Theoretical elastic stiffness, structural stability and thermal conductivity of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlore

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1 Available online at Acta Materialia 58 (2010) Theoretical elastic stiffness, structural stability and thermal conductivity of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlore B. Liu a,b, J.Y. Wang a,c, *, F.Z. Li a,b, Y.C. Zhou a a Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang , China b Graduate School of Chinese Academy of Sciences, Beijing , China c International Centre for Materials Physics, Institute of Metal Research, Chinese Academy of Sciences, Shenyang , China Received 27 July 2009; received in revised form 1 March 2010; accepted 20 April 2010 Available online 18 May 2010 Abstract In order to achieve better understanding of the structural/property relationships of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlore, first-principles calculations were conducted to investigate the bonding characteristics, elastic stiffness, structural stability and minimum thermal conductivity. The results show that the relatively weak La O bonds play a predominant role in determining the structural stability, mechanical and thermal properties of these compounds. In addition, the elastic and thermal properties are influenced when the T atom changes from Ge to Hf. When the bonding strength is enhanced by applying hydrostatic pressure, apart from c 11, c 12, and B, which normally increase at high pressures, it is found that the shear elastic moduli, c 44 and G, which relate to the shear deformation resistance, abnormally remain almost constant. The underlying mechanism may help to explain the damage tolerance of pyrochlore compounds. After comprehensive consideration of the elastic anisotropy, a modified David Clarke-type equation is used to calculate the minimum thermal conductivity of the studied pyrochlore materials, which display an extraordinary low thermal conductivity. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Non-binary oxides; Mechanical properties; Thermal conductivity; Density functional theory; Local density approximations (LDA) 1. Introduction The pyrochlore-type rare-earth oxides (RE 2 T 2 O 7, RE = rare-earth, and T is a tetravalent metallic cation) have attracted considerable attention in recent years owing to their important technological applications. These RE 2 T 2 O 7 oxides exhibit low thermal conductivity, a high melting point, a high thermal expansion coefficient, high stability and the ability to accommodate defects [1]. The unique properties render RE 2 T 2 O 7 compounds promising candidates for high-permittivity dielectrics [2], thermal barrier coatings (TBC) [1], as potential solid electrolytes in high temperature fuel cells [3], as immobilization hosts of the actinides in nuclear waste [4] and as oxidation catalysts [5]. Considerable effort has been made to optimize particular properties (high * Corresponding author. Fax: address: jywang@imr.ac.cn (J.Y. Wang). radioactive resistance, low thermal conductivity, low migration energy), and at the same time to satisfy the requirements, such as the physical stability, and the chemical or thermal compatibility with other materials involved in the design. However, experimental information is quite limited in establishing structure/property relationships for these complex pyrochlore materials. In general, the RE 2 T 2 O 7 pyrochlore structure (as shown in Fig. 1) can be derived from a TO 2 fluorite-type arrangement by two processes [6,7]: (1) half the T atoms are substituted by RE atoms and (2) 1/8 oxygen vacancies are formed to retain electrical neutrality. Besides, oxygen atoms (referred to as O2) around a T atom are relaxed close to the T atom, forming a T O octahedron, while other oxygen atoms (referred to as O1) are located at the initial site. Three types of chemical bonds, RE O1, RE O2 and T O2, can be distinguished inside the complex crystal structure of the RE 2 T 2 O 7 pyrochlore. These chemical bonds penetrate each other /$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi: /j.actamat

2 4370 B. Liu et al. / Acta Materialia 58 (2010) a b TO 6 La O1 O2 T Fig. 1. (a) Crystal structure of La 2 T 2 O 7 pyrochlore and (b) structure of TO 6 octahedron in La 2 T 2 O 7. along three-dimensions, and this may bring out novel properties for ternary pyrochlores compared with their binary TO 2 counterparts. Recently, both atomistic-scale simulations and ab initio calculations have been performed [8 19] on pyrochlores. On the basis of empirical potential methods, atomistic-scale simulations showed that the structural, mechanical and thermal properties, together with defect-formation energy were related to the cationic radius [8 13]: (1) RE 2 T 2 O 7 compound with similar atomic radii for RE and T elements has better radiation resistance than that with dissimilar ones and (2) the T cation radii obviously correlate with the structural, mechanical and thermal properties. Ab initio calculations focused on investigations of the crystal structure, electronic structure and defect-formation mechanism of bismuth, titanate, stannate, hafnate pyrochlores [14 19]. These studies [15,16] reported that the defect-formation energies and the properties were also influenced by the electronic configuration of the RE and T cations. It was concluded that the ionicity increased when the T site was occupied by Sn, Ti, Hf, and Zr atoms [16]. However, many questions about the relationship between the mechanical/thermal properties and the crystal structure or the type of chemical bonding are not clearly answered. One of these questions is how the mechanical or thermal properties depend on the complex features of chemical bonding in the crystal structure. The elastic properties of solids are closely related to many fundamental solid-state properties, such as the equation of state, specific heat, thermal expansion coefficient and Debye temperature [20]. For example, elastic constants provide important information about the bonding strength between adjacent atomic planes, the anisotropy of chemical bonding, as well as the structural stability. As candidates for high-performance TBC materials, the thermal conductivity of pyrochlores is also one of the properties of most concern. Therefore, it is of great technological significance, as well as of theoretical interest, to establish the intrinsic relationships between crystal structure, chemical bonding, mechanical and thermal properties of these ternary pyrochlores. Furthermore, the dependence of properties on T site atoms is still not clear for pyrochlores. To answer the above-mentioned questions, La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores of different chemical compositions and bonding properties were investigated as model materials. The main significances are as follows: first, the pyrochlores containing lanthanum are of wide interest in technological applications. Secondly, Ge and Sn are among the IVA-group elements, while Ti, Zr, and Hf are IVB-group elements. Therefore, changing atoms on the T site is helpful for understanding the relationship between the properties and chemical composition of ternary pyrochlores. By performing plane-wave pseudopotential total energy calculations on La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores, properties such as the full set of elastic constants, the intrinsic mechanical properties and the thermal conductivity were calculated. After detailed consideration of elastic anisotropy, a modified David Clarke-type equation is used to estimate the minimum thermal conductivity of the studied pyrochlore materials. Moreover, by applying pressure from 0 to 50 GPa to enhance the bonding strength, the pressure dependence of the structural stability, the bonding features and the elastic stiffness of the pyrochlores were clarified. The present paper also discusses the differences in bonding strengths, which are important factors for the understanding of the role of chemical bonds in influencing the thermal conductivity of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores compared with their binary TO 2 counterparts. Briefly, the aim was systematically to analyze the relationship between the strengths of interatomic bonds and the materials properties. In addition, the aim was to interpret the responses of crystal structure and elastic stiffness to mechanical disturbance for the La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores. This paper is organized as follows: the crystal structure and computational details are described in Section 2. Section 3.1 presents the results of the equilibrium geometry of the La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores. Elas-

3 B. Liu et al. / Acta Materialia 58 (2010) tic stiffness and intrinsic mechanical properties are discussed in Sections 3.2 and 3.3, respectively. In Section 3.4, the results for the structural stability, bonding features and elastic stiffness of the compounds at different pressures are provided. The extensively used Clarke relation is discussed and revised by considering the anisotropic elastic stiffness of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores in Section 3.5. The modified equation is more accurate for estimating or predicting the minimum thermal conductivity of ternary pyrochlore compounds. Finally, the concluding remarks are given in Section Crystal structure and theoretical methods RE 2 T 2 O 7 pyrochlore crystallizes in the space group of Fd 3m (Fig. 1), which can be described by two independent structural parameters: an internal atomic parameter x for the O 48f position, and the cell parameter a. The RE atom occupies the 16d site, and the T atom resides on the 16c site. There are two oxygen sites: O1 is located on the 8b site, and O2 on the 48f site (x, 1/8, 1/8) [6]. The equilibrium position of O2 on 48f is determined by the x parameter, which ranges from to 0.375, and affects the local coordination around the RE and T cations [7]. In the perfect pyrochlore structure with x = , the coordination of the T atoms and the RE atoms are regular octahedra and distorted cubes, respectively. As x increases to 0.375, the T site polyhedra distort to trigonal antiprisms and the RE-site polyhedra change to regular cubes. The pyrochlore structure is stable when r RE /r T (r RE and r T denote the radii of RE and T cations, respectively) lies between 1.46 and 1.78 [7], while the fluorite structure is stable when r RE /r T is <1.46. The CASTEP code was used in the present DFT calculations [21], and the Vanderbilt-type ultrasoft pseudopotential [22] and the local density approximation [23] were used. The plane-wave basis set cutoff was 450 ev in all the calculations. The special points sampling integration over the Brillouin zone was realized using the Monkhorst Pack method with a 6 6 6specialk-points mesh [24]. The lattice parameters, including the lattice constants and the internal atomic coordinates, were modified independently to minimize the free enthalpy, interatomic forces and unit-cell stresses. The Brodyden Fletcher Goldfarb Shanno minimization scheme [25] was used in the geometry optimization. The tolerances for the geometry optimization were: difference in total energy within ev atom 1, maximum ionic Hellmann Feynman force within 0.01 ev Å 1, maximum ionic displacement within Å and maximum stress within 0.02 GPa. A primitive cell of La 2 T 2 O 7 with 22 atoms was used in all the first-principles calculations. The elastic coefficients were determined from first-principles calculation by applying a set of given homogeneous deformations with a finite value and calculating the resulting stress with respect to optimizing the internal degrees of freedoms, as implemented by Milman and Warren [26]. Detailed computational strategies can be found in previous work, wherein the crystal structure, elastic stiffness and thermal conductivity of complex oxides such as LaPO 4 monazite [27],Y 2 Si 2 O 7 [28] and La 2 Zr 2 O 7 pyrochlore [29] were predicted. 3. Results and discussion 3.1. Structural parameters To investigate ground-state properties, the equilibrium lattice configuration of La 2 T 2 O 7 (T = Ge, Ti, Sn, Hf) was computed first. The optimized lattice constant and the x parameter are listed in Table 1, together with those reported in previous experiments [7,30] and first-principles calculations [16,29]. The computed lattice constant and x parameter deviate from previously reported data within only 1.5%. To validate the accuracy of the present calculations further, the bond lengths and the T O T bond angle between two TO 6 octahedra for La 2 Ti 2 O 7 and La 2 Sn 2 O 7 are also listed in Table 1. The consistency is obviously good, because the theoretical data deviate from experimental value by <1% Elastic constants The computed second-order elastic constants of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) are summarized in Table 2. Itis noted that the elastic constant c 11 decreases when T is changed in the order Ti, Sn, and Zr. It is also found that the c 11 of La 2 Hf 2 O 7 is close to that of La 2 Zr 2 O 7. At the same time, c 12 Table 1 Calculated lattice parameters and bond length for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlore. Source a (Å) x d La O(8a) (Å) d La O(48f) (Å) D B O(48f) (Å) Angle (B O B) La 2 Ge 2 O 7 This work La 2 Ti 2 O 7 This work Theory [16] La 2 Sn 2 O 7 This work Expt. [30] La 2 Zr 2 O 7 Theory [29] La 2 Hf 2 O 7 This work Expt. [7]

4 4372 B. Liu et al. / Acta Materialia 58 (2010) Table 2 Calculated second-order elastic constants (in GPa) and Cauchy pressure for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf). c 11 c 12 c 44 c 12 c 44 La 2 Ge 2 O La 2 Ti 2 O La 2 Sn 2 O La 2 Zr 2 O 7 [29] La 2 Hf 2 O and c 44 decrease in a similar trend. These results can be understood by taking into account the bonding characteristics and properties of T (T = Ti, Sn, Zr, Hf) atoms: first, the radius of the T atom increases from Ti to Zr. Therefore, the T O bond length is increased, and the T O bond strength becomes weaker when T changes from Ti to Zr. Secondly, it has been shown that the stannate and titanate pyrochlores show enhanced covalency and fewer ionic features than hafnates and zirconates do [16]. An enhancement in covalent bonding may lead to higher elastic stiffness. Lastly, since Zr and Hf have the same valence electron configuration and similar atomic radii, hafnate and zirconate pyrochlores have similar elastic constants. Furthermore, the elastic constants of La 2 Ge 2 O 7 are similar to those of La 2 Ti 2 O 7, while GeO 2 yields much higher values than those for TiO 2. This unusual result can be attributed to the expansion and weakening of the Ge O bond in the pyrochlore structure. The details of this are discussed in Section Mechanical properties of polycrystalline aggregates For polycrystalline samples, it is not possible to measure the individual elastic constants c ij, but one can obtain the polycrystalline bulk and shear moduli. The polycrystalline bulk modulus B and shear modulus G can be estimated using the Voigt Reuss Hill approximations [31 33]. Furthermore, Young s modulus E and Poisson s ratio (m) can be calculated from the shear modulus and the bulk modulus [34]. Table 3 summarizes the calculated mechanical properties. There are two characteristics worthy of discussion: on the one hand, the bulk modulus B decreases when T changes from Ti to Zr. It is also found that the bulk moduli of La 2 Ti 2 O 7 and La 2 Hf 2 O 7 are close to those of La 2 Ge 2 O 7 and La 2 Zr 2 O 7, respectively. This result is similar to the trend obtained for elastic constants of the corresponding compounds, which can be understood by considering the bonding characteristics and properties of T (T = Ge, Ti, Sn, Zr, Hf) atoms as discussed in Section 3.2. Furthermore, the trends, including the T cations dependence of elastic constants and mechanical properties, are similar to those observed by others [9]. However, the shear modulus, which describes the resistance of a material to a shape change, and Young s modulus, decrease monotonically, in the sequence La 2 Ge 2 O 7 >La 2 Ti 2 O 7 >- La 2 Sn 2 O 7 >La 2 Zr 2 O 7 > La 2 Hf 2 O 7. It is known that Young s modulus has a close relationship with high temperature thermal conductivity [29]. A lower Young s modulus can help to decrease the thermal conductivity and suggests promising TBC. The theoretical minimum thermal conductivity and its underlying origin are discussed in Section 3.5. The elastic anisotropy of a cubic structure can be indicated by a parameter referred as the Zener anisotropy ratio Z, defined by Z =2c 44 /(c 11 c 12 ) [35]. A material is isotropic if Z = 1, which means that the elastic moduli are independent of crystal orientation. For a cubic lattice, the Zener anisotropy ratio also determines the direction showing the maximum Young s modulus. For Z < 1, the maximum Young s modulus will appear along the [1 0 0] direction; while for Z > 1, the maximum Young s modulus will be in the [1 1 1] direction. In the present work, the elastic anisotropy of La 2 T 2 O 7 is characterized by the Zener anisotropy ratio Z. The variation in Young s modulus as a function of crystallographic orientation is studied within the ð011þ atomic plane, which includes all the principal directions of a cubic crystal: [1 0 0], [0 1 1], and [1 1 1]. The detailed calculation method can be found in Refs. [35,36]. From the Zener anisotropy ratios presented in Table 3 (1.29 for La 2 Ge 2 O 7, 1.24 for La 2 Ti 2 O 7, 1.29 for La 2 Sn 2 O 7, 1.21 for La 2 Zr 2 O 7, 1.18 for La 2 Hf 2 O 7 ), it is known that all these compounds are anisotropic and that the maximum Young s modulus appears along the [1 1 1] direction. Fig. 2 shows the anisotropic Young s modulus in the ð011þ plane. The maximum Young s modulus (along the [1 1 1] direction) is 281 GPa for La 2 Ge 2 O 7, 272 GPa for La 2 Ti 2 O 7, 268 GPa for La 2 Sn 2 O 7, 252 GPa for La 2 Zr 2 O 7, and 240 GPa for La 2 Hf 2 O 7, while the minimum is 226 GPa for La 2 Ge 2 O 7, 227 GPa for La 2 Ti 2 O 7, 214 GPa for La 2 Sn 2 O 7, 214 GPa for La 2 Zr 2 O 7, and 209 GPa for La 2 Hf 2 O 7, which is along the [1 0 0] direction. The aniso- Table 3 Calculated mechanical properties (in GPa) for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf). B G E m Z La 2 Ge 2 O La 2 Ti 2 O La 2 Sn 2 O La 2 Zr 2 O La 2 Hf 2 O Fig. 2. Anisotropic Young s moduli for La 2 T 2 O 7 pyrochlore.

5 B. Liu et al. / Acta Materialia 58 (2010) Fig. 3. The pressure dependence of x coordinate of La 2 T 2 O 7 pyrochlore. tropic Young s modulus can be attributed to the orientations of the La O2, La O1 and T O2 bonds, which are close to the [1 1 1] direction but away from the [1 0 0] direction [29] Structural stability, bonding properties and elastic stiffness at high pressures For La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlore, the value of the x coordinate of the O2 at the 48f site determines whether a lattice configuration shows the pyrochlore structure as presented in Section 1. In order to show the changes in the x parameter under pressure, the equilibrium crystal structures of the La 2 T 2 O 7 pyrochlores were computed at applied hydrostatic pressures ranging from 0 to 50 GPa. At each applied pressure, a complete geometrical optimization of the crystal structure was performed. Fig. 3 shows the changes in the x coordinate versus pressure for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr and Hf) pyrochlores. It is seen that the x coordinate increases under high pressure for La 2 T 2 O 7 (T = Ti, Sn, Zr, Hf), which suggests an approach to the fluorite-type crystal structure; while the x coordinate decreases for La 2 Ge 2 O 7, which shows a tendency towards a pyrochlore-type crystal structure. This result can be explained by different bond-length contractions, which will be further explained in the following paragraph. The resistance of interatomic bonds against external pressure characterizes the relative bond strengths [29]. Following the same idea, the degree of bond-length contractions under various pressures were examined for La 2 T 2 O 7 (T = Ge, Ti, Sn, Hf) pyrochlores, and the results are illustrated in Fig. 4a c and e. Similar characteristics are shown in Fig. 4b, c, and e for La 2 T 2 O 7 (T = Ti, Sn, Hf). First, the lowest lying curve is associated with the La O2 bond, which is the most compressible bond. Above it is the curve for the La O1 bond, and the upper curve corresponds to the stiffest T O2 bond. The results shown in Fig. 4b, c, and e indicate that the bond strengths increase in the order La O2 < La O1 < T O2 bonds in the La 2 T 2 O 7 pyrochlore, which was previously reported for La 2 Zr 2 O 7 (Fig. 4d) [29]. Based on the different compression ratios of the La O and/or the T O bond under various pressures, Fig. 4. Relative bond-length contractions at various pressures for La 2 T 2 O 7 pyrochlore.

6 4374 B. Liu et al. / Acta Materialia 58 (2010) Fig. 5. The T O bond increment in La 2 T 2 O 7 pyrochlore compared with its related binary compounds TO 2. the origin of the structural change of La 2 T 2 O 7 (T = Ti, Sn, Zr, Hf) pyrochlore under pressure can be recognized. Nikiforov [37] assumed that the interatomic distance between O2 and the cations (T and La) was equal to the sum of their relative ionic radii. Within the applied pressure range, the La O2 bond length shrinks more rapidly than the T O2 bond, as shown in Fig. 4b, c, and e. This suggests that the relative ionic radius of La decreases more than that of T and that the relative ionic radii ratio (r La /r T ) reduces gradually. A lower relative ionic radii ratio (r La /r T ) will lead to an increase in the x coordinate (x? 0.375) and results in a tendency to change from the pyrochlore structure to the fluorite structure for the La 2 T 2 O 7 (T = Ti, Sn, Hf) compounds. The case is different for La 2 Ge 2 O 7. As shown in Fig. 4a, the curves from the top to the bottom are for the La O2, La O1 and Ge O2, and are seen to be quite close in magnitude. According to Nikiforov s discussion [37], the relative ionic radii ratio (r La /r Ge ) increases, and La 2 Ge 2 O 7 approaches the perfect pyrochlore structure. This result indicates that the Ge O bond is slightly weaker than the La O bonds, but that the strengths deviate slightly. To understand why La 2 Ge 2 O 7 is different from the other pyrochlores, Fig. 5 demonstrates the variations in the T O bond lengths in La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) compared with those in binary TO 2. The pyrochlore structure can be recognized as a three-dimensional network of T O2 octahedra with La and O1 located at the interstitial sites. The atomic size of Ge is small in comparison with those of Ti, Sn, Zr, and Hf. Therefore, the Ge O bond in La 2 Ge 2 O 7 will expand more (4%) than that in other compounds (1% and less) in order to provide enough interstitial space to accommodate the La and O1 atoms. Elongation of the Ge O bond decreases its bonding strength, which degrades the elastic stiffness. In conclusion, the Ge O bond is slightly weaker than La O bonds in La 2 Ge 2 O 7, which is different from those observed for other pyrochlores, whereas the T O bond is stronger than La O bonds. To the authors knowledge, there has been no work which discusses the influence of structural changes and/or chemical bonding on the mechanical properties of pyrochlores. The present authors expect that, after enhancing the bonding strengths by applying hydrostatic pressure, the Fig. 6. Calculated pressure dependent elastic constants (c 11, c 12, and c 44 ) and mechanical properties (B, G, E) ofla 2 T 2 O 7 pyrochlore.

7 B. Liu et al. / Acta Materialia 58 (2010) mechanical parameters will change in a way that can be helpful for understanding the intrinsic relationship between mechanical properties, chemical bond and deformation mechanisms. Fig. 6 presents the variation in elastic constants (c 11, c 12, c 44 ) and mechanical properties (B, G, E) of La 2 T 2 O 7 (T = Ge, Ti, Zr, Sn, Hf) pyrochlore with respect to variations in pressure. Some common tendencies are observed for these compounds: first, the c 11, c 12, and the bulk modulus B increase when the pressure is increased. It is known that these mechanical parameters are directly determined by the features of the average bond strength. So, the c 11, c 12, and bulk modulus B increase as a result of bond strength (La O and T O) enhancement at elevated pressures. This result also suggests that these pyrochlore structures remain stable at high pressures up to 50 GPa. Furthermore, the shear moduli c 44 and G are changed less when the applied pressure is increased. The results indicate that the enhancement of bond strength has little influence on the shear deformation resistance. This observation can be attributed to the fact that the shear deformation mechanism is determined by the distortion of the soft La O polyhedra and the rotation of the rigid T O octahedra. Wang et al. [27,28] studied the atomistic deformation mechanisms of LaPO 4 and Y 2 SiO 7, and proposed that the distortion of the soft La O polyhedra and the rotation of the rigid PO 4 tetrahedra, or the distortion of the soft Y O polyhedra and the rotation of the rigid Si 2 O 7 pyrosilicate, dominate the shear deformation of the two compounds, respectively. And in addition, the observed deformation mechanism is helpful for improving the damage tolerance of these complex phosphates and silicates owing to their low shear strain energy. The shear deformation mechanism of La 2 T 2 O 7 (T = Ge, Ti, Zr, Sn, Hf) may also be beneficial for enhancing the damage tolerance by activating deformation twining in deformed pyrochlore compounds. This conclusion can be further strengthened by a discussion of the Cauchy pressure c 12 c 44. Generally speaking, a positive Cauchy pressure suggests damage tolerance and ductility of a material; while a negative value indicates enhanced brittleness [38]. The calculated Cauchy pressures for La 2 T 2 O 7 (T = Ge, Ti, Zr, Sn, Hf) pyrochlores are shown in Table 2. All the Cauchy pressures for La 2 T 2 O 7 are positive, which suggests an intrinsic damage tolerance and quasi-ductility of these materials. Finally, Young s moduli also do not obviously change at high pressures. Because 3B G for these compounds, Young s modulus (E = 9BG/(3B + G)) is approximated as 3G. Therefore, a trend is found for Young s modulus E similar to that for the shear modulus G with respect to the variation in pressure Thermal properties of polycrystalline La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlore The thermal properties of these compounds need to be investigated, as they are of considerable interest as novel TBC materials. Based on the Debye model, Clarke [39] suggested that theoretical minimum thermal conductivity can be calculated after replacing different atoms by an equivalent atom with a mean atomic mass of M/n: M j min! k B m m nqn A 2 3 ð1þ where k B is the Boltzmann s constant, v m the average sound velocity, N A the Avogadro s number, q the density, M the molecular weight, and n is the number of atoms in the molecule. Clarke further simplified Eq. (1) into M 3 E j min! 0:87k B nqn A q qffiffi E using the relationship m m ¼ 0:87. Generally, the Clarke relation has been successful in calculating the minimum thermal conductivity for many binary oxides, such as TiO 2, SiC, ZrO 2 (YSZ) and many others [39]. However, few investigations have focused on ternary oxides with a complex crystal structure and chemical bonding such as the La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores. It should be noted that the Clarke relation averages the anisotropic elastic stiffness of a material. In order to estimate precisely the theoretical minimum thermal conductivity of La 2 T 2 O 7 pyrochlore with elastic anisotropy, the Clarke relation was modified by considering the anisotropic elasticity, and the results were compared with those obtained with Eq. (2). The present calculation starts from the calculation of the average sound velocity with respect to a new consideration of elastic anisotropy: v m ¼ 1 2 þ 1 1=3 ð3þ 3 v 3 s v 3 l wherein shear (v s ) and longitudinal (v l ) sound velocities can be defined by the following equations: sffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G v s ¼ and v l ¼ B þ 4 q 3 G q ð4þ A new equation for calculating the average sound velocity is defined by ( " 3 #) v m ¼ þ 2m 2 þ 1 3 6m þ 2 3 þ 3m q ð2þ sffiffiffi E q ð5þ Therefore, the theoretical minimum thermal conductivity can be obtained from Eqs. (1) and (5) based on theoretical elastic parameters. The calculated thermal properties for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf), including the sound velocities and minimum thermal conductivity (j min ), are summarized in Table 4. The minimum thermal conductivity obtained in the present calculation is 20% lower than those obtain by the

8 4376 B. Liu et al. / Acta Materialia 58 (2010) Table 4 Calculated phonon sound velocity and thermal properties for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf), in which j * is obtained using the Clarke relation. v s (10 5 cm s 1 ) v l (10 5 cm s 1 ) v m (10 5 cm s 1 ) j (W mk 1 ) j * (W mk 1 ) La 2 Ge 2 O La 2 Ti 2 O La 2 Sn 2 O La 2 Zr 2 O La 2 Hf 2 O Clarke relation. The results demonstrated that a noticeable decrement on the minimum thermal conductivity appears after the consideration of elastic anisotropy of pyrochlores with complex crystal structures. In experiment, only the value of La 2 Zr 2 O 7 (1.5 Wm K 1 ) can be obtained for comparison [1]. It is noted that the experimental data are slightly higher than the theoretical result. In addition, the present minimum thermal conductivity is also lower than the corresponding ones obtained using the molecular dynamics simulations of Schelling et al. [9]. The reason is that the value obtained in this work is the lower limit of the thermal conductivity, while that obtained in Ref. [9] is the value at a certain temperature. Nevertheless, a trend similar to that reported by Schelling et al. [9] was obtained for the decrement of thermal conductivity with increment of the B-cation size. In addition, the minimum thermal conductivity of rutiletype TO 2 (T = Ge, Ti, Sn, Zr, Hf) was calculated by Eqs. (1) and (5) for comparison, in order to understand the origin of the lower thermal conductivity of pyrochlores. The results are shown in Fig. 7a, together with those for the La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores. Fig. 7a indicates that the minimum thermal conductivity of La 2 T 2 O 7 pyrochlores is lower than that of the related binary TO 2. The reduction can be attributed mainly to the difference in Young s modulus between the binary compounds and their ternary counterparts. Young s modulus E is a measure of the second derivative of the bonding energy at the equilibrium interatomic distance x 0. For the La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf), two different cases need to be discussed. First, the La O bonds are weaker than the T O bond in La 2 T 2 O 7 (T = Ti, Sn, Zr, Hf) pyrochlore. In other words, weak La O bonds appear in the ternary compounds La 2 T 2 O 7, while only strong T O bonds are present in TO 2. The La O bonds weaken the average bond strengths of La 2 T 2 O 7 and lead to a lower Young s modulus E (Fig. 7b). As a result, La 2 T 2 O 7 shows a lower j min than TO 2 when the T site is occupied by Ti, Sn, Zr, Hf. Second, the lower E and j min of La 2 Ge 2 O 7 originate from the effects of the La O and Ge O bonds, which are both weak bonds. Furthermore, the discrepancies for E and j min are large for the binary TO 2, but small for the ternary La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf). This fact definitely indicates that the weak La O bond plays the predominant role in determining the mechanical and thermal properties of La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores. 4. Conclusions Fig. 7. (a) Calculated minimum thermal conductivity and (b) Young s modulus for La 2 T 2 O 7 pyrochlore and its related binary compound TO 2 (T = Ge, Ti, Sn, Zr, Hf). In summary, the bonding characteristics, elastic stiffness, structural stability and minimum thermal conductivity for La 2 T 2 O 7 (T = Ge, Ti, Sn, Zr, Hf) pyrochlores were investigated by first-principles calculations and the crystal structural/property relationship of the materials was established. The La O bonds have been demonstrated to be weaker than the T O bond in La 2 T 2 O 7 (T = Ti, Sn, Zr, Hf). In the case of La 2 Ge 2 O 7, it has been shown that the Ge O bond strength is similar to that of La O bonds as a result of the expansion of the Ge O bond. The relatively weak La O bonds play a predominant role in determining the structural stability, and the mechanical and thermal properties. For example, La 2 T 2 O 7 has a lower elastic modulus and minimum thermal conductivity than its binary TO 2 counterpart, which is attributed to the effects of the weak La O bonds presented in the ternary pyrochlore. In addition, the elastic and thermal properties are also influenced when the T atom changes from Ge to Hf. It is found that the crystal structures of La 2 T 2 O 7

9 B. Liu et al. / Acta Materialia 58 (2010) (T = Ti, Zr, Sn, Hf) approach the fluorite-type structure at high pressures. However, the structure of La 2 Ge 2 O 7 tends to the perfect pyrochlore structure as a result of Ge O bond length shrinking. The present work also provides the pressure dependences of elastic constants and mechanical properties. Besides the c 11, c 12, and B, which normally increase at high pressures, the shear-deformation-related elastic moduli c 44 and G abnormally remain almost constant. The underlying mechanism enhances the damage tolerance and quasi-ductility of studied pyrochlores. After new consideration of elastic anisotropy, a modified Clarke relation is presented to calculate the minimum thermal conductivity of the pyrochlores. The La 2 T 2 O 7 pyrochlores display extraordinary low thermal conductivity, which demonstrates their technological application as novel TBC materials. Acknowledgements The authors acknowledge Professor Oates at the University of Salford (UK) for improving the English. This work was supported by the National Outstanding Young Scientist Foundation for Y.C. Zhou under Grant No , and the Natural Sciences Foundation of China under Grant Nos , , , and References [1] Vassen R, Cao X, Tietz F, Basu D, Stover D. J Am Ceram Soc 2000;83:2023. [2] Cann DP, Randall CA, Shrout TR. Solid State Commun 1996;100:529. [3] Vries KJD, Dijk TV, Burggraaf AJ. Fast ion transport in solids. Amsterda: Elsevier, North Holland; [4] Ewing RC, Weber WJ, Lian J. J Appl Phys 2004;95:5949. [5] Korf SJ, Koopmans HJA, Lippens BC, Burggrasst AJ, Gellings PJ. J Chem Soc Faraday Trans 1987;83:1485. [6] Lian J, Wang LM, Wang SX, Chen J, Boatner LA, Ewing RC. Phys Rev Lett 2001;87: [7] Subramanian MA, Aravamundan G, Rao GVS. Solid State Chem 1983;15:55. [8] Sickafus KE, Minervini L, Grimes RW, Valdez JA, Ishimaru M, Li F, et al. Science 2000;289:748. [9] Schelling PK, Phillpot SR, Grimes RW. Philos Mag Lett 2004;84:127. [10] Tabira Y, Withers RL, Minervini L, Grimes RW. J Solid State Chem 2000;153:16. [11] Pirzada M, Grimes RW, Minervini L, Maguire JF, Sickafus KE. Solid State Ionics 2001;140:201. [12] Minervini L, Grimes RW, Tabira Y, Withers RL, Sickafus KE. Philos Mag A 2002;82:123. [13] Minervini L, Grimes RW, Sickafus KE. J Am Ceram Soc 2000;83:1873. [14] Hinojosa BB, Nino JC, Asthagiri A. Phys Rev B 2008;77: [15] Panero WR, Stixrude LP, Ewing RC. Phys Rev B 2004;70: [16] Pruneda JM, Artacho E. Phys Rev B 2005;72: [17] Xiao HY, Wang LM, Zu XT, Lian J, Ewing RC. J Phys Condens Matter 2007;19: [18] Chen ZJ, Xiao HY, Zu XT, Wang LM, Gao F, Lian J, et al. Comp Mater Sci 2008;42:653. [19] Liu D, Tse K, Robertson J. Appl Phys Lett 2007;90: [20] Bouhemadou A, Khenata R, Chegaar M. Eur Phys J B 2007;56:209. [21] Segall MD, Lindan PLD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, et al. J Phys Condens Matter 2002;14:2717. [22] Vanderbilt D. Phys Rev B 1990;41:7892. [23] Ceperley DM, Alder BJ. Phys Rev Lett 1980;45:566. [24] Monkhorst HJ, Pack JD. Phys Rev B 1977;16:1748. [25] Pfrommer BG, Côté M, Louie SG, Cohen ML. J Comp Physiol 1997;131:233. [26] Milman V, Warren MC. J Phys Condens Matter 2001;13:241. [27] Wang JY, Zhou YC, Lin ZJ. Appl Phys Lett 2005;87: [28] Wang JY, Zhou YC, Lin ZJ. Acta Mater 2007;55:6019. [29] Liu B, Wang JY, Zhou YC, Liao T, Li FZ. Acta Mater 2007;55:2949. [30] Kennedy BJ, Hunter BA, Howard CJ. J Solid State Chem 1997;130:58. [31] Voigt W. Lehrbuch der Kristallphysik. Leipzig: Taubner; [32] Reuss A, Angew Z. Math Mech 1929;9:55. [33] Hill R. Proc Phys Soc Lond Sect A 1952;65:350. [34] Holm B, Ahuja R, Yourdshahyan Y, Johansson B, Lundqvist BI. Phys Rev B 1999;59: [35] Ingel RP, Lewis III D. J Am Ceram Soc 1988;71:265. [36] Love AEH. A treatise on the mathematical theory of elasticity. 4th ed. New York: Dover; [37] Nikiforov LG. Sov Phys Crystallogr 1972;47:347. [38] Pettifor DG. Mater Sci Technol 1992;8:345. [39] Clarke DR. Surf Coat Technol 2003; :67.