Impact of isovalent and aliovalent substitution on the mechanical and thermal properties of Gd2Zr2O7

In this study, a density functional theory method is employed to investigate the effects of isovalent and aliovalent substitution of Sm3+ on the phase stability, thermo-physical properties and electronic structure of Gd2Zr2O7. It is shown that the isovalent substitution of Sm3+ for Gd3+ results in the formation of Gd2Zr2O7-Sm2Zr2O7 solid solution, which retains the pyrochlore structure and has slight effects on the elastic moduli, ductility, Debye temperature and band gap of Gd2Zr2O7. As for the aliovalent substitution of Sm3+ for Zr4+ site, a pyrochlore-to-defect fluorite structural transition is induced, and the mechanical, thermal properties and electronic structures are influenced significantly. As compared with the Gd2Zr2O7, the resulted Gd2SmyZr2-yO7 compositions have much smaller elastic moduli, better ductility and smaller Debye temperature. Especially, an amount of electrons distribute on the fermi level and they are expected to have larger thermal conductivity than Gd2Zr2O7. This study suggests an alternative way to engineer the thermo-physical properties of Gd2Zr2O7 and will be beneficial for its applications under stress and high temperature.

In this study, a density functional theory method is employed to investigate the effects of isovalent and aliovalent substitution of Sm 3+ on the phase stability, thermo-physical properties and electronic structure of Gd 2 Zr 2 O 7 . It is shown that the isovalent substitution of Sm 3+ for Gd 3+ results in the formation of Gd 2 Zr 2 O 7 -Sm 2 Zr 2 O 7 solid solution, which retains the pyrochlore structure and has slight effects on the elastic moduli, ductility, Debye temperature and band gap of Gd 2 Zr 2 O 7 . As for the aliovalent substitution of Sm 3+ for Zr 4+ site, a pyrochlore-to-defect fluorite structural transition is induced, and the mechanical, thermal properties and electronic structures are influenced significantly. As compared with the Gd 2 Zr 2 O 7 , the resulted Gd 2 Sm y Zr 2-y O 7 compositions have much smaller elastic moduli, better ductility and smaller Debye temperature. Especially, an amount of electrons distribute on the fermi level and they are expected to have larger thermal conductivity than Gd 2 Zr 2 O 7 . This study suggests an alternative way to engineer the thermo-physical properties of Gd 2 Zr 2 O 7 and will be beneficial for its applications under stress and high temperature.
The rare-earth zirconates, with chemical formula A 2 Zr 2 O 7 (A = Y or another rare earth elements) 1, 2 , exhibit ordered pyrochlore-type structure or defect fluorite-type structure, which is mainly governed by the ionic radii of A 3+ and Zr 4+3 . They have attracted the attention of many researchers, due to their good chemical and mechanical stability, excellent catalytic activity, high ionic conductivity, ferromagnetism, luminescence as well as strong resistance to amorphization under irradiation [4][5][6][7][8] . Owing to these outstanding properties, the rare-earth zirconates have a wide range of technical applications, e.g., ceramic thermal barrier coating 9, 10 , oxidation catalyst 5,11 , solid electrolyte 12 , hosts of actinides in nuclear waste 13 and oxygen gas sensor 14 .
Of the zirconate pyrochlores, Gd 2 Zr 2 O 7 is of special interest due to its good thermo-physical properties [15][16][17][18][19] . Shimamura et al. have measured the thermal expansion of a series of zirconate pyrochlores employing high-temperature X-ray diffraction and found that the thermal expansion coefficient of Gd 2 Zr 2 O 7 is larger than other zirconates during the temperature range of 400-1600 °C 20 . The thermal conductivity of rare-earth zirconates has been investigated by Wang et al. 21 19 . and Pan et al. 23 reported that the thermal diffusivity of (Sm x Gd 1-x ) 2 Zr 2 O 7 (0 ≤ x ≤ 1) are lower than those of pure Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 . Especially, Sm 2 Zr 2 O 7 -Gd 2 Zr 2 O 7 solid solutions have lower Young's modulus than (La x Gd 1-x ) 2 Zr 2 O 7 (0 < x < 1) at room temperature and larger thermal expansion coefficients than (Gd 1-x Yb x ) 2 Zr 2 O 7 (x = 0, 0.1, 0.3, 0.5, 0.7) from 300 °C to 900 °C 18,19,23,26 . These investigations are mainly experimental studies, and the related theoretical investigations are relatively much fewer 27,28 .
Very recently, the Th 4+ ion incorporation into Gd 3+ and Zr 4+ sites in Gd 2 Zr 2 O 7 was investigated by first-principles calculations 28 . Unexpectedly, the aliovalent substitution of Th 4+ for Gd 3+ turns out to be thermodynamically stable and such substitution even results in better thermo-physical properties than the pure Gd 2 Zr 2 O 7 28 . This thus arouses our interest that whether the aliovalent substitution of Ln 3+ (Ln = lanthanide elements) for Zr 4+ sites are energetically and mechanically stable or not? If yes, will the substitution of Ln 3+ for Zr 4+ sites cause different thermo-mechanical properties from the isovalent substitution of Ln 3+ for Gd 3+ sites? In this study, we choose Sm 3+ as a model and investigate the phase stability and thermo-physical properties of Gd 2 Zr 2 O 7 with isovalent and aliovalent substitution of Ln 3+ for Gd 3+ and Zr 4+ sites by employing the density functional theory (DFT) method. It reveals that the Sm y Gd 2-y Zr 2 O 7 retains the pyrochlore structure and the isovalent substitution of Sm 3+ for Gd 3+ sites influences slightly the mechanical and thermal properties of Gd 2 Zr 2 O 7 . On the other hand, the aliovalent substitution of Sm 3+ for Zr 4+ sites induces pyrochlore-to-fluorite structural transition and affects significantly the elastic moduli, Debye temperature and thermal conductivity. The presented results provide a new way to tune the thermo-physical properties of Gd 2 Zr 2 O 7 and will have important implications in advancing the further related experimental and theoretical studies for its applications under high temperature.       agreeing well with the experimental observation 19 Table 1. On the other hand, structural transformation from ordered pyrochlore to disordered defect-fluorite structure is induced by the substitution of Sm for Zr-site, which is accompanied by the weakened interaction of < Sm-O 48f > and < Gd-O 8b > bonds (see Table 1).

Results and Discussion
In the literature, it has been reported that the Madelung binding energy reduces with the increasing x O48f and the thermal expansion increases with the decreasing Madelung binding energy 20 . Considering that Gd 2 Sm y Zr 2-y O 7 has larger x O48f than Sm y Gd 2-y Zr 2 O 7 , we thus suggest that smaller Madelung binding energy exists in the Gd 2 Sm y Zr 2-y O 7 and they probably have larger thermal expansion coefficient. Meanwhile, the disordering of oxygen ions in Gd 2 Sm y Zr 2-y O 7 may increase the phonon scattering, which will reduce the mean free path of the phonon and result in small phonon thermal conductivity.  properties, agreeing well with the experiment carried out by Shimamura et al. 20 . The elastic constants reported by Lan et al. employing GGA method are generally smaller than our LDA results, while they also found that the results of Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 are very similar to each other 36 . We find that the mechanical stability criteria, i.e., (C 11 -C 12 ) > 0, C 44 > 0, and (C 11 + 2C 12 ) > 0 1 , are satisfied for all compounds, indicating that Sm-substituted Gd 2 Zr 2 O 7 compounds are mechanically stable.
The calculated elastic constants for both Sm y Gd 2-y Zr 2 O 7 and Gd 2 Sm y Zr 2-y O 7 (0 ≤ y ≤ 2) as a function of Sm content are plotted in Fig. 3. It is noted that Sm substitution for Gd-site results in very small changes in the elastic constants and the mixed Gd 2 Zr 2 O 7 -Sm 2 Zr 2 O 7 phases have similar values with the pure states. As the Sm substitutes for Zr-site, the values of C 11 are strikingly decreased with the increasing Sm content, implying that the compositions with high content of Sm are more likely to undergo the uniaxial deformation along the [001] direction. On the other hand, the C 12 and C 44 slightly increase and decrease with the Sm incorporation, respectively. Generally, the changes in the elastic constants of Gd 2 Sm y Zr 2-y O 7 are more significant than those of Sm y Gd 2-y Zr 2 O 7 , meaning that the effects of Sm incorporation into Gd-site on the mechanical properties of Gd 2 Zr 2 O 7 are nearly negligible whereas Sm incorporation into Zr-site has remarkable influence. This may be mainly due to the fact that Gd 3+ and Sm 3+ have similar mass (Gd 3+ : 157.25 amu; Sm 3+ : 150.3 amu) and cation size (Gd 3+ : 1.053 Å; Sm 3+ : 1.079 Å) and Sm substitution for Gd-site affects the structural properties of Gd 2 Zr 2 O 7 slightly, while the mass and radius for Gd 3+ are largely different from the mass of 91.22 amu and the radius of 0.72 Å for Zr 4+19, 27 and an order-disorder phase transition has occurred in Gd 2 Sm y Zr 2-y O 7 .
Based on the three elastic constants, the bulk modulus (B), Young's modulus (E) and shear modulus (G) can be deduced under Voigt-Reuss-Hill (VRH) approximation 37 , i.e., B = (C 11 + 2C 12 )/3, E = 9BG/(3B + G), G = ((C 11 -C 12 + 3C 44 )/5 + 5(C 11 -C 12 )C 44 /(4C 44 + 3(C 11 -C 12 )))/2 1,[38][39][40] . The values of bulk modulus, Young's modulus and shear modulus are shown in Table 2, together with available experimental and theoretical results in the literature 20,36 . As compared with the experimental measurement, our calculated elastic moduli for Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 are overestimated slightly. This may be resulted from the employed LDA method, which generally underestimates the lattice constant whereas overestimates the mechanical modulus 41 . Besides, the defects and impurities in the sample experimentally may also lead to the underestimated values of the B and G 41 . Variation of the elastic moduli for Sm y Gd 2-y Zr 2 O 7 and Gd 2 Sm y Zr 2-y O 7 with the Sm content is illustrated in Fig. 4. As expected, the elastic moduli for Sm y Gd 2-y Zr 2 O 7 vary slightly with the Sm content since the elastic constants for all compositions are very similar to each other. Different from the case of Sm y Gd 2-y Zr 2 O 7 , the bulk modulus, Young's modulus and shear modulus for Gd 2 Sm y Zr 2-y O 7 all decrease sharply with the increasing Sm content, especially the Young's modulus. Consequently, the Gd 2 Sm 2 O 7 has the minimum Young's modulus of 87 GPa, minimum shear modulus of 31 GPa and minimum bulk modulus of 151 GPa. These results indicate that the Gd 2 Sm 2 O 7 has good compliance 2 due to the lowest bulk modulus and the lowest Young's modulus, which will produce relatively smaller residual stresses in the coating system under the severe conditions and result in better thermo-mechanical stability 23 .
Elastic anisotropy, ductility and Debye temperature of Sm y Gd 2-y Zr 2 O 7 and Gd 2 Sm y Zr 2-y O 7 (0 ≤ y ≤ 2). The elastic anisotropy, which is correlated with the possibility of the appearance of microcracks, is an important mechanical property of materials 42,43 . For a crystal, the elastic anisotropy of materials can be evaluated by the A U (universal elastic anisotropy index), which can be calculated by A U = 5 G V /G R + B V /B R -6 44 , with V and R representing the Voigt and Reuss approximation, respectively 38,40 . The A U value of zero refers to isotropic  Table 3. Obviously, the values of A U are very close to zero in the case of Sm y Gd 2-y Zr 2 O 7 (0 ≤ y ≤ 2), which indicates that all compounds are of elastic isotropy. However, the crystals show strong anisotropy in the Gd 2 Sm y Zr 2-y O 7 (0.5 ≤ y ≤ 2) system, as the A U values have large deviation from zero and they increase with the increasing Sm content. The sharp increase from 0.920 for Gd 2 Sm 1.5 Zr 0.5 O 7 to 4.738 for Gd 2 Sm 2 O 7 may be caused by the pyrochlore-to-defect fluorite structural transition.
Another important mechanical property of materials is the ductility, which is often evaluated by the Pugh's indicator (G/B ratio) 45 . The Pugh's indicator of 0.5 is a boundary of brittleness or ductility, i.e., if G/B > 0.5, the material tends to be brittle; otherwise, the material is ductile 46 . The calculated Pugh's indicators are presented in Table 3. For the compounds of Sm y Gd 2-y Zr 2 O 7 , the values are close to 0.5, and are located within the range of 0.499 to 0.505. As for Gd 2 Sm y Zr 2-y O 7 (y = 0.5, 1, 1.5, 2), the G/B values of 0.205-0.428 are much smaller. Obviously, the Gd 2 Sm y Zr 2-y O 7 compositions have better ductility than the Sm y Gd 2-y Zr 2 O 7 compounds. The Poisson's ratio (σ) can also be used to evaluate the relative ductility of materials. Generally, the σ values are close to 0.1 and 0.33 for brittle covalent material and ductile metallic material, respectively 46,47 . The σ values are 0.286 and 0.284 for Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 , respectively, which are comparable with the experimental data of 0.274-0.276 for Gd 2 Zr 2 O 7 and 0.277-0.278 for Sm 2 Zr 2 O 7 20, 48 . As shown in Table 3, the Poisson's ratio are ~0.285 for Sm y Gd 2-y Zr 2 O 7 and vary from 0.313 to 0.404 for Gd 2 Sm y Zr 2-y O 7 , meaning that the latter compositions are more ductile, which is consistent with the results obtained from the Pugh's indicator.
In this study, the Debye temperature that is related to the hardness and thermal expansion coefficient of materials 20, 49 is also estimated for Sm-contained is the Boltzmann's constant, n is the number of atoms in molecular, N A is the Avogadro's constant, ρ is the density, M is the molecular mass and v m is the sound wave velocity. The v m can be deduced by  Table 3, Sm substitution for Gd-site has slight effects on the Debye temperature of Gd 2 Zr 2 O 7 and all the compositions have very similar results. However, as Sm substitutes for the Zr-site, the Debye temperature decreases considerably with the increasing Sm content. The Gd 2 Sm 2 O 7 has the lowest Debye temperature of 341.7 K, which is about 44.3% lower than that of Gd 2 Zr 2 O 7 . These results suggest that Sm incorporation into Zr-site causes weaker interaction of chemical bonds and the Gd 2 Sm y Zr 2-y O 7 with high content of Sm will have much larger thermal expansion coefficient than Gd 2 Zr 2 O 7 .
Electronic structure of Sm y Gd 2-y Zr 2 O 7 and Gd 2 Sm y Zr 2-y O 7 (0 ≤ y ≤ 2). In order to investigate how Sm incorporation influences the electronic structure of Gd 2 Zr 2 O 7 , the atomic projected density of state (DOS) distribution of Sm y Gd 2-y Zr 2 O 7 and Gd 2 Sm y Zr 2-y O 7 are analyzed and plotted in Fig. 5 and Fig. 6, respectively. In Fig. 7a and b, the orbital projected DOS for Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 are also presented. For Gd 2 Zr 2 O 7 , the valence band maximum (VBM) are mainly contributed by O 2p states hybridized with Zr 4d, Gd 5d and Gd 5p orbitals, and the conduction band minimum (CBM) are mainly contributed by Zr 4d states and O 2p states. The obtained band gap of 2.55 eV is comparable with the value of 2.71 eV reported by Wang et al. 50 . For Sm 2 Zr 2 O 7 , the DOS distribution is similar to that of Gd 2 Zr 2 O 7 , i.e., the VBM are mainly contributed by O 2p states hybridized with Zr 4d, Sm 5d and Sm 5p states, and the CBM are mainly dominated by Zr 4d states and O 2p states. Considering that Sm and Gd are heavy atoms and the spin-orbit coupling (SOC) may affect the band gap and electronic structure of the investigated systems, we further calculate the density of state distribution of Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 employing the LDA + SOC method. A comparison of LDA and LDA + SOC results for both compounds is illustrated in Fig. 8. For Gd 2 Zr 2 O 7 , the band gap value of 2.64 eV by LDA + SOC is close to the value of 2.55 eV by LDA method. In the case of Sm 2 Zr 2 O 7 , the band gap values are 2.9 eV and 2.83 eV for LDA + SOC and LDA calculations, respectively. As shown in Fig. 8, the atomic projected and total DOS obtained by LDA with and without spin-orbit coupling exhibit very similar characters for both compounds. These results suggest that the spin-orbit coupling has slight effects on our results and such effects are thus not considered for the mixed states. When Sm substitutes for Gd-site, the Sm 5d orbitals also contribute to the VBM and interacts with the oxygen. Meanwhile, the insulating character of Gd 2

Summary
In this work, a systematic study based on the DFT method is carried out to investigate the effects of Sm substitution for Gd-site and Zr-site in Gd 2 Zr 2 O 7 on its structural stability, mechanical properties, Debye temperature and electronic structures. It is shown that the Sm y Gd 2-y Zr 2 O 7 compositions keep the pyrochlore structure and their lattice parameters follow well the Vegard's law, indicative of the formation of Gd 2 Zr 2 O 7 -Sm 2 Zr 2 O 7 solid solution. On the other hand, Sm substitution for Zr-site influences the structure significantly and a pyrochlore-to-defect fluorite structural transition occurs. The Sm y Gd 2-y Zr 2 O 7 compositions are of elastic isotropy and their elastic moduli, ductility, Debye temperature and band gap vary slightly with the Sm content. However, the Gd 2 Sm y Zr 2-y O 7 compounds show strong elastic anisotropy and their bulk, Young's and shear moduli all decrease sharply with the increasing Sm content. Consequently, the Gd 2 Sm 2 O 7 has the minimum Young's modulus of 87 GPa, minimum shear modulus of 31 GPa and minimum bulk modulus of 151 GPa. Meanwhile, both the Pugh's indicator and Poisson's ratio suggested that the Gd 2 Sm y Zr 2-y O 7 have better ductility than the Sm y Gd 2-y Zr 2 O 7 . As the Sm substitutes for Zr-site, the Debye temperature decreases considerably with the increasing Sm content and the Debye temperature of 341.7 K for Gd 2 Sm 2 O 7 is about 44.3% lower than that of Gd 2 Zr 2 O 7 . In addition, the insulating character of Gd 2 Zr 2 O 7 is kept in the system of Sm y Gd 2-y Zr 2 O 7 , while the Gd 2 Sm y Zr 2-y O 7 compositions exhibit metallic characters. Our calculations demonstrate that substituting Sm for Zr-site is an effective approach to tailor the mechanical and thermal properties of Gd 2 Zr 2 O 7 .

Methods
In this work, first-principles total energy calculations within the DFT framework are carried out. All calculations are performed with the Vienna Ab-initio Simulation Package (VASP) 51,52 . The interaction between electrons and ions is described by the projector augmented wave method 52,53 . All computations are based on a supercell containing 88 atoms. The convergence criteria for total energies and forces are 10 −6 eV and 10 −6 eV/Å, respectively. The structural relaxation is carried out at variable volume. In order to determine the values of cutoff energy and k-point sampling, a series of test calculation has been carried out. Figure 9 shows the variation of total energy of Gd 2 Zr 2 O 7 and Sm 2 Zr 2 O 7 with cutoff energy and k-point sampling, which leads to our calculation being performed with a 2 × 2 × 2 Monkhorst-Pack k-mesh for Brillouin-zone integrations and a cutoff energy of 600 eV for plane wave. For Sm y Gd 2-y Zr 2 O 7 and Gd 2 Sm y Zr 2-y O 7 (y = 0.5, 1, 1.5), the structure models are constructed by the special quasirandom structure approach [54][55][56] .