Elastic anisotropy and thermodynamics properties of BiCu2PO6, BiZn2PO6 and BiPb2PO6 ceramics materials from first-principles calculations

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Elastic anisotropy and thermodynamics properties of BiCu2PO6, BiZn2PO6 and BiPb2PO6 ceramics materials from first-principles calculations Jing Chena, Xudong Zhanga,∗, Shiyu Zhua, He Maa, Xiaoyu Lib, Hui Yub, Feng Wangc a

School of Science, Shenyang University of Technology, Shenyang, 110870, China School of Information Science and Engineering, Shenyang University of Technology, Shenyang, 110870, China c School of Materials Science and Engineering, Shenyang University of Technology, Shenyang, 110870, China b

ARTICLE INFO

ABSTRACT

Keywords: BiX2PO6 compounds Structural stability Mechanical properties Thermodynamics properties

In present work, the elastic properties, anisotropy in elasticity and thermodynamics properties for BiCu2PO6, BiZn2PO6 and BiPb2PO6 ceramics materials were investigated using the first-principles calculation. The formation enthalpy and phonon frequencies confirm that three BiX2PO6 (X = Cu, Zn, and Pb) compounds exhibit the structural stability. The calculated elastic constants and elastic moduli indicate that BiZn2PO6 has the better mechanical properties than BiCu2PO6 and BiPb2PO6 at ground state. The values of B/G confirm that three BiX2PO6 compounds all exhibits the ductile behavior. The values of anisotropic parameters, three-dimensional surface constrctions and two-dimensional projection curves of the Young's modulus reveal the anisotropic degree of three BiX2PO6 compounds. The thermodyanmic parameters indicate that three BiMn2XO6 materials show the thermal stability from 0 to 1000 K. The obtained physical parameters can provide the useful data for the further experimental investigations.

1. Introduction With the development of science and technology, Bismuth oxides BiM2XO6 compounds have attracted the extensive attention of scientists, due to their outstanding physical and chemical properties [1–7]. For the BiM2XO6 family, M elements include Mg, Cu, Zn, Pb, Ca, Mn and Cd, X elements include P, As, and V etc. [8–14]. These BiM2XO6 compounds exhibit good physical and chemical properties, and they can be used as the bright yellow pigment, potential oxidation catalysts and the better microwave dielectric ceramics. They can efficiently emit in the red spectral region for lighting devices with LED technology [15]. For BiX2PO6 (X = Cu, Zn, and Pb) compounds, they also belong to the BiM2XO6 family [16–18]. But, the related reports of physical properties of BiX2PO6 (X = Cu, Zn, and Pb) compounds are very few up to now [19–22]. BiCu2PO6 microwave dielectric ceramic was prepared using a solid-state reaction method. As the sintering temperature increased from 800 °C to 880 °C, the bulk density of BiCu2PO6 ceramic increased from 6.299 g/cm3 to 6.366 g/cm3. The best microwave dielectric properties and a temperature coefficient of resonant frequency were obtained in the ceramic sintered at 860 °C for 2 h. This system could potentially be used for low-temperature co-fired ceramics technology applications. The literature mainly reported the crystal structure and microwave dielectric properties [19]. The crystalline BiX2PO6 (X = Cu, ∗

Zn, and Pb) exhibits the orthorhombic structure with Pnma group (No.62). The atoms of BiX2PO6 (X = Cu, Zn, and Pb) compounds occupy the following Wyckoff positions [16–18]: Bi 4c(0.1071, 0.25, 0.0236), X 4c(0.0898, 0.75, 0.6864) and (0.0722, 0.75, 0.3156), P 4c (0.1972, 0.25, 0.4658), O 8d(-0.0056, 0.0040, 0.1760) and (0.1232, 0.4973, 0.4945), 4c (0.2986, 0.25, 0.5860) and (0.2345,0.25, 0.2782). Up to now, the first-principles methods have been used to successfully investigate the physical and chemical properties of many materials. In order to investigate the overall performances of BiX2PO6 (X = Cu, Zn, and Pb) compounds, we used the first-principles methods to clarify the physical properties of BiX2PO6 compounds. To develop and use BiX2PO6 materials, the present work can provide the data support for experimental investigations. 2. Computational methods In this study, we employed the Cambridge sequential total energy package [23], which is based on density functional theory (DFT), to implement the first-principles calculations. Ultrasoft pseudo-potentials were applied to illustrate the interactions between the ionic cores and valence electrons. The Perdew-Burke-Eruzerh (PBE) form of the generalized gradient approximation (GGA) was used to represent the exchange correlation energy [24]. The corresponding plane wavebasis

Corresponding author. E-mail address: [email protected] (X. Zhang).

https://doi.org/10.1016/j.ceramint.2019.12.089 Received 19 November 2019; Received in revised form 5 December 2019; Accepted 8 December 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.

Please cite this article as: Jing Chen, et al., Ceramics International, https://doi.org/10.1016/j.ceramint.2019.12.089

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thermodynamically stable owing to their negative values of Ec and ΔH. Fig. 2 displays the phonon dispersion and densities of phonon states of BiX2PO6 (M = Cu, Zn and Pb). In the whole Brillouin zone, the phonon frequencies of three BiM2PO6 (M = Cu, Zn, and Pb) compounds are all positive, which indicating that they all possess dynamical stability. In general, the cohesive energy, formation enthalpy and phonon frequencies confirm that three BiM2PO6 (M = Cu, Zn, and Pb) compounds possess the structural stability at ground state. The elastic constants Cijs of BiX2PO6 (X = Cu, Zn, Pb) ceramics were studied by the stress-strain method [40–46]. Due to the orthorhombic structure, there are nine independent elastic constants Cijs for the three BiX2PO6 (X = Cu, Zn, Pb) ceramics. The independent Cijs and the compliance matrices Sijs are tabulated in Table 2. The mechanical stability of BiX2PO6 (X = Cu, Zn,/Pb) ceramics evaluated by Born criteria [47]. Because the elastic constants of the BiX2PO6 (X = Cu, Zn, Pb) ceramics are in coincidence with the Born criteria, which indicates they all possess mechanical stability. For elastic constants C11 and C33, the larger values they are, the higher resistance against the linear compression. The linear compression corresponds to uniaxial stress along the a axis and c axis orientations, respectively. On the other side, elastic constants C44 and C66 are better indirect reflections of resistance capacity against shear force along [110] and [001] directions on the (100) plane [41,42]. The values of elastic constants C11 of both BiCu2PO6 and BiZn2PO6 are smaller than those of elastic constants C33, indicating that the two ceramics can be compressed more easily along the a-axis than along the c-axis. While BiPb2PO6 exhibits more incompressible properties along the a-axis than c-axis, as a result of the larger value of elastic constant C11 than that of elastic constant C33. And its difference between the compressible extents along the two directions is very small compared with the other two ceramics. Among three BiX2PO6 (M = Cu, Zn, Pb) ceramics, BiM2PO6 possesses the largest elastic constants C44 and C66, which indicates that it has the largest shear modulus and the best resistance to shear stress. The ductile/brittle feature of crystalline materials can be effectively judged by the Cauchy pressure (C12–C44). The three BiM2PO6 (M = Cu, Zn, Pb) ceramics are all ductile because of their positive values for Cauchy pressure [48–51]. Utilizing the elastic constants, the elastic moduli (including Young's modulus E, shear modulus G and bulk modulus B) can be calculated using the VRH method [52–56] and were shown in Table 3. The bulk modulus can roughly reveal the incompressible degree of crystalline materials. Therefore, the better incompressibility under the hydrostatic pressure for solid materials corresponds to the larger values of bulk modulus. BiZn2PO6 is more incompressible under the hydrostatic pressure than the other two ceramics according to its largest values of bulk modulus among three BiX2PO6 ceramics. And it can be inferred from Table 3 that BiPb2PO6 owns the easiest compressibility among three BiX2PO6 ceramics. And the incompressibility of BiCu2PO6 approaches to that of BiZn2PO6. The deformation caused by the shear force is indirectly reflected by the shear modulus. And the larger values of the shear modulus are, the higher resistance against the deformation the solid material has. BiZn2PO6 possesses the highest ability to resist deformation among three BiM2PO6 ceramics owing to its largest values of shear modulus. In general, the larger Young's modulus and shear modulus correspond to the higher elastic hardness of solid materials. The increasing order of the Young's modulus is BiPb2PO6 < BiCu2PO6 < BiZn2PO6 in Table 3. And the shear modulus also exhibits the same order, which means that BiZn2PO6 owns the biggest hardness among three ceramics, BiCu2PO6 takes the second place, and BiPb2PO6 is the softest one among them. Table 3 also tabulated the values of B/G and Poisson's ratio ν. The critical values are 1.75 and 0.26 for B/G and ν, respectively. When the values of B/G and ν of the solid materials are larger than the critical values, it is ductile, otherwise it is brittle. Based on the above point, all three BiX2PO6 ceramics possess ductility, and the outcome is the same with that inferred the Cauchy pressure. Usually, Poisson's ratio stands for the resistance to shear deformation of solid materials. And its scope is

Fig. 1. Crystal structures of BiX2PO6 compounds. The red balls are O atoms, the green balls are P atoms, the purple balls are X (X = Cu, Zn, Pb) atoms and the brown balls are Bi atoms.

cutoff energy was determined to be 650 eV for the present calculations. The Monkhorst Pack k-point was 8 × 4 × 6 was chosen for BiX2PO6 (X = Cu, Zn, and Pb) compounds. The geometry optimization tolerances were as follows: the total energy, maximum ionic force, maximum ionic displacement, and maximum stress were controlled so that they remained less than 5 × 10−6 eV/atom, 0.01 eV/Å, 5.0 × 10−4 Å, and 0.02 GPa, respectively. Broyden-Fletcher-Goldfarb-Shanno minimization was applied to optimize the structure [25–28]. The self-consistent field tolerance was 5.0 × 10−7 Å. The phonon calculations were conducted by employing the linear response method in the PHONON code [29–32]. 3. Results and discussion Fig. 1 plots the orthorhombic structure with Pnma space group of BiX2PO6 (X = Cu, Zn, and Pb) ceramics materials. Table 1 lists the calculated lattice parameters of BiX2PO6 (X = Cu, Zn, and Pb) and the experimental ones [16–18]. The values of the optimized lattice parameters are slightly larger compared to the experimental values, which intrinsically characterizes the general gradient approximation. The deviations are less than 3.5% for lattice parameters, it can be concluded that the computational method is credible. The correlative equations were applied to calculate the cohesive energy Ec and formation enthalpy ΔH [33–38] of BiX2PO6. And the results are tabulated in Table 1. It can be inferred that three BiX2PO6 (X = Cu, Zn, Pb) ceramics are all Table 1 The structural constants (a, b, c in Å), cohesive energy Ec(in ev/atom) and formation enthalpy ΔH (in kJ/mol atom) for BiX2PO6 ceramics. Compounds

a

b

c

Ec

ΔH

BiCu2PO6

12.029 11.776 12.0942 11.8941 11.726 11.473

5.185 5.173 5.3309 5.2754 5.974 5.930

7.956 7.7903 7.9293 7.8161 9.370 9.079

−6.562

−15.894

−6.718

−16.008

−6.992

−16.321

BiZn2PO6 BiPb2PO6

This work Expt [16]. This work Expt [17]. This work Expt [18].

2

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Fig. 2. The phonon dispersion and density of phonon states for BiCu2PO6, BiZn2PO6 and BiPb2PO6. Table 2 Elastic constants Cijs (GPa) and compliance matrices Sijs for BiX2PO6 ceramics.

BiCu2PO6 BiZn2PO6 BiPb2PO6

BiCu2PO6 BiZn2PO6 BiPb2PO6

C11

C22

C33

C44

C55

C66

C12

C13

C23

114.424 127.602 119.018

234.617 240.437 185.927

145.401 176.897 112.302

38.379 32.445 53.426

42.781 56.433 60.582

20.646 23.748 30.591

51.843 44.068 65.087

64.566 64.079 61.648

69.540 69.613 63.286

S11

S22

S33

S44

S55

S66

S12

S13

S23

0.0119 0.0097 0.0126

0.0051 0.0047 0.0072

0.0099 0.0074 0.0134

0.0261 0.0308 0.0187

0.0234 0.0177 0.0165

0.0484 0.0421 0.0327

−0.0012 −0.0009 −0.0026

−0.0047 −0.0032 −0.0055

−0.0019 −0.0016 −0.0026

normally from −1 to 0.5, if the solid is stable and linear elasticity [50–52]. A large Poisson's ratio of solid materials usually conforms to its good plasticity. The values of Poisson's ratio ν range from 0.274 to 0.320, within the scope from −1 to 0.5, so all the three compounds have stability and linear elasticity. It can be concluded from the largest Poisson's ratio ν of BiCu2PO6 that it ought to be the most plastic among the three BiX2PO6 ceramics. In this work, the universal constant AU and shear constants A1, A2, A3 for anisotropy are adopted to study the elastic anisotropy for the three compounds. The following equations were used to calculate these anisotropic parameters and the outcomes are listed in Table 3 [57–59].

AU = 5

GV B + V GR BR

6

(1)

4C44 4C55 , A2 = , C11 + C33 2C13 C22 + C33 2C23 4C66 A3 = C11 + C22 2C12 A1 =

(2) U

In general, if the material is isotropic, its value of A should be equal to 0 and the ones of A1, A2, and A3 all approach to 1. Hence, the greater degree deviating from the isotropic values corresponds to higher elastic anisotropy. The lowest and largest values of AU belong to

Table 3 Elastic modulus (in GPa) and anisotropic constants for BiX2PO6 ceramics. Model

BV

BR

B

GV

GR

G

E

B/G

v

Au

A1

A2

A3

BiCu2PO6 BiZn2PO6 BiPb2PO6

96.260 100.051 60.501

88.965 93.451 55.863

92.613 96.751 58.182

40.928 47.004 31.976

34.664 39.255 29.886

37.796 43.129 30.931

99.810 112.648 78.825

2.450 2.243 1.881

0.320 0.306 0.274

0.985 1.057 0.433

1.175 0.736 1.978

0.710 0.812 1.412

0.337 0.339 0.701

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BiPb2PO6 and BiZn2PO6, respectively, indicating that the elastic anisotropy of BiPb2PO6 is the highest and that of BiZn2PO6 is the lowest among three investigated compounds. The constants A1, A2, A3 describe the anisotropic degrees on the (100), (010) and (001) planes, respectively. Among three BiX2PO6 ceramics, the smallest value of (1-A3)

Fig. 4. Projections of Young's moduli on the (001), (010) and (100) crystal planes of BiX2PO6 ceramics. Fig. 3. 3D surface contours of the Young's modulus for BiCu2PO6 (a), BiZn2PO6 (b) and BiPb2PO6 (c) ceramics. 4

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belongs to BiPb2PO6, while BiPb2PO6 possesses the largest absolute values of (1-A1) and (1-A2), suggesting that the shear anisotropic degree of BiPb2PO6 is the lowest on the (001) plane and its shear anisotropic degrees are the largest on the (100) and (010) planes. From the absolute values of (1-A3) of BiCu2PO6 and BiZn2PO6, it can be concluded that their shear anisotropic degrees are not only very close to each other but also are greatly higher than the one of BiPb2PO6 on the (001) plane. In addition, owing to the smallest value of (1-A1) for BiCu2PO6 and the smallest one of (1-A2) for BiZn2PO6, the weaker shear anisotropy on (100) and (010) planes belongs to BiCu2PO6 and BiZn2PO6, respectively. The graphs of three-dimensional (3D) surface construction for Young's modulus can deeply depict the anisotropy of BiX2PO6 (X = Cu, Zn, and Pb) ceramics. The reciprocal expression of orthorhombic structure Young's modulus for BiX2PO6 (X = Cu, Zn, and Pb) ceramics is as following [60–66]:

conductivity on the (100) and (010) planes, respectively. These anisotropic results accord well with those inferred from the values of A1, A2, A3. After discusing the dynamical stability, we calculated the

1/E = Ɩ14S11 + Ɩ24S22 + Ɩ34S33 + 2Ɩ12Ɩ22S12 + 2Ɩ12Ɩ32S13 + 2Ɩ22Ɩ32S23 + Ɩ22Ɩ32S44 + Ɩ12Ɩ32S55 + Ɩ12Ɩ22S66 (3) The constants l and Sijs correspond to the direction cosine and compliance matrix constants, respectively. The 3D figures of Young's modulus for BiX2PO6 (X = Cu, Zn, and Pb) ceramics are shown in Fig. 3. If the solid material is isotropic, its 3D surface construction should be a sphere. Furthermore, if the 3D surface construction of a material owns the higher degree of deviating from sphere, the material possesses the higher anisotropy. It can be concluded that BiX2PO6 (X = Cu, Zn, and Pb) ceramics are evidently anisotropic. Obviously, the lowest degree of 3D surface construction deviating from sphere belongs to BiPb2PO6 among BiX2PO6 (X = Cu, Zn, and Pb), which indicates that BiPb2PO6 has the lowest anisotropic properties, and the anisotropy of BiCu2PO6 and BiZn2PO6 approach to each other. Fig. 4 plots the projections of Young's moduli on different planes. The projections can more details describe the anisotropic properties of BiX2PO6 (X = Cu, Zn, and Pb) ceramics. In general, the larger degree of deviation from the circle is, the higher anisotropy on the homologous plane is. For BiPb2PO6, its degree of deviation circle is the lowest on the (001) plane, so its shear anisotropy is weakest among the three ceramics. However, BiPb2PO6 owns the largest deviation from circle on the (100) and (010) planes, which corresponds to the highest shear anisotropy on the two planes. The projection of Young's modulus on the (010) plane for BiZn2PO6 is most close to circle, indicating that BiZn2PO6 is the least shear anisotropic on the (010) plane. It indicates that BiCu2PO6 is more shear anisotropic than BiZn2PO6 on the (010) plane, but it is less shear anisotropic than BiZn2PO6 on the (100) plane. Moreover, it is inferred that BiZn2PO6 and BiCu2PO6 have similar shear anisotropy on the (001) plane and the higher shear anisotropic extent is pronounced compared to BiPb2PO6. The results analyzed from the projections are consistent with the ones inferred from the absolute values of (1-A1), (1-A2), and (1-A3). For ceramic materials, the high temperature applications can be effectively investigated by the minimum thermal conductivity ĸmin. In this work, Clarke's model was applied in the calculation for anisotropic minimum thermal conductivity ĸmin and its expression is as following [60–66]: ĸmin = 0.87kBMa−2/3ρ1/6E1/2

(4)

For BiX2PO6 (X = Cu, Zn, and Pb), Fig. 5 shows the shapes of the minimum thermal conductivities on (100), (010) and (001) planes. Evidently, the minimum thermal conductivities of the three ceramics all possess anisotropy, as a result of the diverse speeds of the heat conduction in the diverse orientations. In Fig. 5, it is observed that the lowest anisotropic thermal conductivity belongs to BiPb2PO6 on the (001) plane. However, the anisotropy of minimum thermal conductivity for BiPb2PO6 is the highest on the (100) and (010) planes. Besides, BiCu2PO6 and BiZn2PO6 has the lowest anisotropic thermal

Fig. 5. Projections of minimum thermal conductivities on the (001), (010) and (100) crystal planes of BiX2PO6 ceramics. 5

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Fig. 8. The calculated heat capacity of BiX2PO6 compounds.

Fig. 6. The calculated Gibbs free energy of BiX2PO6 compounds.

4. Conclusions In this work, the elastic anisotropy and thermodynamic of BiX2PO6 (X = Cu, Zn, and Pb) ceramics were systematically studied using the first-principle method. The orthorhombic BiX2PO6 ceramics are structurally stable, due to the negative values of Ec and ΔH and positive phonon frequencies. Owing to the biggest values of mechanical moduli (B, G, E) for BiZn2PO6, it has highest mechanical strength among three ceramics. According to the values of B/G and Cauchy pressure for BiX2PO6 (X = Cu, Zn, and Pb), they are all ductile materials. The anisotropic constants, anisotropy of Young's modulus and minimum thermal conductivity exhibit the anisotropic degree. The thermodyanmic parameters confirm that three BiX2PO6 compounds are thermally stable from 0 to 1000 K. The calculated thermodynamic parameters can provide the useful data for the further experimental investigations. Declaration of competing interest Fig. 7. The calculated Debye temperature of BiX2PO6 compounds.

The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

thermodynamic properties using the CASTEP code for further applications [67–70]. Fig. 6 exhibits the Gibbs free energy of BiX2PO6 compounds from 0 to 1000 K. The values are all negative in the whole temperature range, indicating that three BiX2PO6 compounds all exhibit thermal stability from 0 to 1000 K. The Gibbs free energy become more negative, the thermal stability of solid materials becomes better [70]. The Gibbs free energy at the given temperature for three BiX2PO6 compounds follow the following order: BiCu2PO6 < BiPb2PO6 < BiZn2PO6. The order indicates that BiZn2PO6 compound exhibit the best thermal stability at the given temperature among three BiX2PO6 compounds. Fig. 7 Shows the calculated Debye temperature for BiX2PO6 compounds. The Debye temperature for BiX2PO6 compounds basically keep unchanged, when the temperature is over 300 K. Generally speaking, the higher Debye temperature corresponds to the better mechanical strength at high temperature [67–70]. So BiPb2PO6 compound will show the best mechanical strength with the increase temperature due to the largest value of Debye temperature (~1000 K). Fig. 8 Shows the calculated heat capacity at constant volume (Cv) for three BiX2PO6 compounds. As the temperature is below 300 K, the values of heat capacity rapidly increase with elevated temperature. As the temperature increases, the heat capacity for three BiX2PO6 compounds slowly becomes increasing and approaches to the Petit limit. Three BiX2PO6 compounds show the similar value (220 J/(mol·K)).

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