A DFT study of mechanical properties, thermal conductivity and electronic structures of Th-doped Gd2Zr2O7

Acta Materialia 121 (2016) 299e309

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A DFT study of mechanical properties, thermal conductivity and electronic structures of Th-doped Gd2Zr2O7 F.A. Zhao a, H.Y. Xiao a, *, Z.J. Liu b, **, Sean Li c, X.T. Zu a, d a

School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 610054, China Department of Physics, Lanzhou City University, Lanzhou 730070, China c School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia d Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 610054, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 January 2016 Received in revised form 4 August 2016 Accepted 12 September 2016

A systematic density functional theory study is performed to investigate the mechanical stability, elastic moduli, Debye temperature, thermal conductivity and electronic structures of Gd2-yThyZr2O7 and Gd2Zr2yThyO7. All the Th-doped Gd2Zr2O7 compositions are found to be structurally and mechanically stable. As compared with the pure Gd2Zr2O7, Th incorporation into both Gd-site and Zr-site results in generally better ductility, lower Debye temperature, and reduced thermal conductivity. The reduction in thermal conductivity can be as high as 25e32%, depending on the content of the Th substituent. Our calculations suggest that Th-doped Gd2Zr2O7, especially Gd2Zr2-yThyO7, exhibits better mechanical and thermal properties that are beneficial to its application as thermal barrier coating material at high temperatures than the pure state. © 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: DFT Gd2Zr2O7 Mechanical properties Thermal conductivity

1. Introduction With the development of high-temperature gas turbine and aircraft engines, thermal barrier coatings (TBCs) are introduced to protect the hot section of the metallic components and to increase the engine inlet gas temperature, which will ensure the system functioning properly under harsh environment and even improve the system efficiency and performance [1e4]. The severe working condition requires the TBC materials to have low thermal conductivity, good comprehensive mechanical properties, high phase stabilities and strong chemical inertness [5e8]. Up to now, the most commonly used TBCs are 6e8 wt% Y2O3-ZrO2 (6-8YSZ), with a thermal conductivity of 2.3 W/(mK) at 1000
C [9]. However, the 68YSZ ceramics still have some disadvantages like low sintering resistance and poor phase stability at high temperature (>1200
C) for the long-term application, which will increase the thermal conductivity and influence the coating durability [10e12]. In recent years, many new refractory ceramic materials have been investigated to develop the new TBCs with high thermal insulation capability and strong chemical stability at even higher

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H.Y. Xiao), [email protected] (Z.J. Liu). http://dx.doi.org/10.1016/j.actamat.2016.09.018 1359-6454/© 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

temperatures. Rare-earth zirconates, with a general formula Re2Zr2O7, where Re is Y or lanthanide element, have been identified as a class of low-thermal-conductivity ceramics for a new generation of TBCs, due to their relatively high melting point, large thermal expansion coefficient and strong chemical stability [1,13e16]. The thermal conductivities for Re2Zr2O7 (Re¼ Gd, Eu, Sm, Nd, Er, Dy, Yb, or La) have been reported to range from 1.1 to 1.8 W/(mK) at temperatures between 973 and 1473 K [1,12,17e19], which are even lower than those for Y2O3-ZrO2 ceramics. Particularly, Gd2Zr2O7, which shows ordered pyrochlore structure below 1550
C, exhibits relatively lower thermal conductivity than other zirconates [20,21]. In fact, Gd2Zr2O7 has been successfully grown as a barrier coating on a Ni-based super-alloy with traditional bond coats [22]. In order to improve the thermo-physical properties of Gd2Zr2O7 for its potential application as high-temperature thermal insulating material, considerable experimental investigations of chemical doping at Gd-site or Zr-site have been carried out. Wan et al. have studied the thermal transport properties of (LaxGd1-x)2Zr2O7 (0  x  1) ceramics employing laser-flash methods, who suggested that LaGdZr2O7 had the lowest thermal conductivity among all the investigated compositions and the Young's moduli of all the solid solutions were generally lower than those of pure states, i.e., La2Zr2O7 and Gd2Zr2O7[23]. The thermo-physical properties of (SmxGd1-x)2Zr2O7 (0  x  1) were investigated by Liu et al.

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employing high-temperature dilatometer and laser flash diffusivity measurements from room temperature to 1400
C, in which it was found that the thermal conductivities for Sm-doped Gd2Zr2O7 gradually decreased with the increasing temperature up to 800
C and increased very slightly thereafter [24]. Pan et al. also reported that the Young's moduli of Sm2Zr2O7-Gd2Zr2O7 solid solutions were lower than those of pure Sm2Zr2O7 and Gd2Zr2O7[25]. Liu et al. have found that the thermal conductivity of Gd2Zr2O7 was significantly reduced by the substitution of Nd3þ for Gd-site [26]. The (Gd1xYbx)2Zr2O7 (0  x  0.1) compounds were synthesized by Wang et al. via solid state reaction, who found that Yb2O3 doping reduced the thermal conductivity of Gd2Zr2O7 by ~25% and the lowest thermal conductivity was obtained at x ¼ 0.06 [27]. The influence of Zr-site substitution on the thermo-physical properties of Gd2Zr2O7 has been investigated as well. Wan et al. found that the thermal conductivity of Gd2Zr2O7 was significantly reduced by Ti4þ doping and a minimum value of the Young's modulus was obtained for Gd2(Zr0.7Ti0.3)2O7[28]. Although Gd2Zr2O7 with partial and complete substitution at Gd-site or Zr-site has been extensively studied experimentally, there are few theoretical investigations of the mechanical and thermo-physical properties of Gd2Zr2O7 or Re-doped Gd2Zr2O7. Schelling et al. have computed the thermal transport properties of (Gd1-xRex)2Zr2O7 (Re¼La, Y or Sm) and a series of pure pyrochlores using the molecular dynamics (MD) simulations [29,30]. For Gd2Zr2O7, the calculated thermal conductivities of 2.08 W/(mK) at 873 K and 1.91 W/(mK) at 1473 K were larger than the experimental measurement of 1.0e1.6 W/(mK) at the temperature ranging from 973 to 1150 K [1,19,22,27]. Recently, Feng et al. have conducted a complete investigation of the thermo-physical properties for rare earth zirconate pyrochlores by density functional theory (DFT) method [31] and the obtained minimum high-temperature thermal conductivity for Gd2Zr2O7 of 1.2 W/(mK) was in reasonable agreement with experimental results [27]. Up to now, the mechanical properties of zirconate pyrochlores doped with other chemical elements have not been reported yet. Besides, there are very few thermo-physical data on actinide-doped Gd2Zr2O7 in the literature. In order to further improve the performance of Gd2Zr2O7 for its potential application at high temperature, it is necessary to perform more detailed and in-depth investigation of the fundamental properties of the related systems. In this study, a systematic study of Th incorporation into Gd2Zr2O7 is carried out to explore how Th incorporation influences the mechanical, thermo-physical and electronic properties of Gd2Zr2O7. The solubility of Th that substitutes for Gd-site in Gd2Zr2O7 has been reported by Mandal et al. using a ceramic sintering route method [32], whereas no experimental or theoretical investigation of Th substitution for Zr-site has been carried out. Computer simulation of Pu3þ and Pu4þ substitution in Gd2Zr2O7 performed by Williford et al. suggested that the substitution of Pu for Gd- and Zr-site was most likely to occur under reducing and oxidizing conditions, respectively [33]. Cleave et al. have investigated U and Pu immobilization in A2B2O7 pyrochlores using the classical molecular dynamics method, and found that the solution energies for Pu substitution for A-site were lower than those for Bsite [34]. In order to explore if Th will exhibit similar accommodation behavior to U and Pu, Th incorporation into both Gd-site and Zr-site has been taken into account in this work. The site preference, mechanical properties, thermal conductivities, as well as electronic structures all have been determined. This study proposes Th-Gd2Zr2O7 to be new TBC candidates and provides new insights into the performance of TBC alternates, which may promote further theoretical and experimental investigations of the related topic. Besides thermal barrier coating, zirconate pyrochlores can also serve as the inert matrix for Th nuclear fuel [32,35], the host for

immobilization of nuclear wastes like Pu and Am (Th is often used as a surrogate for them) [36,37], the solid oxide fuel cell electrodes [38], high-temperature heating elements [39], oxidation catalyst [39], and refractory ceramics for use in magneto-hydrodynamic power generation [40]. In most of these applications the zirconate pyrochlores are exposed to the extreme condition of high temperature [41]. The predicted mechanical and thermos-physical results for Th-doped Gd2Zr2O7 will thus have important implications for the application of zirconate pyrochlores in different fields. 2. Computational details All the calculations are carried out based on the density functional theory (DFT) within the Vienna Ab initio Simulation Package (VASP). The projector augmented-wave (PAW) method [42,43] is employed to treat the interaction between ions and electrons. The generalized gradient approximation (GGA) parameterized by Perdew and Wang [44] are used to describe the exchange-correlation effects. All the calculations are conducted with a 4  4  4 kpoint grid in reciprocal space and a cutoff energy of 500 eV for the plane wave basis set, with spin-polarized effects considered. The uncertainty on the energy of the simulation box is about 0.03 eV. The structural relaxation is carried out at variable volume. The electronic configurations for the PAW potentials are 6s25p65d1 for Gd, 5s26s27s25p65d0 for Th, 5s15p04d3 for Zr and 2s22p4 for O. In this study, the substitution of Th for both Gd and Zr lattice sites in Gd2Zr2O7 with different concentrations, i.e., Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 (y ¼ 0, 0.5, 1.0, 1.5, 2.0), has been taken into account. In pyrochlore, an unoccupied interstitial site 8a exists [45]. For comparison, Th incorporation into Gd-site with oxygens occupying the vacant 8a site as charge compensators, i.e., Gd2-yThyZr2O7þO8a, is also considered. A supercell consisting of 88 atoms is used. The structural models for Gd1.5Th0.5Zr2O7, GdThZr2O7, Gd0.5Th1.5Zr2O7, Gd2Th0.5Zr1.5O7, Gd2ThZrO7 and Gd2Th1.5Zr0.5O7 are built by employing the special quasi-random structure approach [46e49]. The schematic view of the considered configurations is illustrated in Fig. 1. 3. Results and discussion 3.1. Structural properties and the site preference of Th-doped Gd2Zr2O7 In our calculations, the Gd 4f electrons are treated as core states. In order to explore how the consideration of Gd 4f electrons as valence states influences the density of state (DOS) distribution of Gd2Zr2O7, we compare the atomic projected DOS distribution by DFT with Gd 4f electrons as core states (DFT_f core), DFT with Gd 4f electrons as valence states (DFT_f valence), and DFT þ U (Ueff ¼ 6 eV) [31] with Gd 4f electrons as valence states (DFT þ U_f valence), as well as the orbital projected DOS for Gd in Fig. 2. When the Gd 4f electrons are treated as core states, the valence band maximum (VBM) are mainly contributed by O 2p orbitals hybridized with Zr 4d and Gd 5d orbitals. On the other hand, when the Gd 4f electrons are treated as valence states but no Hubbard U correction is made for them, the Gd 4f orbitals contribute significantly to the VBM and hybridize with the O 2p and Zr 4d states. The conduction band minimum (CBM) are also mainly contributed by the Gd 4f orbitals. In spite of these differences, the obtained band gap of 2.55 eV by DFT_f core and 2.57 eV by DFT_f valence are very similar to each other. As for the DOS distribution obtained by DFT þ U_f valence, we find that the Gd 4f orbitals at the VBM and CBM are pushed far away from each other and shifted to the lower and higher energy levels, respectively. Similar to the results obtained by DFT_f core, the VBM are mainly contributed by O 2p

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301

Fig. 1. Illustration of schematic views of (a)e(e) Gd2-yThyZr2O7; (f)e(i) Gd2Zr2-yThyO7; (j)e(m) Gd2-yThyZr2O7þO8a. The purple, grey, blue and red spheres represent Gd, Th, Zr and O atoms, respectively. The O8a is represented by the dark red spheres. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

hybridized with Zr 4d and Gd 5d orbitals. The resulted band gap of 2.74 eV is slightly larger than those obtained by DFT_f core and DFT_f valence methods. To elucidate if consideration of Gd 4f electrons will affect the elastic moduli, we further calculate the elastic constants. The calculated results are C11 ¼ 316.9 GPa, C12 ¼ 115.5 GPa, C44 ¼ 93.4 GPa by DFT_f core method, C11 ¼ 317 GPa, C12 ¼ 123.8 GPa, C44 ¼ 90.0 GPa by DFT_f valence method, and C11 ¼ 316.9 GPa, C12 ¼ 121.2 GPa, C44 ¼ 91.6 GPa by DFT þ U_f valence method. It is shown that the elastic constants obtained by DFT_f core deviate from the one calculated by DFT_f valence by only 0.1e8.3 GPa and deviate from the DFT þ U results by only 0e5.7 GPa. Feng et al. also found that the elastic constants determined by DFT and DFT þ U are similar to each other [31]. Since the focus of this work is the mechanical and thermal properties of Thcontaining Gd2Zr2O7, and consideration of Gd 4f electrons as valence states affect slightly the elastic constants, the subsequent calculations are thus performed by DFT with Gd 4f electrons as core states. A full structural relaxation is first performed for Gd2-yThyZr2O7, Gd2-yThyZr2O7þO8a and Gd2Zr2-yThyO7. The calculated and experimental lattice parameters, including lattice constant a0 and positional parameter xO48f, are listed in Table 1. For the pure Gd2Zr2O7, the calculated lattice constant of 10.452 Å and the positional parameter xO48f of 0.342 are in good agreement with experimental [50] and other theoretical [31,33,51,52] results. The variation of the lattice parameters for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 is presented in Fig. 3. As Th is doped at Gd-site, the unit cell length increases from 10.452 to 10.602 Å and the positional parameter xO48f increases from 0.342 to 0.360 when the composition varies from Gd2Zr2O7 to Th2Zr2O7. Oxygen occupation at the vacant 8a site results in the lattice parameters for Gd2-yThyZr2O7þO8a being slightly

larger than those for Gd2-yThyZr2O7. Experimentally, Mandal et al. also found a lattice expansion for Gd2-yThyZr2O7þy/2 with the increasing Th content [32]. As compared with Gd2-yThyZr2O7, Th substitution for Zr-site results in a more linear increase in the unit cell length, which follows well the Vegard's law [52]. Besides, the lattice expansion for Gd2Zr2-yThyO7 is more significant than Gd2yThyZr2O7, i.e., from 10.452 Å for Gd2Zr2O7 to 10.941 Å for Gd2Th2O7. These results are reasonable since the ionic radius of 1.05 Å for Th4þ is much larger than the value of 0.72 Å for Zr4þ[53]. As for xO48f, it increases non-linearly with the increasing Th concentration. In order to elucidate the charge state of Th incorporated into Gdsite and Zr-site in Gd2Zr2O7, the Bader charge of each ion in Gd2yThyZr2O7 and Gd2Zr2-yThyO7 has been analyzed and summarized in Table 2. Upon Th substitution for Zr-site, Th exhibits an average Bader charge of þ2.56 jej, which is close to the value of þ2.63 jej in ThO2. The result for ThO2 is reasonable since the classical model of an ionic material with nominal charges is oversimplified [54]. As Th substitutes for Gd-site, the average Bader charge for Th is þ2.62 jej. In the case of Th incorporation into Gd-site with oxygen occupying the vacant 8a site, the Bader charge for Th is þ2.64 jej. Obviously, Th has only one charge state, i.e., Th4þ. When no oxygen occupies the vacant 8a site, the excess electrons introduced by Th incorporation into Gd-site are transferred to the Zr ions, resulting in the Bader charge of Zr much smaller than those in Gd2Zr2-yThyO7. The Bader charge of Zr in Gd2-yThyZr2O7þO8a is comparable to that in Gd2Zr2yThyO7, since the O8a acts as charge compensator here. To explore whether Gd-site or Zr-site is more preferable for Th incorporation into Gd2Zr2O7, we calculate the reaction energies for Gd2y0 Thy0 Zr2 O7 þ nThO2 /Gd2y Thy Zr2 O7 þ n2Gd2 O3 þ n4O2 (1) and Gd2 Zr2y0 Thy0 O7 þ nThO2 /Gd2 Zr2y Thy O7 þ nZrO2 (2). In reaction (1), the Th ions exclusively occupy the Gd-site and it allows for the rejection of Gd ions to form excess Gd2O3. The reaction

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Fig. 2. Atomic projected DOS distributions for Gd2Zr2O7 and orbital projected DOS for Gd. The elastic constants (C11, C12, C44, in GPa) obtained by each method are also shown.

energies are defined as EReac ðGd2y Thy Zr2 O7 Þ ¼ EðGd2y Thy Zr2 O7 Þ where E(Gd2EðGd2y0 Thy0 Zr2 O7 Þ þ n2mGd2 O3  nmThO2 þ n4mO2 , yThyZr2O7) is the total energy of Gd2Zr2O7 with n Th ions

incorporation into Gd-site, E(Gd2-y’Thy’Zr2O7) is the total energy of the system before n Th ions substitutes for Gd, with y0 ¼ y  n, and mGd2 O3 ,mThO2 and mo2 are the chemical potentials of Gd2O3, ThO2 and

F.A. Zhao et al. / Acta Materialia 121 (2016) 299e309 Table 1 The lattice constant a0 (Å) and O48f positional parameter x for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 (y ¼ 0, 0.5, 1.0, 1.5, 2.0). The values in the parenthesis are obtained for Gd2-yThyZr2O7þO8a, in which Th substitutes for Gd-site with oxygen occupation at the vacant 8a site as charge compensators.

Gd2Zr2O7 Exp [50] Cal [33] Cal [51] Cal [52] Cal [31] Gd1.5Th0.5Zr2O7 Gd1.0Th1.0Zr2O7 Gd0.5Th1.5Zr2O7 Th2Zr2O7 Gd2Zr1.5Th0.5O7 Gd2Zr1.0Th1.0O7 Gd2Zr0.5Th1.5O7 Gd2Th2O7

a0

xO48f

10.452 10.540 10.521 10.66 10.446 10.52 10.459 10.481 10.533 10.602 10.545 10.671 10.798 10.941

0.342 0.344 0.339 0.342 (10.482) (10.512) (10.553) (10.602)

0.346 0.351 0.355 0.360 0.338 0.342 0.333 0.358

(0.346) (0.352) (0.357) (0.363)

303

mGd2 O3 ¼ 2mGd þ 3mO , O2 molecule, respectively. Here, andmThO2 ¼ mTh þ 2mO ; and mGd , mTh and mO are the chemical potentials of Gd, Th and O, respectively. The reaction (2) can be explained by the exclusive occupation of the Zr-site by Th ions, the rejection of Zr ions and the formation of excess ZrO2. The corresponding reaction energies are defined as EReac ðGd2 Zr2y Thy O7 Þ ¼ EðGd2 Zr2y Thy O7 Þ  EðGd2 Zr2y0 Thy0 O7 Þ  nmThO2 þ nmZrO2 . Here, E(Gd2Zr2-yThyO7) is the total energy of Gd2Zr2O7 with n Th ions incorporated into Zr-site, E(Gd2Zr2-y’Thy’O7) is the total energy of the system before n Th ions occupy Zr-site, with y0 ¼ y  n, and mZrO2 is the chemical potential of ZrO2. Here, mZrO2 ¼ mZr þ 2mO and mThO2 ¼ mTh þ 2mO . The reaction energies for Th incorporation into Gd-site with oxygen occupying the vacant 8a site, i.e., ðGd2y0 Thy0 Zr2 O7 þ O8a Þ þ nThO2 þ O2 /ðGd2y Thy Zr2 O7 þ O8a Þþ nGd O þ nO (3), are also calculated, where the reaction energies 2 3 2 4 2

are defined as EReac Gd2y Thy Zr2 O7 þ O8a ¼ E Gd2y Thy Zr2 O7 þ   
 O8a  E Gd2y0 Thy0 Zr2 O7 þ O8a þ n2mGd2 O3 nmThO2 þ n4  1 mO2 . Obviously, the reaction energies for reactions (1)e(3) are strongly dependent on the chemical potentials of O (mO ), Gd (mGd ) and Zr (mZr ). For the chemical potential of oxygen, it satisfies   T mO2 ðpO2 ; TÞ ¼ 2mO ðpO2 ; TÞ ¼ E0 ðO2 Þ þ ðm0O2  E0 Þ TT0  5k T ln 2 T0 ! þkT ln

PO2 PO0

2

Fig. 3. Variation of lattice parameters for (a) Gd2-yThyZr2O7 with and without O8a and (b) Gd2Zr2-yThyO7 with Th content.

. At T ¼ 0 K, mO ðpO2 ; TÞ ¼ 12E0 ðO2 Þ, where E0(O2) is the

total energy of a free, isolated O2 molecule at 0 K. In our calculations of reaction energies, the most oxygen-rich condition is assumed and the oxygen chemical potential is taken as a constant, i.e., half of the total energy of an isolated O2 molecule at 0 K. To obtain mGd and mZr , four different situations, i.e., Zr-rich, ZrO2-rich, Gd-rich, as well as Gd2O3-rich, are considered. The corresponding mGd is calculated to be 26.31, 14.85, 5.10 and 14.89 eV, respectively, and the corresponding mZr is determined to be 9.03, 20.49, 30.24 and 20.45 eV, respectively. Taking mGd from 26.31 eV (Zr-rich) to 5.10 eV (Gd-rich), and mZr from 30.24 eV (Gd-rich) to 9.03 eV (Zr-rich), we plot the reaction energies for Gd2-yThyZr2O7 (and Gd2-yThyZr2O7þO8a) as a function of mGd and the reaction energies for Gd2Zr2-yThyO7 as a function of mZr in Fig. 4. It is shown that under Zr-rich conditions the reaction energies for Gd2-yThyZr2O7 are generally smaller than those for Gd2Zr2-yThyO7, i.e., Th substitution for Gd-site is more preferable. Especially, Th incorporation into Gd-site with oxygen occupying the 8a site is energetically more favorable than the pure Th substitution. However, under Gd-rich conditions, Th favors to occupy the Zr-site. These results suggest that the site preference of Th for Gd or Zr lattice site depends on the chemical environments. Since the current work mainly focuses on how Th incorporation into Gd-site and Zr-site influences the mechanical and thermal properties of Gd2Zr2O7, only Gd2-yThyZr2O7 without oxygen

Table 2 Bader charge (jej) for each ion in Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 (y ¼ 0, 0.5, 1.0, 1.5, 2.0). The values in the parenthesis are obtained for Gd2-yThyZr2O7þO8a, in which Th substitutes for Gd-site with oxygen occupation at the vacant 8a site as charge compensators.

Gd2Zr2O7 Gd1.5Th0.5Zr2O7 Gd1.0Th1.0Zr2O7 Gd0.5Th1.5Zr2O7 Th2Zr2O7 Gd2Zr1.5Th0.5O7 Gd2Zr1.0Th1.0O7 Gd2Zr0.5Th1.5O7 Gd2Th2O7

Gd

Th

Zr

O48f

O8b

O8a

2.07 2.04(2.04) 2.03(2.03) 2.02(2.01) e 2.02 2.02 2.00 2.07

e 2.63(2.64) 2.62(2.64) 2.63(2.63) 2.62(2.65) 2.59 2.56 2.59 2.50

2.22 2.12 (2.25) 2.01(2.26) 1.94(2.28) 1.88(2.32) 2.23 2.23 2.21 e

1.22 1.22 (1.22) 1.23(1.22) 1.25(1.23) 1.28(1.24) 1.23 1.25 1.28 1.30

1.32 1.29(1.2) 1.29(1.28) 1.310(1.30) 1.32(1.31) 1.30 1.31 1.31 1.33

e (1.12) (1.11) (1.14) (1.17) e e e e

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Fig. 4. Variation of reaction energy for Gd2-yThyZr2O7 with and without O8a as a function of mGd and for Gd2Zr2-yThyO7 as a function of mZr .

occupying the 8a site and Gd2Zr2-yThyO7 are considered in the subsequent calculations. 3.2. Mechanical stability of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 Based on the optimized structures, the elastic constants of Gd2and Gd2Zr2-yThyO7 are further calculated. For cubic phase, there are three independent elastic constants, i.e., C11, C12 and C44, where C11 represents the uniaxial deformation along the [001] direction, C12 is the pure shear stress at (110) crystal plane along the [110] direction and C44 is a pure shear deformation on (100) crystal plane [55]. The calculated values of C11, C12 and C44 along with available theoretical values are given in Table 3. The calculated C11 of 324.7 GPa, C12 of 125.3 GPa and C44 of 94.0 GPa for Gd2Zr2O7 are comparable with the other theoretical values [31]. Fig. 5 presents the variation of elastic constants of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 with the increasing Th content. As the y value increases from 0 to 2.0, the C11 for Gd2-yThyZr2O7 first increases, reaching a maximum value at y ¼ 1.5, then decreases again; the C12 and C44 smoothly increases and decreases with the increasing y value, respectively. In the case of Gd2Zr2-yThyO7, the elastic constants exhibit somewhat different character from Gd2-yThyZr2O7. The C11 first decreases with the increasing Th content to y ¼ 1.0, then rises up to y ¼ 1.5 and finally decreases to y ¼ 2.0. As for the C12

yThyZr2O7

Table 3 Elastic constants (C11, C12, C44, in GPa), bulk modulus (B in GPa), shear modulus (G in GPa) and Young's modulus (E in GPa) for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 (y ¼ 0, 0.5, 1.0, 1.5, 2.0).

Gd2Zr2O7 Exp [60] Exp [41] Cal [31] Gd1.5Th0.5Zr2O7 Gd1.0Th1.0Zr2O7 Gd0.5Th1.5Zr2O7 Th2Zr2O7 Gd2Zr1.5Th0.5O7 Gd2Zr1.0Th1.0O7 Gd2Zr0.5Th1.5O7 Gd2Th2O7

C11

C12

C44

B

G

E

324.7

125.3

94

277 341.9 365.2 370 365.3 309.3 274.7 280.9 246.7

110 127.8 134.4 137.1 145.1 122.9 113 111.8 99.1

52 86.6 82 80.7 77.5 79.1 70.6 70.7 66.1

191.8 153 174 165 199.2 211.3 214.7 218.5 185.0 166.9 168.2 148.3

96.2 80 93 63 94.3 94.0 93.5 89.2 84.5 74.5 75.9 69.1

247.3 205 236 214 244.3 245.7 244.9 235.6 219.9 194.6 198.0 179.4

Fig. 5. Variation of the elastic constants C11, C12 and C44 for (a) Gd2-yThyZr2O7 and (b) Gd2Zr2-yThyO7 with Th content.

and C44, they decrease gradually with the increasing Th content. These results indicate that Th incorporation into Gd-site and Zr-site will cause different influences on the mechanical properties of Gd2Zr2O7. For a cubic crystal, the generalized elastic stability criteria are defined as C11þ2C12 > 0; C44 > 0;C11-C12>0 [55]. The first two criteria are obviously necessary for stability, because C11þ2C12 and C44 are related to bulk modulus (B) and shear modulus (G) along the [100] direction, respectively, i.e., B¼(C11þ2C12)/3 and G100 ¼ C44. The third criterion, which is related to the shear modulus along the [110] direction, i.e., G110¼(C11-C12)/2, is violated in a number of situations such as melting, pressure-induced amorphization and polymorphism [55]. In the case of zero applied stress, these criteria are adequate to describe the stability limits of perfect crystals at finite strain [55]. In our calculations, the determined elastic constants for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 are all positive and the three criteria stated above are all satisfied. These results suggest that as Th substitutes for both Gd-site and Zr-site, the resulted Gd2yThyZr2O7 and Gd2Zr2-yThyO7 compositions are all mechanically stable. 3.3. Bulk modulus, shear modulus and Young's modulus of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 Under Voigt-Reuss-Hill (VRH) approximation, the bulk modulus, shear modulus and Young's modulus (E) can be described as B¼(C11þ2C12)/3, G¼((C11-C12þ3C44)/5 þ 5(C11-C12)C44/(4C44þ3(C11C12)))/2, E ¼ 9BG/(3B þ G) [56e58]. Here, the B and G can be used to

F.A. Zhao et al. / Acta Materialia 121 (2016) 299e309

measure the resistance to volume change with invariable proportions and plastic deformation, respectively [59]. The calculated elastic moduli for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 together with experimental and other theoretical values are listed in Table 3. For Gd2Zr2O7, the determined bulk modulus of 199.2 GPa, shear modulus of 94.3 GPa and Young's modulus of 244.3 GPa are generally larger than the calculated results reported by Feng et al. [31]. The differences should be resulted from the differences in the employed exchange-correlation potential and the pseudopotential. In our work, a GGA method is used to describe the exchangecorrelation potential between electrons and a PAW pseudopotential is employed to describe the ion-electron interactions, whereas Feng et al. employed a localized density approximation method and ultrasoft pseudopotentials. Experimentally, Shimamura et al. synthesized the Gd2Zr2O7 samples through a solid-state reaction and characterized the elastic moduli via ultrasound pulse-echo technique [41]. The obtained experimental data was in good agreement with our results. Dijk et al. also reported the elastic moduli for Gd2Zr2O7, where the bulk modulus and Young's modulus were much smaller than our results [60]. Since no details about the measurement of the elastic moduli were provided in their work, it is difficult to explore the origin of this discrepancy. The variation of the elastic moduli for all the compounds with Th content is presented in Fig. 6. The bulk moduli increase with the increasing Th content for Gd2-yThyZr2O7, while for Gd2Zr2-yThyO7 they initially decrease with the increasing Th content to y ¼ 1.0, then rise up to y ¼ 1.5 and finally decrease to y ¼ 2.0. As compared with the bulk moduli, the shear moduli vary more smoothly with

305

the increasing Th content. In the case of the Young's modulus, it is interesting to find that the values for Gd2-yThyZr2O7 and Gd2Zr2yThyO7 are about 1.6e67.9 GPa lower than those of pure Gd2Zr2O7. Similar phenomena have also been observed experimentally for Gd2-xSmxZr2O7[25], Gd2-xLaxZr2O7[23] and Gd2Zr2-xTixO7[28]. One possible explanation is that the size and coupling force misfit induced by the substitutional atoms have “softened” the lattice and the lattice relaxation resulted from the strain field fluctuation decreases the Young's modulus of these compounds [23]. A low Young's modulus is preferable for TBC materials, which produces small residual stresses in the coating system under working conditions and results in good thermo-mechanical stability [61,62]. Our calculations show that the Young's modulus for Gd2-yThyZr2O7 is generally larger than that for Gd2Zr2-yThyO7, suggesting that Gd2Zr2O7 pyrochlores with Th substitution for Zr-site may be more favorable to be used as TBC materials than those with Th incorporation into Gd-site. 3.4. Malleability and elastic anisotropy of Th-doped Gd2Zr2O7 The brittle or ductile behavior is an important mechanical property of materials, which closely correlates with their reversible compressive deformation and fracture ability [59]. The Pugh's indicator (G/B ratio) is widely employed to measure the brittleness or ductility of a material and the critical value of this ratio is 0.5 [63]. If Pugh's indicator is smaller than 0.5, the material is ductile; otherwise, it demonstrates brittleness [63]. In our calculations, the G/B ratio for Gd2Zr2O7 is determined to be 0.5, which is consistent with the experimental value of 0.53 [41]. As shown in Table 4, Th incorporation into both Gd-site and Zr-site results in smaller G/B ratio than Gd2Zr2O7, suggesting that Th-doped Gd2Zr2O7 is principally more ductile than the pure state. Another widely used malleability indicator is Poisson's ratio (s). The Poisson's ratio is about 0.1 for strongly covalent crystals and 0.33 for ductile metallic materials [64]. The calculated Poisson's ratio for Gd2Zr2O7 is 0.285, which is comparable with the experimental [41,60] and other theoretical [31] results. As shown in Table 4, the ratios for Gd2yThyZr2O7 and Gd2Zr2-yThyO7 are generally larger than 0.285, which is consistent with the results obtained from Pugh's indicator. Comparing the G/B ratio and Poisson's ratio of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7, we find that the malleability of them are comparable to each other. Another important mechanical property is the elastic anisotropy of materials, which is related to the possibility of microcracks appearing in materials [59]. Here we use Zener's anisotropic index to characterize the elastic anisotropy of cubic crystals, which is defined as AZ ¼ 2C44/(C11-C12) [65]. If AZ equals to unity, the crystals are elastically isotropic; otherwise, the materials have elastic Table 4 Poisson's ratio (s), Pugh's indicator (G/B), anisotropic indices (AZ), average sound wave velocity (vm), and Debye temperature (QD) for Gd2-yThyZr2O7 and Gd2Zr2yThyO7 (y ¼ 0, 0.5, 1.0, 1.5, 2.0).

Fig. 6. Variation of the bulk modulus (B), shear modulus (G) and Young's modulus (E) for (a) Gd2-yThyZr2O7 and (b) Gd2Zr2-yThyO7 with Th content.

Gd2Zr2O7 Exp [60] Exp [41] Cal [31] Gd1.5Th0.5Zr2O7 Gd1.0Th1.0Zr2O7 Gd0.5Th1.5Zr2O7 Th2Zr2O7 Gd2Zr1.5Th0.5O7 Gd2Zr1.0Th1.0O7 Gd2Zr0.5Th1.5O7 Gd2Th2O7

s

G/B

AZ

vm (m/s)

QD (K)

0.285 0.276 0.273 0.284 0.296 0.306 0.310 0.320 0.302 0.306 0.304 0.298

0.50

0.94

4832.7 3772.1 4658.5 3374.2 4667.8 4567.6 4474.5 4322.9 4374.8 3988.1 3914.6 3646.2

612.6 474.9

0.53 0.47 0.45 0.44 0.41 0.46 0.45 0.45 0.47

0.62 0.81 0.71 0.69 0.70 0.85 0.87 0.84 0.90

423.8 591.2 577.3 562.8 540.2 549.6 495.1 480.3 441.5

306

F.A. Zhao et al. / Acta Materialia 121 (2016) 299e309

Table 5 The molar mass (M, in g/mol) and thermal conductivity (k, in W/(mK)) for Gd2yThyZr2O7 and Gd2Zr2-yThyO7 (y ¼ 0, 0.5, 1.0, 1.5, 2.0). M

kClarke min

kCahill min

kExp:

kCal:

Gd2Zr2O7

1217.9

1.29

1.42

1.6 [1] 1.5 [27] 1.0 [19] 1.1e1.4 [22]

1.2 [31] 1.91 [29] 2.08 [30]

Gd1.5Th0.5Zr2O7 Gd1.0Th1.0Zr2O7 Gd0.5Th1.5Zr2O7 Th2Zr2O7 Gd2Zr1.5Th0.5O7 Gd2Zr1.0Th1.0O7 Gd2Zr0.5Th1.5O7 Gd2Th2O7

1292.7 1367.5 1442.3 1517.0 1358.7 1499.5 1640.3 1781.1

1.24 1.21 1.17 1.12 1.14 1.02 0.97 0.88

1.38 1.35 1.31 1.26 1.27 1.13 1.09 0.98

solid, which is used to characterize many properties of materials, such as thermal vibration of atoms, heat capacity, and thermal expansion coefficient, etc. [66] The relatively high thermal expansion coefficient is favorable for the fabrication of superior TBC materials. Employing the standard equation in Ref. [67], the Debye    1 3 temperature can be calculated by QD ¼ hk 43np NMA r vm , where h and k are the Planck's and Boltzmann's constants, respectively, n is the number of atoms per unit cell, NA is the Avogadro's number, M is the unit-cell molecular weight and r is the density. In the above formula, vm is the average sound wave velocity, which can be " !#1 3

deduced from vm ¼

anisotropy. The calculated AZ values for Gd2-yThyZr2O7 and Gd2Zr2yThyO7 are summarized in Table 4. For Th incorporation into Gdsite, the AZ values are all smaller than unity, indicating that Gd2Zr2O7 pyrochlores with Th occupation at Gd-site are anisotropic. In the case of Th substitution for Zr-site, it is found that the AZ values are larger than those for Gd2-yThyZr2O7, indicating that Gd2Zr2-yThyO7 are less elastically anisotropic than Gd2-yThyZr2O7. Since the anisotropic materials have the greater possibility of microcracks, our calculations predict that Gd2Zr2-yThyO7 may be more mechanically stable than Gd2-yThyZr2O7. 3.5. Debye temperature and thermal conductivity of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 The Debye temperature is an important physical parameter of a

1 3

2 v3l

þ v13 t

. Here, vl (longitudinal sound

wave velocity) and vt (transverse sound wave velocity) are calcusffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   pffiffiffiffiffiffiffiffi lated by vl ¼ B þ 43 G r and vt ¼ G=r, respectively [31]. The calculated results of vm and QD for all the compositions together with available experimental and theoretical data are presented in Table 4. For Gd2Zr2O7, the determined vm of 4832.7 m/s agrees very well with the experimental value reported by Shimamura and Arima [41], while it is larger than the experimental result of Dijk et al. [60] and theoretical result of Feng et al. [31]. Consequently, our calculated Debye temperature of 612.6 K for Gd2Zr2O7 is larger than the experimental and theoretical results [31,60]. Comparing the results for Gd2Zr2O7 with and without Th doping, we find that Th doping at both Gd-site and Zr-site causes Debye temperatures to be 21.4e171.1 K smaller than the pure Gd2Zr2O7, indicating that the inter-atomic interactions are weakened by Th incorporation and the thermal expansion coefficient will be increased. Moreover, Th incorporation into Zr-site leads to generally smaller Debye

Fig. 7. Band structures for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7.

F.A. Zhao et al. / Acta Materialia 121 (2016) 299e309

307

Fig. 8. Projected density of states for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7.

temperatures than those into Gd-site at each certain Th content, suggesting that the Gd2Zr2-yThyO7 compounds have higher thermal expansion coefficient than Gd2-yThyZr2O7.

The minimum high-temperature thermal conductivities of Gd2and Gd2Zr2-yThyO7 are calculated by Clarke's model

yThyZr2O7

 23

[68,69], i.e., kClarke min ¼ 0:87kMa

E2 r6 , and Cahill's model, i.e., 1

1

308

k kCahill ¼ 2:48 min

F.A. Zhao et al. / Acta Materialia 121 (2016) 299e309

 23 n V



ðvl þ 2vt Þ[70], where M a is the average mass per

atom and V is the cell volume. The determined results are summarized in Table 5. For the pure Gd2Zr2O7, the thermal conductivities are calculated to be 1.29 W/(mK) by Clarke's model and 1.42 W/(mK) by Cahill's model, which is located within the reported experimental range of 1.0e1.6 W/(mK) [1,19,22,27]. Our calculated results also agree well with the value of 1.2 W/(mK) obtained by DFT calculations [31], but differ greatly from the value of 1.91 W/ (mK) at 1473 K [29] and 2.08 W/(mK) at 873 K [23] predicted by classical MD method. For each certain Th content in Gd2-yThyZr2O7 and Gd2Zr2-yThyO7, the thermal conductivities obtained by Cahill's model are generally larger than those by Clarke's model, while both models predict that Th incorporation into Gd2Zr2O7 reduces the thermal conductivities. Based on Clarke's model, the thermal conductivity reduction can be as high as 25% for 75% Th incorporation into Zr-site, and even reach 32% for complete Th substitution. The reduction in thermal conductivities was also observed in other Gd2Zr2O7-based solid solutions. For example, the thermal conductivity of (Yb0.06Gd0.94)2Zr2O7 was nearly 25% lower at 1073 K than that of Gd2Zr2O7[27]. In (LaxGd1-x)2Zr2O7 series, La2O3 doping reduced the k value of Gd2Zr2O7 by ~ 17% at 873 K [23]. MD simulations performed by Schelling suggested that the thermal conductivity of Gd2Zr2O7 was reduced by ~4%, 6% and 1% for La, Y and Sm substitution for Gd-site at 873 K, respectively [30]. As compared with Gd2-yThyZr2O7, Gd2Zr2-yThyO7 pyrochlores exhibit much lower thermal conductivity. The possible reason is that the thermal conduction is related to the phonon scattering, which is strongly dependent on the variation in the ionic size and mass difference between the substituent and host atoms [11]. In the present cases, the size mismatch and mass differences between Th4þ and Zr4þ are larger than those between Th4þ and Gd3þ ion, which can introduce more phonon scattering sources and result in lower thermal conductivity for Gd2Zr2-yThyO7. These results suggest that Gd2Zr2O7 with Th substitution for Zr-site may be better candidates for hightemperature thermal insulating materials than Gd2-yThyZr2O7. 3.6. Electronic properties of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 The above calculations suggest that Gd2Zr2-yThyO7 are more preferable as TBC materials than Gd2-yThyZr2O7. In order to explore the origin of the difference in mechanical and thermal properties, we further analyze the band structures and partial density of states distribution, as shown in Figs. 7e8. For Gd2Zr2O7, the band structure shows an energy band gap of 2.72 eV and exhibits insulating character. The DOS reveals that the hybridization of O 2p with Gd 5d and Zr 4d orbitals dominates at the VBM. As Th substitutes for Gdsite, the VBM and CBM shift towards lower energy levels, resulting in much smaller band gap, i.e., 1.035 eV for Gd1.5Th0.5Zr2O7, 1.477 eV for GdThZr2O7, and 0.319 eV for Gd0.5Th1.5Zr2O7. For complete substitution, the Th2Zr2O7 even exhibits metallic character. The band structures of Gd2Zr2-yThyO7 compounds show relatively large band gaps, which are determined to be 3.066 eV for Gd2Zr1.5Th0.5O7, 3.098 eV for Gd2ZrThO7, 2.715 eV for Gd2Zr0.5Th1.5O7 and 0.958 eV for Gd2Th2O7. It should be pointed out here that the band gap predicted from our calculations is KohnSham-band gap, which is generally significantly smaller than the real band gap[71]. The Th 5d orbitals appear at the VBM and hybridize with the O 2p orbitals and few Gd 5d and Zr 4d states. With the increasing Th content, more and more Th 5d orbitals appear at the VBM and the interaction between O 2p and Th 5d becomes much stronger. At the CBM, the hybridization of Th 5d, O 2p and Zr 4d states shifts to lower energy levels, leading to reduced band gap with the increasing Th content. In the cases of A2Zr2-yPuyO7 (A ¼ Gd and La), the band gaps are also narrowed as the compositions vary

from A2Zr2O7 to A2Pu2O7[72,73]. Comparing the electronic structures of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7, we find that the VBM for Gd2-yThyZr2O7 is mainly contributed by the Zr 4d and O 2p orbitals, whereas for the VBM of Gd2Zr2-yThyO7 the O 2p states dominate and hybridize with the Zr 4d, Th 5d and Gd 5d orbitals. Besides, the 〈Zr-O〉 bonds in Gd2Zr2-yThyO7 are more covalent than those in Gd2yThyZr2O7. Such differences may contribute significantly to their different thermo-mechanical properties. 4. Conclusion In summary, the mechanical properties, Debye temperatures, thermal conductivities and electronic structures of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 pyrochlores have been investigated by the DFT method. The calculated elastic constants are all positive and satisfy the generally mechanical stability criteria. The Young's moduli for Gd2-yThyZr2O7 and Gd2Zr2-yThyO7 are about 1.6e67.9 GPa smaller than Gd2Zr2O7, and the Debye temperature for Gd2Zr2O7 is decreased by 21.4e171.1 K by Th incorporation, indicative of larger thermal expansion coefficient for Th-doped Gd2Zr2O7. The Th substitution for Gd-site and Zr-site also causes better malleability and lower thermal conductivity. Based on Clarke's model, the thermal conductivity reduction can be as high as 25% for 75% Th incorporation into Zr-site, and even reach 32% for complete Th substitution. Comparing the mechanical and thermal-physical properties of Gd2-yThyZr2O7 and Gd2Zr2-yThyO7, we find that the Young's modulus, Debye temperature and thermal conductivity for Gd2Zr2-yThyO7 are generally lower. Our calculations suggest that Th-incorporated Gd2Zr2O7, particularly Gd2Zr2-yThyO7, are potentially good candidates as TBC materials at high temperature. Acknowledgements H.Y. Xiao was supported by the NSAF Joint Foundation of China (Grant No. U1530129) and the scientific research starting funding of University of Electronic Science and Technology of China (Grant No. Y02002010401085). Z.J. Liu was supported by National Natural Science Foundation of China (Grant No. 11464025) and the New Century Excellent Talents in University under Grant no. NCET-110906. The theoretical calculations were performed using the supercomputer resources at TianHe-1 located at National Supercomputer Center in Tianjin. References [1] J. Wu, X.Z. Wei, N.P. Padture, P.G. Klemens, M. Gell, Low-thermal-conductivity rare-earth zirconates for potential thermal-barrier-coating applications, J. Am. Ceram. Soc. 85 (2002) 3031e3035. [2] R.A. Miller, Current status of thermal barrier coatings-an overview, Surf. Coat. Technol. 30 (1987) 1e11. [3] A.G. Evans, D.R. Mumm, J.W. Hutchinson, G.H. Meier, F.S. Pettit, Mechanisms controlling the durability of thermal barrier coatings, Prog. Mater. Sci. 46 (2001) 505e553. [4] N.P. Padture, M. Gell, E.H. Jordan, Thermal barrier coatings for gas-turbine engine applications, Science 296 (2002) 280e284. [5] J.S. Gardner, M.J.P. Gingras, J.E. Greedan, Magnetic pyrochlore oxides, Rev. Mod. Phys. 82 (2010) 53. [6] Y. Jiang, J.R. Smith, G.R. Odette, Prediction of structural, electronic and elastic properties of Y2Ti2O7 and Y2Ti2O5, Acta Mater 58 (2010) 1536. [7] M.R. Winter, D.R. Clarke, Thermal conductivity of yttria-stabilized zirconiaehafnia solid solutions, Acta Mater 54 (2006) 5051. [8] J.Y. Wang, Y.C. Zhou, Z.J. Lin, Mechanical properties and atomistic deformation mechanism of g-Y2Si2O7 from first-principles investigations, Acta Mater 55 (2007) 6019. [9] W. Pan, S.R. Phillpot, C.L. Wan, A. Chernatynskiy, Z.X. Qu, Low thermal conductivity oxides, MRS Bull. 37 (2012) 917e922. [10] D.R. Clarke, M. Oechsner, N.P. Padture, Thermal-barrier coatings for more efficient gas-turbine engines, MRS Bull. 37 (2012) 891. [11] D.R. Clarke, S.R. Phillpot, Thermal barrier coating materials, Mater. Today 8 (2005) 22. [12] R. Vassen, X.Q. Cao, F. Tietz, D. Basu, D. Stover, Zirconates as new materials for

F.A. Zhao et al. / Acta Materialia 121 (2016) 299e309 thermal barrier coatings, J. Am. Ceram. Soc. 83 (2000) 2023e2028. [13] N.P. Padture, M. Gell, E.H. Jordan, Thermal barrier coatings for gas-turbine engine applications, Science 296 (2002) 280e284. [14] R. Vassen, X.Q. Cao, F. Tietz, D. Basu, D. Stover, Theoretical elastic stiffness, structure stability and thermal conductivity of La2Zr2O7 pyrochlore, J. Am. Ceram. Soc. 83 (2000) 2023e2028. [15] B. Liu, J.Y. Wang, Y.C. Zhou, T. Liao, F.Z. Li, Theoretical elastic stiffness, structure stability and thermal conductivity of La2Zr2O7 pyrochlore, Acta Mater 55 (2007) 2949e2957. [16] Z.X. Qu, C.L. Wan, W. Pan, Thermophysical properties of rare-earth stannates: effect of pyrochlore structure, Acta Mater 60 (2012) 2939e2949. [17] Q. Xu, W. Pan, J.D. Wang, C.L. Wan, L.H. Qi, H.Z. Miao, K. Mori, T. Torigoe, rareearth zirconate ceramics with fluorite structure for thermal barrier coatings, J. Am. Ceram. Soc. 89 (2006) 340. [18] G. Suresh, G. Seenivasan, M.V. Krishnaiah, P.S. Murti, Investigation of the thermal conductivity of selected compounds of lanthanum, samarium and europium, J. Alloys. Compd. 269 (1998) L9. [19] G. Suresh, G. Seenivasan, M.V. Krishnaiah, P.S. Murti, Investigation of the thermal conductivity of selected compounds of gadolinium and lanthanum, J. Nucl. Mater 249 (1997) 259. [20] K.V.G. Kutty, S. Rajagopalan, C.K. Mathews, Thermal expansion behaviour of some rare earth oxide pyrochlores, Mater. Res. Bull. 29 (1994) 759e766. [21] A.N. Radhakrishnan, P.P. Rao, K.S.M. Linsa, M. Deepa, P. Koshy, Influence of disorder-to-order transition on lattice thermal expansion and oxide ion conductivity in (CaxGd1ex)2(Zr1exMx)2O7 pyrochlore solid solutions, Dalton. Trans. 40 (2011) 3839e3848. [22] M.J. Maloney, US Patent and Trademark Office (United Technologies Corporation, Hartford, CT) (2001a.b) Pat. 6,177,200 Pat. 176,231,991. [23] C.L. Wan, W. Pan, Q. Xu, Y.X. Qin, J.D. Wang, Z.X. Qu, M.H. Fang, Effect of point defects on the thermal transport properties of (LaxGd1ex)2Zr2O7: Experiment and theoretical model, Phys. Rev. B 74 (2006) 144109. [24] Z.G. Liu, J.H. Ouyang, Y. Zhou, Structural evolution and thermophysical properties of (SmxGd1ex)2Zr2O7 (0 x 1.0) ceramics, J. Alloys. Compd. 472 (2009) 319e324. [25] W. Pan, C.L. Wan, Q. Xu, J.D. Wang, Z.X. Qu, Thermal diffusivity of samariumegadolinium zirconate solid solutions, Thermochim. Acta 455 (2007) 16. [26] Z.G. Liu, J.H. Ouyang, Y. Zhou, Preparation and thermophysical properties of (NdxGd1ex)2Zr2O7 ceramics, J. Mater. Sci. 43 (2008) 3596e3603. [27] X.Z. Wang, L. Guo, H.L. Zhang, S.K. Gong, H.B. Guo, Structural evolution and thermal conductivities of (Gd1exYbx)2Zr2O7 (x ¼ 0, 0.02, 0.04, 0.06, 0.08, 0.1) ceramics for thermal barrier coatings, Ceram. Int. 41 (2015) 12621e12625. [28] C.L. Wan, Z.X. Qu, A.B. Du, W. Pan, Influence of B site substituent Ti on the structure and thermophysical properties of A2B2O7-type pyrochlore Gd2Zr2O7, Acta Mater 57 (2009) 4782e4789. [29] P.K. Schelling, S.R. Phillpot, R.W. Grimes, Optimum pyrochlore compositions for low thermal conductivity, Philos. Mag. Lett. 84 (2004) 127e137. [30] P.K. Schelling, Thermal conductivity of A-site doped pyrochlore oxides studied by molecular-dynamics simulation, Comput. Mater. Sci. 48 (2010) 336e342. [31] J. Feng, B. Xiao, C.L. Wan, Z.X. Qu, Z.C. Huang, J.C. Chen, R. Zhou, Electronic structure, mechanical properties and thermal conductivity of Ln2Zr2O7 (Ln ¼ La, Pr, Nd, Sm, Eu and Gd) pyrochlore, Acta. Mater 59 (2011) 1742e1760. [32] B.P. Mandal, N. Garg, S.M. Sharma, A.K. Tyagi, Solubility of ThO2 in Gd2Zr2O7 pyrochlore: XRD, SEM and Raman spectroscopic studies, J. Nucl. Mater 392 (2009) 95. [33] R.E. Williford, W.J. Webere, Computer simulation of Pu3þ and Pu4þ substitutions in gadolinium zirconate, J. Nucl. Mater 299 (2001) 140e147. [34] A. Cleave, R.W. Grimes, K. Sickafus, Plutonium and uranium accommodation in pyrochlore oxides, Philos. Mag. 85 (2005) 967e980. [35] B. Raj, M. Vijayalakshmi, P.V. Rao, K.B.S. Rao, Challenges in materials research for sustainable nuclear energy, MRS Bull. 33 (2008) 327e337. [36] H.Y. Xiao, W.J. Weber, Y. Zhang, Ab initio molecular dynamics simulations of ionesolid interactions in zirconate pyrochlores, Acta Mater. 87 (2015) 273e282. [37] R.J.M. Konings, K. Bakker, J.G. Boshoven, Transmutation of actinides in inertmatrix fuels: fabrication studies and modelling of fuel behaviour, J. Nucl. Mater 274 (1999) 84e90. [38] R.N. Basu, S. Basu, Materials for Solid Oxide Fuel Cells, in Recent Trends Fuel Cell Science and Technology, Springer and Anamaya, New Delhi, India, 2007, pp. 286e331. [39] A.V. Radha, S.V. Ushakov, A. Navrotsky, Thermochemistry of lanthanum zirconate pyrochlore, J. Mater. Res. 24 (2009) 3350. [40] R.A. Chapman, D.B. Meadowcroft, A.J. Walkden, Some properties of zirconates and stannates with the pyrochlore structure, J. Phys. D. Appl. Phys. 3 (1970) 307. [41] K. Shimamura, T. Arima, Thermophysical properties of rare-earth-stabilized zirconia and zirconate pyrochlores as surrogates for actinide-doped zirconia, Int. J. Thermophys. 84 (2007) 1074e1084.

309

[42] G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996) 15. [43] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169. [44] J. Perdew, K. Burke, Y. Wang, Erratum: generalized gradient approximation for the exchange-correlation hole of a many-electron system, Phys. Rev. B 57 (1998) 14999. [45] H.Y. Xiao, X.T. Zu, F. Gao, First-principles study of energetic and electronic properties of A2Ti2O7 (A¼ Sm, Gd, Er) pyrochlore, J. Appl. Phys. 104 (2008) 073503. [46] C. Jiang, C.R. Stanek, K.E. Sickafus, B.P. Uberuaga, First-principles prediction of disordering tendencies in pyrochlore oxides, Phys. Rev. B 79 (2009) 104203. [47] C. Jiang, C. Wolverton, J. Sofo, L.Q. Chen, Z.K. Liu, First-principles study of binary bcc alloys using special quasirandom structures, Phys. Rev. B 69 (2004) 214202. [48] S.H. Wei, L.G. Ferreira, J.E. Bernard, A. Zunger, Electronic properties of random alloys: special quasirandom structures, Phys. Rev. B 42 (1990) 15. [49] A. Zunger, S.H. Wei, L.G. Ferreira, J.E. Bernard, Special quasirandom structures, Phys. Rev. Lett. 65 (1990) 3. [50] B.P. Mandal, A. Banerji, V. Sathe, S.K. Deb, A.K. Tyagi, Orderedisorder transition in Nd2yGdyZr2O7 pyrochlore solid solution: an X-ray diffraction and Raman spectroscopic study, J. Solid. State. Chem. 180 (2007) 2643e2648. [51] X.J. Wang, H.Y. Xiao, X.T. Zu, W.J. Weber, Study of cerium solubility in Gd2Zr2O7 by DFTþ U calculations, J. Nucl. Mater 419 (2011) 105e111. [52] F.A. Zhao, H.Y. Xiao, X.T. Zu, A first-principles study of Th immobilization in Gd2Zr2O7, 2015 submitted. [53] R.D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta. Cryst. A 32 (1976) 751e767. [54] C.W.M. Castleton, J. Kullgren, K. Hermansson, Tuning LDAþ U for electron localization and structure at oxygen vacancies in ceria, J. Chem. Phys. 127 (2007) 244704. [55] J.H. Wang, S.R. Phillpot, D. Wolf, Crystal instabilities at finite strain, Phys. Rev. Lett. 71 (1993) 25. [56] R. Hill, The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. A 65 (1952) 349. €tsconstanten [57] W. Voigt, Über die Beziehung zwischen den beiden Elasticita € rper, Ann. Phys. 38 (1889) 573. isotroper Ko [58] A. Reuss, Z. Angew, Berechnung der Fließgrenze von Mischkristallen auf €tsbedingung für Einkristalle, Math. Phys. 9 (1929) 49. Grund der Plastizita [59] V.V. Bannikov, I.R. Shein, A.L. Ivanovskii, Elastic and electronic properties of hexagonal rhenium sub-nitrides Re3N and Re2N in comparison with hcp-Re and wurtzite-like rhenium mononitride ReN, Phys. Status. Solidi. B 248 (2011) 1369. [60] M.P.V. Dijk, K.J.D. Vries, A.J. Burggraaf, Oxygen ion and mixed conductivity in compounds with the fluorite and pyrochlore structure, J. Solid. State. Ionics 9 (1983) 913. [61] H. Ibegazene, S. Alperine, C. Diot, Yttria-stabilized hafnia-zirconia thermal barrier coatings: the influence of hafnia addition on TBC structure and hightemperature behaviour, J. Mater. Sci. 30 (1995) 938. [62] J.A. Thompson, T.W. Clyne, The effect of heat treatment on the stiffness of zirconia top coats in plasma-sprayed TBCs, Acta. Mater 49 (2001) 1565. [63] S.F. Pugh, Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Phil. Mag. 45 (1954) 823. [64] J. Haines, J.M. Leger, G. Bocquillon, Synthesis and design of superhard materials, Ann. Rev. Mater. Res. 31 (2001) 1. [65] C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, Chicago (, 1948. Elasticity and Anelasticity of Metals. [66] J.R. Childress, C.L. Chien, M.Y. Zhou, P. Sheng, Lattice softening in nanometersize iron particles, Phys. Rev. B 44 (1991) 11689. [67] C. Kittel, Introduction to Solid State Physics, Wiley, New York, 1997. [68] D.R. Clarke, Materials selection guidelines for low thermal conductivity thermal barrier coatings, Surf. Coat. Technol. 163 (2003) 67. [69] D.R. Clarke, C.G. Levi, Materials design for the next generation thermal barrier coatings, Annu. Rev. Mater. Res. 33 (2003) 383. [70] D.G. Cahill, S.K. Watson, R.O. Pohl, Lower limit to the thermal conductivity of disordered crystals, Phys. Rev. B 46 (1992) 6131. [71] N. Li, H.Y. Xiao, X.T. Zu, L.M. Wang, R.C. Ewing, J. Lian, F. Gao, First-principles study of electronic properties of La2Hf2O7 and Gd2Hf2O7, J. Appl. Phys. 102 (2007) 063704. [72] H.Y. Xiao, M. Jiang, F.A. Zhao, Z.J. Liu, X.T. Zu, Thermal and mechanical stability, electronic structure and energetic properties of Pu-containing pyrochlores: La2-yPuyZr2O7 and La2Zr2-yPuyO7 (0 y 2), J. Nucl. Mater 466 (2015) 162e171. [73] F.A. Zhao, H.Y. Xiao, X.T. Zu, A first-principles study of Th immobilization in Gd2Zr2O7,, J. Nucl. Mater 467 (2015) 937e948.