The Properties of Elasticity , Thermology , and Anisotropy in Pd-Based Alloys

This work is devoted to investigate the elasticity, anisotropy, plastic properties, and thermal conductivity of PdSnYb, PdSn2Yb and Heusler alloy Pd2SnYb via employing the first-principles. The magnetic properties of Pd2SnYb, PdSnYb and PdSn2Yb are obtained by the geometry optimization combining with spin polarization. And the stability of these three kinds of materials is ensured by comparing with the enthalpy of formation and binding energy. The Fermi energy has same trend with stability. The details of bulk and Young’s modulus are demonstrated in 3D plots, embodied the elastic anisotropies of PdSnYb, PdSn2Yb, and Pd2SnYb. The calculations of plastic properties are also anisotropic. And the minimum thermal conductivities are small enough for these three materials to be used as thermal barrier coatings.


Introduction
Heusler alloys are composed of a series of intermetallics.In recent years, plenty of magnetic properties that Heusler alloys presented and their applications in spintronic devices had aroused wide concern [1] [2] [3].The properties of Heusler alloys were well diversified, such as non-ferromagnetic elements could exhibit ferromagnetism after highly ordered [4] [5], 100% spin polarization were presented in materials which were called half-metallic ferromagnets (HMF) [6] [7], only a minority of Heusler alloys containing rare earth had been reported to be superconductors [8] [9] [10], etc.These above features elucidated their potential for future applications in different fields.
As early as 1903, Cu 2 MnAl became the prototype of Heusler alloys, since F.
Heusler [11] firstly reported Cu 2 MnAl and high magnetic ordered alloy of Cu 2 MnAl series.In 1969, P. Webster [12] discussed magnetic and structure properties of Heusler alloys systematically.Liu et al. [13] discovered another highly ordered Heusler alloy, which named Hg 2 CuTi.Up to now, more than one hundred kinds of Heusler alloys were found both in theory and experiments, such as Mn-based alloys [14] [15] [16], Co-based alloys [17] [18], Cu-based alloys [19] [20] [21], Ni-based alloys [22] [23].But the reports about Pd group Heusler alloys were relatively less and the majority of them were related to experiments.Kierstead et al. [24], Aoki et al. [25] [26] and Stanley et al. [8] studied the Heusler compound Pd 2 SnYb and Pd 2 SnEr, whose superconductivity and antiferromagnetism were concomitant.Novel properties of thermodynamics and transmission were shown in Pd 2 SnYb obviously.And the superconductivity presented at T c = 2.3 K, along with a synchronous phase of antiferromagnetism and superconductivity yielding at T N = 220 mK.The testing of elastic and inelastic neutron scattering for Pd 2 SnEr was carried out, which proved that Pd 2 SnEr turned into superconductor at T c = 1.17 K.Only when temperature conditions met T > T c , the antiferromagnetic correlations would occur.The maximum critical temperature was found in Pd 2 YSn, which was revealed as the Heusler alloy [27].
However, in the aspect of theoretical calculation, there is no systematic research on elasticity, thermal properties and anisotropy of Pd-based alloys PdSnYb, PdSn 2 Yb and Pd 2 SnYb so far.In this work, we provide the overall calculation and analysis of these properties.Especially, once the thermal conductivity is smaller, the heat-shielding performance will be better.The computed minimum thermal conductivities of Pd 2 SnYb, PdSnYb and PdSn 2 Yb are all less than 0.5 W•m −1 •K −1 .This minimum thermal conductivity is small enough to be applied to thermal barrier coatings and many other fields.Hence, the thorough discussion carried on the three materials is essential, which inspires our passion on studying these materials.And it makes great sense to explore the microstructure and properties of Pd-based alloys.

Parameters Setting
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 4d 10 , Sn 5s 2 5p 2 , and Yb 4f 14 5s 2 5p 6 6s 2 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 Pd 2 SnYb, PdSnYb and PdSn 2 Yb 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.

Magnetic Property
The equilibrium lattice constants of Pd 2 SnYb, PdSnYb, and PdSn 2 Yb 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  Energy (eV)

Structural Parameters
Based on the calculation of magnetic ground state in 3.  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.

Fermi Energy
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

Elastic Property
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 (Pd 2 SnYb): ( ) For orthorhombic phase (PdSnYb and PdSn 2 Yb): ( ) ( ) 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 C 11 and C 33 are depicted as the ability to resist linear compression along x and z-axis [45].The present C 11 is equal to C 33 in Pd 2 SnYb, indicating that the compression of x and z-axis is isotropy.The largest C 11 of PdSnYb implies that it is the most incompressible material along x-axes obviously.For PdSn 2 Yb, the value of C 33 is slightly higher than the C 11 , which indicates that the z-axis is less compressible than x-axis.The calculated elastic constants of Pd 2 SnYb follow the order: 11 Thus, the bonding strength of adjacent atoms are the highest in (1 0 0) plane.
The present C 22 is higher than the C 11 and C 22 for PdSn 2 Yb.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, C 44 , which measures the ability to resist monoclinic shear strain 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 (Pd 2 SnYb): ( ) ( ) For orthorhombic phase (PdSnYb and PdSn 2 Yb): ( ) ( ) where V B , R B and V G , R G , 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: ( ) Pd 2 SnYb 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 G B 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   The share modulus G on different plane along different directions can be expressed as [51]: (001) plane from [100] to [010]: (100) plane from [100] to [010]: (010) plane from [100] to [010]: (1 0 1 ) plane from [001] to [110]: in which θ represent the angle between [uvw] direction and [HKL] direction.
As we can see in Figure 3(a), the shear modulus of Pd 2 SnYb in (001), (100), and (010) trajectory planes are similar to the quarter circles, which imply that Pd 2 SnYb shows almost isotropy in these planes.On the curves of (1 0 1 ) plane from [001] to [110] and (110) plane from [001] to [1 0 1 ], shear modulus show obviously anisotropy of Pd 2 SnYb.For PdSnYb and PdSn 2 Yb, shear modulus are all anisotropy due to the noncircular plots on the mentioned planes along different directions.To take a panoramic view of Figure 3, shear modulus is the smallest on (110) plane, which suggests it may be the glide plane of Pd-based alloys.
In order to clearly illustrate the anisotropies of mechanical modulus for Pd 2 SnYb, PdSnYb, and PdSn 2 Yb, we plot three dimensional surfaces of modulus in Figure 4.For bulk modulus and Young's modulus, the 3D plots can be more intuitive to determine the ability to withstand external stress.Their formulas are as follow: bulk modulus [52]: Young's modulus [53]: where nmpq S is the elastic compliance coefficient and 1n l , 1m l , 1 p l , 1q l re- present the directional cosine.
In Figure 4, the bulk modulus of Pd  planes show a more pronounced anisotropy.Therefore, a stronger directional dependence of Young's modulus has displayed on these planes.

Ideal Strength of Tensile and Shear Deformation
It is essential to comprehend the causation of the structural stability for the design and application of these Pd-based alloys, especially the response of lattice stress to the applied strain.To analysis the mechanism of mechanical deformation, the stress-strain curves of tensile and shear deformation are performed in It can be seen in the Figures 5(d)-(f) that the shear moduli can be obtained from the strains less than 2% [54].On the basis of this linear parts, the computed ( ) for PdSn 2 Yb, respectively.In contrast with the results using Voigt-Reuss-Hill method, they are not accord, which proves that they are all anisotropic in the whole crystal of the three structures.

The Minimum Value of Thermal Conductivity Kmin
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 Pd 2 SnYb, PdSnYb, and PdSn 2 Yb is calculated on the basis of Clark's model [56] and Cahill's model [57]: Clark's Model: Cahill's Model: ( ) ( ) 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 ZrO 2 is also calculated, aiming to compare the value with the experimental value to confirm the accuracy of the calculation method.
As shown in which show Pd 2 SnYb, PdSnYb and PdSn 2 Yb can be used for high-temperatureresistant materials, aerospace field, and many other fields.

Conclusions
The calculated results showed that the AFM-2 state of Pd

3 - 2
shown, the energy of AFM-2 (each layer of Yb atoms spin in the opposite manner along the crystal orientation [001]) in Pd 2 SnYb is higher than other magnetic orders.And this proves spin polarization displaying in Pd 2 SnYb.However, the energy of NM in PdSnYb and PdSn 2 Yb are the highest.It demonstrates the ground state of these three materials, which is in accordance NM FM AFM-1 AFM-2 AFM- theory, each fermion obeys Fermi-Dirac statistics.According to Pauli exclusionprinciple, 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 Pd 2 SnYb, PdSnYb, and PdSn 2 Yb 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 (Pd 2 SnYb) > E f (PdSn 2 Yb) indicate that PdSnYb is most likely to lose electrons, while PdSn 2 Yb is difficult.Their corrosion potential and complexity of corroding are in the order of PdSn 2 Yb > Pd 2 SnYb > PdSnYb.

Figure 3 ,
Figure 3, which are labeled in black solid line, mean isotropy and play a supporting role in estimating the anisotropic degree.

2
SnYb is spherical, reflecting isotropy of the bulk modulus.The three dimensional graphs of bulk modulus of PdSnYb and PdSn 2 Yb, and Young's modulus of Pd 2 SnYb are irregularly.Thus, they express anisotropic nature, as well as the PdSn 2 Yb performs the strongest anisotropy.Conversely, Pd 2 SnYb is isotropy, which is in good agreement with the

Figure 5 .
Figure 5.For tensile deformation, the strain directions [100], [010], [001] are parallel to the coordinate axis of the corresponding unit cell.From Figures 5(a)-(c), the tensile strengths of PdSnYb and PdSn 2 Yb show anisotropy.And the strongest ideal tensile strengths of these two alloys exist in the strain direction [001].Due to the different symmetry of Pd 2 SnYb compared with PdSnYb and PdSn 2 Yb, it's isotropic along the strain direction [100], [010], and [001].The yielding stage of Pd 2 SnYb, PdSnYb, and PdSn 2 Yb in different orientations all occurs in 2% strains.

Figure 5 .
Figure 5.The tensile and shear stress-strain curves of Pd-based alloys along different directions.
[38]re 1.The energy curve of NM, FM and AFM in Pd 2 SnYb, PdSnYb and PdSn 2 Yb.with the experiments[37][38].The calculated magnetic moment value of each Yb is 2.4 μ B for the alloy composition Pd 2 SnYb.The total magnetic moment of per formula unit for Pd 2 SnYb and atoms Yb, Pd, Sn are all 0 μ B .As we can see,

Table 1
. Mulliken's bond population and length (Ǻ) of the Pd-based alloys.

Table 2
. The calculated lattice constants a, b, c (Å), volume V (Å3), density ρ (g•cm −3 ), E tot (eV/atom), ΔH (eV), ΔE coh (eV) and partial experiment values of Pd-Sn-Yb.tot −2022.47−2430.19−1846.57∆H −0.844 −1.567 −2.750 ∆E coh −4.113 −4.986 −6.012 . 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 Table2are negative: 0 > Pd 2 SnYb > PdSnYb > PdSn 2 Yb.The results show that Pd 2 SnYb, PdSnYb and PdSn 2 Yb are all thermally stable.Among them, PdSn 2 Yb is the easiest to synthesis and the most stable alloy.And Pd 2 SnYb, which has the poorest stability and reacts easily with Cl − or H + resulting in corrosion, is just on the contrary.

Table 3 .
Elastic constants (GPa) of the three Pd-Sn-Yb.The highest C 44 for PdSnYb indicates that it has the strongest resistance to shear deformation in (1 0 0) plane.The equation (C 12 -C 44 ) 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) > PdSn 2 Yb (93 GPa) > PdSn 2 Yb (60 GPa) > 0. The largest value of Cauchy pressure for PdSnYb and the smallest one for PdSn 2 Yb manifest PdSnYb is the most ductile structure and PdSn 2 Yb is the least one.

Table 4 .
[50]calculated values for elastic modulus (GPa), bulk modulus (GPa), shear modulus (GPa), Poisson's ratio and Pugh modules ratio G B , melting temperature T m (˚C).SnYb, PdSnYb and PdSn 2 Yb is higher than 0.25, which shows the atomic forces are remarkably central forces.Pd 2 SnYb presents the largest v, reflecting its resistance of shear strain is the weakest.In accordance with Pugh's criterion[50],

Table 5
As for Pd 2 SnYb, the minimum thermal conductivity is largest, and that for PdSn 2 Yb 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 Y 2 O 3 -stabilized ZrO 2 (~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 Pd 2 SnYb, PdSnYb and PdSn 2 Yb are all at least a quarter less than ZrO 2 , [58]e 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 ZrO 2 is closer to the experimental value, which confirms this calculation method is credible.
2 SnYb and the NM state of PdSnYb, PdSn 2 Yb are found to be the ground state, which are agreed with experimental reports.The obtained enthalpy of formation and binding energy are in the order: 0 > Pd 2 SnYb > PdSnYb > PdSn 2 Yb, indicating that the Pd-based alloys are mechanically stable.The Fermi energy (E f ) values of these compounds with E f (PdSnYb) > E f (Pd 2 SnYb) > E f (PdSn 2 Yb) imply that PdSnYb is most likely to lose electrons while PdSn 2 Yb is difficult.In line with the Cauchy pressure, values of Poisson's ratio ν, and Pugh modules ratio G B , these three alloys are deemed to be ductility.The three compounds are all elastic anisotropic, and the anisotropic sequence is PdSn 2 Yb > PdSnYb > Pd 2 SnYb.The ideal strength of tensile and shear deformation are inconformity in different crystal orientations, implying that Pd 2 SnYb, Pd 2 SnYb, and Pd 2 SnYb are plastic anisotropic.Moreover, the calculated minimum thermal conductivities of Pd 2 SnYb, PdSnYb and PdSn 2 Yb are all at least a quarter less than that of ZrO 2 , the usual thermal barrier coatings materials.That implies these Pd-based alloys can be candidates for high-temperature-resistant materials.