cahill’s Model: (29)

where k_B represents Boltzmann’s constant, Ma is the average mass of atoms, E is the Young’s modulus, ρ is density, v_n (n = 1, 2, 3) is acoustic wave velocity, p is the number of atoms in unit volume. All the indexes are calculated in Table 5. The thermal conductivity for cubic ZrO₂ is also calculated, aiming to compare the value with the experimental value to confirm the accuracy of the calculation method.

As shown in Table 5, the calculated thermal conductivity using the Cahill’s model is lightly greater than that computed by the Clark’s model. This is due to the atom number density and phonon spectrum are both considered in Cahill’s model, whereas the Clark’s model does not [58] . Thus, the Clark’s model underestimates the thermal conductivity, however the value adopting Cahill’s model gets closer to the real values of thermal conductivity. In comparison with Clark’s model, the Cahill’s value of ZrO₂ is closer to the experimental value, which confirms this calculation method is credible. As for Pd₂SnYb, the minimum thermal conductivity is largest, and that for PdSn₂Yb is the smallest. Compared to the results in the present work, the increasing content of Sn atoms cause the decreasing in minimum thermal conductivity, when the proportion of Pd/Sn ratios modify. As is known to all, the Y₂O₃-stabilized ZrO₂ (~2.2 W・m⁻¹・K⁻¹) are investigated for application as materials for thermal barrier coatings. Based on the accuracy of the calculation method, the calculated minimum thermal conductivities of Pd₂SnYb, PdSnYb and PdSn₂Yb are all at least a quarter less than ZrO₂,

which show Pd₂SnYb, PdSnYb and PdSn₂Yb can be used for high-temperature- resistant materials, aerospace field, and many other fields.