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This work is devoted to investigate the elasticity, anisotropy, plastic properties, and thermal conductivity of PdSnYb, PdSn
_{2}Yb and Heusler alloy Pd
_{2}SnYb via employing the first-principles. The magnetic properties of Pd
_{2}SnYb, PdSnYb and PdSn
_{2}Yb 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, PdSn
_{2}Yb, and Pd
_{2}SnYb. 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.

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 [

As early as 1903, Cu_{2}MnAl became the prototype of Heusler alloys, since F. Heusler [_{2}MnAl and high magnetic ordered alloy of Cu_{2}MnAl series. In 1969, P. Webster [_{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 [_{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 [

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.

Pd-based system used in this work includes three alloys: Pd_{2}SnYb, PdSnYb and PdSn_{2}Yb. The symmetry group and international table number of Pd_{2}SnYb are _{2}Yb are orthorhombic system. Their space groups are PNMA (No. 62) and CMCM (No. 63). In Pd_{2}SnYb, 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 PdSn_{2}Yb, 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 [^{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 [_{2}SnYb, PdSnYb and PdSn_{2}Yb were optimized successively by using the BFGS scheme [

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

As the _{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

with the experiments [_{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, Pd_{2}SnYb is barely the magnetic one among the three Pd-based alloys. According to Mulliken’s bond population and length shown in _{2}SnYb 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 (Pd_{2}SnYb, Pd_{2}SnYb) to anti-ferromagnetism (Pd_{2}SnYb). 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 Pd_{2}SnYb.

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 Pd_{2}SnYb, PdSnYb and PdSn_{2}Yb are listed in

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

Pd_{2}SnYb | PdSnYb | PdSn_{2}Yb | |||||
---|---|---|---|---|---|---|---|

Pd-Sn | Pd-Yb | Pd-Sn | Pd-Sn | ||||

population | 1.36 | −0.18 | 0.61 | 1.42 | 0.09 | 0.94 | 0.07 |

length | 2.93 | 2.93 | 2.72 | 2.73 | 2.85 | 2.80 | 2.83 |

Pd_{2}SnYb | Exp [ | PdSnYb | Exp [ | PdSn_{2}Yb | Exp [ | |
---|---|---|---|---|---|---|

SG | Pnma | Cmcm | ||||

a | 6.755 | 6.658 | 7.247 | 7.191 | 4.420 | 4.424 |

b | 6.755 | 6.658 | 4.638 | 4.588 | 11.059 | 11.086 |

c | 6.755 | 6.658 | 8.040 | 7.961 | 7.603 | 7.384 |

V | 308.264 | 295.142 | 270.231 | 262.652 | 371.627 | 362.144 |

10.913 | 11.354 | 9.842 | 10.068 | 9.290 | 9.479 | |

E_{tot} | −2022.47 | −2430.19 | −1846.57 | |||

∆H | −0.844 | −1.567 | −2.750 | |||

∆E_{coh} | −4.113 | −4.986 | −6.012 |

as the calculated Equation (1) and (2) [

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.

It is clear that the calculated formation enthalpy and the cohesive energy given in _{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.

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 Pd_{2}SnYb, PdSnYb, and PdSn_{2}Yb are shown in _{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.

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.

For cubic phase (Pd_{2}SnYb):

For orthorhombic phase (PdSnYb and PdSn_{2}Yb):

As mentioned in

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 [_{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:_{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

C_{11} | C_{22} | C_{33} | C_{44} | C_{55} | C_{66} | C_{12} | C_{13} | C_{23} | |
---|---|---|---|---|---|---|---|---|---|

Pd_{2}SnYb | 127 | 127 | 127 | 34 | 34 | 34 | 95 | 95 | 95 |

PdSnYb | 270 | 119 | 117 | 38 | 15 | 25 | 41 | 44 | 64 |

PdSn_{2}Yb | 70 | 112 | 79 | 10 | 53 | 20 | 11 | 30 | 37 |

in (1 0 0) plane, is a vital parameter indirectly affecting the indentation hardness [_{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 [_{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.

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. [

For cubic phase (Pd_{2}SnYb):

For orthorhombic phase (PdSnYb and PdSn_{2}Yb):

where

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 [

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. [

_{2}SnYb, PdSnYb, and PdSn_{2}Yb.

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 _{2}SnYb (106 GPa) > PdSnYb (71 GPa) > PdSn_{2}Yb (45 GPa), indicating 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

Pd_{2}SnYb | PdSnYb | PdSn_{2}Yb | |
---|---|---|---|

E (GPa) | 70 | 72 | 63 |

E[ | 47 | 90 | 59 |

E[ | 47 | 82 | 95 |

E[ | 47 | 78 | 57 |

B (GPa) | 106 | 71 | 45 |

G (GPa) | 25 | 27 | 25 |

0.426 | 0.198 | −0.030 | |

0.426 | 0.181 | −0.049 | |

0.426 | 0.234 | 0.379 | |

0.426 | 0.270 | 0.387 | |

0.426 | 0.478 | 0.483 | |

0.426 | 0.455 | 0.291 | |

0.390 | 0.330 | 0.267 | |

0.238 | 0.383 | 0.551 | |

T_{m} (˚C) | 984 | 1230 | 747 |

for Pd_{2}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 [_{2}SnYb, PdSnYb, and PdSn_{2}Yb 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 _{2}SnYb, PdSnYb, and PdSn_{2}Yb are 984˚C, 1230˚C, 747˚C, respectively, which further verifies the stability.

It is well known that single crystal is anisotropic, which has great influence on the performance of thin-film materials. So how to characterize the degree of anisotropic is necessary. To obtain the anisotropic degree of Pd-based alloys, the two-dimensional images of shear modulus of Pd_{2}SnYb, PdSnYb, and PdSn_{2}Yb are described in

The share modulus G on different plane along different directions can be expressed as [

(

(110) plane from [

in which θ represent the angle between [uvw] direction and [HKL] direction.

As we can see in _{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 (_{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

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

bulk modulus [

Young’s modulus [

where

In _{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

calculated anisotropy of shear modulus in the present work. Compared with bulk modulus, the projections of Young’s modulus on the (100), (010) and (001) planes show a more pronounced anisotropy. Therefore, a stronger directional dependence of Young’s modulus has displayed on these planes.

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

For tensile deformation, the strain directions [_{2}Yb show anisotropy. And the strongest ideal tensile strengths of these two alloys exist in the strain direction [_{2}SnYb compared with PdSnYb and PdSn_{2}Yb, it’s isotropic along the strain direction [_{2}SnYb, PdSnYb, and PdSn_{2}Yb in different orientations all occurs in 2% strains.

It can be seen in the Figures 5(d)-(f) that the shear moduli can be obtained from the strains less than 2% [

shear moduli values are

_{2}SnYb,

_{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 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 [

In the condition of high temperature, the value of thermal conductivity will decrease with increasing temperature [_{2}SnYb, PdSnYb, and PdSn_{2}Yb is calculated on the basis of Clark’s model [

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 _{2} is also calculated, aiming to compare the value with the experimental value to confirm the accuracy of the calculation method.

As shown in _{2} is closer to the experimental value, which confirms this calculation method is credible. 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},

v_{t} | v_{l} | v_{m} | Chill | Clark | ||
---|---|---|---|---|---|---|

K_{min} | K_{min} | |||||

Pd_{2}SnYb | 1519 | 3572 | 1717 | 0.53 | 0.42 | |

PdSnYb | 1163 | 3304 | 1865 | 0.48 | 0.41 | |

PdSn_{2}Yb | 1635 | 2902 | 1819 | 0.43 | 0.39 | |

ZrO_{2} | 4188 | 7774 | 4675 | 1.86 | 1.62 | 2.2 [ |

which show Pd_{2}SnYb, PdSnYb and PdSn_{2}Yb can be used for high-temperature- resistant materials, aerospace field, and many other fields.

The calculated results showed that the AFM-2 state of Pd_{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_{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.

This work was supported by Fundamental Research Funds for the Central Universities (XDJK2016D043).

Chen, K.K., Hu, M., Li, C.M., Li, G.N. and Chen, Z.Q. (2017) The Properties of Elasticity, Thermology, and Anisotropy in Pd-Based Alloys. Journal of Materials Science and Chemical Engineering, 5, 17-34. https://doi.org/10.4236/msce.2017.53002