Anisotropic elasticity and thermal conductivities of (α, β, γ)-LiAlSi2O6 from the first-principles calculation

Journal of Alloys and Compounds 756 (2018) 40e49

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Anisotropic elasticity and thermal conductivities of (a, b, g)-LiAlSi2O6 from the first-principles calculation Hongwei Shou, Yonghua Duan* Faculty of Material Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 March 2018 Received in revised form 2 May 2018 Accepted 4 May 2018 Available online 5 May 2018

In this work, the first-principles calculation was employed to perform the investigations of the structural properties, elastic constants and moduli, anisotropy in elastic properties and thermal conductivities of (a, b, g)-LiAlSi2O6. The obtained cohesive energies and formation enthalpies show that all LiAlSi2O6 are energetically stable and a-C2/c LiAlSi2O6 is the most stable phase. The elastic properties, including bulk, shear and Young's moduli (B, G and E), Poisson's ratio v and hardness HV were computed based on the single-crystal elastic constants. a-C2/c LiAlSi2O6 has the largest G, E and HV values. The elastic anisotropy was characterized by the elastic anisotropic indexes, surface constructions and projections of elastic modulus. It can be found that all LiAlSi2O6 show anisotropic elasticity and the order is a-P21/c > g > b > aC2/c. The thermal conductivities and their anisotropy of LiAlSi2O6 were also discussed by Long's and Cahill's models, and the results indicate that all LiAlSi2O6 exhibit potential for insulation materials. © 2018 Elsevier B.V. All rights reserved.

Keywords: First-principles calculation LiAlSi2O6 Elastic properties Thermal conductivity

1. Introduction Spodumene, which has the formula of LiAlSi2O6, is a natural silicate belonging to the family of pyroxenes [1]. In the aspect of application, spodumene is usually used as a raw material for lithium products including lithium-based chemicals, glasses and ceramics [2,3]. At ambient conditions, spodumene is available in three structurally stable forms such as monoclinic, tetragonal and hexagonal symmetries, which are known as a, b and g-spodumenes, respectively [4e6]. a-LiAlSi2O6 crystallizes as a monoclinic system with the space groups (SG) of C2/c [7] and P21/c [8], bLiAlSi2O6 is found to be tetragonal structure with the space group of P41212 (or P43212) [5], and g- LiAlSi2O6 is hexagonal with the space group of P6222 [6]. For a-LiAlSi2O6, it has been proved by experiments that there is a C2/c / P21/c transition at the pressure of 3.19 GPa [8]. Moreover, at temperature below 875 K, a-LiAlSi2O6 is the stable phase, while the temperature is above 875 K, bLiAlSi2O6 is stable. The a/b phase transition is of first order [9]. gLiAlSi2O6 is the other high temperature phase. When the glass of spodumene composition is heat treated, g-LiAlSi2O6 is the first product. As the treatment temperature or time increases, g-LiAlSi2O6 transforms to b-LiAlSi2O6 [6]. These results indicate that,

* Corresponding author. E-mail address: [email protected] (Y. Duan). https://doi.org/10.1016/j.jallcom.2018.05.040 0925-8388/© 2018 Elsevier B.V. All rights reserved.

when the temperature increases, the a/g/b phase transition will occur in spodumene. Nowadays, spodumene is mainly used in glasses and ceramics. Therefore, the current interest of spodumene is mainly focused on the experimental synthesis and characterization, and the theoretical investigations of electronic and optical properties. The optically stimulated luminescence (OSL) response and the potential use for gamma radiation dosimeter of a-LiAlSi2O6 have been investigated experimentally, and the results indicated that a-LiAlSi2O6 can be a candidate for OSL dosimeter of high gamma doses [10]. The natural a-spodumene was successfully used to synthesize glasses and glass ceramics in Li2O-Al2O3-SiO2 system and the transformation of bspodumene occurred at 850e900
C by the X-ray diffraction analysis [11]. By the replacement of Al2O3 by ZnO, b-spodumenewillemite glass ceramics were produced, and the results of mechanical properties showed that the crystallization and morphology of the glass ceramics can affect the flexural strength and fracture toughness [12]. A recent investigation of crystallization kinetics of a b-spodumene glass ceramic revealed that the bspodumene layer could create the residual stress, which can increase fracture toughness and energy absorption remarkably [13]. The fabricated b-spodumene glass-ceramics containing B2O3, P2O5 and TiO2 showed low strength and modulus due to a possible premature crystallization [14]. A theoretical study of a-LiAlSi2O6 indicated that in C2/c-spodumene the charge concentration of three non-equivalent oxygen atoms are different [1]. The electronic

H. Shou, Y. Duan / Journal of Alloys and Compounds 756 (2018) 40e49

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Fig. 1. Crystal structures of a, b and g-LiAlSi2O6: (a) a-LiAlSi2O6 in SG of C2/c, (b) a-LiAlSi2O6 in SG of P21/c, (c) b-LiAlSi2O6, (d) g-LiAlSi2O6, (e) the structure of SiO4 tetrahedron and AlO6 octahedron with a sharing O atom, and (f) the structure of SiO4 tetrahedron and AlO4 tetrahedron with a sharing O atom. The red and purple balls are oxygen and lithium atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

structures and basic optical properties of a-spodumene were investigated by the first-principles calculation [15,16]. Here, aspodumene was found to be anisotropic in absorption along the three optical axes, and had the large absorption coefficient in the wavelength ranging from 60 nm to 125 nm. Moreover, we have theoretically investigated the electronic structures and optical properties of a, b and g-spodumenes in the previous works [17,18]. On the elastic properties of a, b and g-spodumenes, only the elastic moduli of a-spodumene were investigated experimentally [19]. However, it still lacks the systematic investigation of the elastic properties and thermal conductivities of a, b and g-spodumenes. Besides, the anisotropic elastic properties are quite important to improving the mechanical durability of ceramics due to the microcracks in ceramics introduced by the anisotropic elasticity. Moreover, the thermal conductivity is of vital importance in high temperature applications of ceramics. As is known, the physical properties of solids are related to their electron configurations. The first-principles calculation, which is based on the density functional theory, is an effective theoretical tool to investigate the physical properties and to solve the experimental limitations of bulk or single-crystal syntheses. To ensure the feasibility of the computational method and the reliability of the new data in our work, we performed a revisit of structural properties and phase stability of a, b and g-spodumenes, and made a comparison with the previously reported works [8,10,17,18]. Most importantly, our greater interest in this work lied in the anisotropic elastic properties and thermal conductivities of these spodumenes. Therefore, the bulk, shear and Young's moduli and their

anisotropies were investigated. Moreover, the thermal conductivities of these spodumenes were also computed from two models. We hope the obtained results will provide guidance for the future works of spodumenes. 2. Computational methods In our work, we performed the first-principles calculation to discuss the phase stabilities, anisotropy in elastic properties and thermal conductivities of a, b and g-spodumenes within the CASTEP code [20]. The ultrasoft pseudo-potentials (USPPs) were employed to characterize the valence-electrons and ionic-core interaction. The Perdew-Burke-Ernzerhof (PBE) scheme of the generalized gradient approximation (GGA) was used to represent the exchange-correlation energy [21]. The applied valence electron configurations of Li, Al, Si and O were 1s22s1, 3s23p1, 3s23p2 and 2s22p4, respectively. The cut-off energy was 450 eV, and the k points were 3  3  4, 8  8  6 and 14  14  14 for a, b and gspodumenes, respectively. During the geometry optimization, the values of 5  106 eV and 0.01 eV/Å were set for the total energy variation and the forces per atom, respectively. 3. Results and discussion 3.1. Structural properties and phase stability Fig. 1 plots the crystal structures of a, b and g-spodumenes. In aspodumene (Fig. 1(a) and (b)), the SiO4 tetrahedrons form chains

Table 1 Experimental and theoretic lattice parameters (a, b and c in Å, V0 in Å3), cohesive energies Ec (eV/atom) and formation enthalpies DH (eV/atom) of LiAlSi2O6. Spodumene

a

Space group

C2/c

P21/c

b

P41212

g

P6222

Lattice parameters a

b

c

9.495 9.456 9.507 9.316 9.31 9.330 7.558 7.541 7.564 5.220 5.217 5.221

8.393 8.386 8.390 8.382 8.36 8.393

5.218 5.216 5.220 5.138 5.11 5.130 9.122 9.156 9.116 5.472 5.464 5.470

V0

Ec

DH

Refs.

390.43 388.35 390.82 377.33 375.55 377.83 521.08 520.67 521.55 129.13 128.79 129.28

7.523

0.228

7.512 7.144

0.225 0.215

7.132 6.564

0.211 0.188

6.550 5.817

0.187 0.177

5.805

0.173

Present [10] [17] Present [8] [17] Present [24] [18] Present [25] [18]

42

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parameters used in the present work differ from those used in the previous works, which can reach a better convergence. As is known, the phase stability of ceramics is very significant for establishing the heating and sintering processing. Therefore, the phase stability of ceramics needs to be fully taken into account during the designation and fabrication of ceramics. In this work, we evaluated the phase stabilities of spodumenes by means of cohesive energy Ec and formation enthalpy DH. Generally, a solid with more negative values of cohesive energy and formation enthalpy has a better phase stability. The computed values of Ec and DH for a, b and g-spodumenes are listed in Table 1 and plotted in Fig. 2. Our previous results of Ec and DH [17,18] are also listed in Table 1 for a comparison. The present results agree well with our previous results. The difference between the present and our previous results is due to the different computational parameters used in the corresponding works. However, it can be found in Table 1 and Fig. 2 that the most negative cohesive energy corresponds to the most negative formation enthalpy for a-spodumene with C2/c SG. It implies that a-C2/c spodumene is the most energetically stable. Similarly, g-spodumene is the least energetically stable. As a result, a conclusion can be drawn from the obtained cohesive energies and formation enthalpies that the order in phase stability of these spodumenes is a-C2/c > a-P21/c > b > g. This conclusion is confirmed by the facts that a-LiAlSi2O6 is the stable phase below 875 K and b-LiAlSi2O6 is stable above 875 K [9], and the other hightemperature phase g-LiAlSi2O6 will transform to b-LiAlSi2O6 with the increase of the treatment temperature or time [6]. It should be noted that the present calculations of structural parameters, cohesive energies and formation enthalpies of spodumenes are revisits of the earlier reported results [17,18]. The present calculations agree well with the previous results, indicating that the present computational methods are valid and the presented results are reliable.

Fig. 2. Calculated cohesive energies Ec and formation enthalpies DH of LiAlSi2O6 in this work. a-C and a-P represent a-LiAlSi2O6 with C2/c SG and P21/c SG, respectively.

along the c axis and the AlO6 octahedrons link laterally these chains. Besides, the SiO4 tetrahedrons and AlO6 octahedrons share one O atom (Fig. 1(e)). The difference in structure between C2/c and P21/c structures is the positions of O and Si atoms that O and Si atoms occupy the 8f site in the C2/c structure [22], while O and Si atoms hold the 4e site in the P21/c structure [23]. In b- and gspodumenes (Fig. 1(c) and (d)), there are SiO4 and AlO4 tetrahedrons linked by a sharing O atom (as shown in Fig. 1(f)). In b and gspodumene, the SiO4 and AlO4 tetrahedrons parallel to the ab plane. We firstly optimized the structural parameters before the calculations of energies and elastic properties. Table 1 lists the optimized equilibrium structural parameters, together with the experimental data [8,10,24,25] and our previous works [17,18]. It can be seen in Table 1 that symmetries of these spodumenes do not change after optimization due to the unchanged space groups. Moreover, the present structural parameters are slightly overestimated by the first-principles calculation. These overestimations are due to the GGA approximation used in this work. However, the present structural parameters of spodumenes coincide well with the available experimental data with a deviation within 1%, indicating that the computational method and set parameters in this work should be feasible and reliable. Moreover, the structural parameters in the present work are closer to the experimental values than our previous results. This is because that the computational

3.2. Single-crystal elastic constants We calculated the single-crystal elastic constants Cij of LiAlSi2O6 according to the stress-strain method, and the results, together with the experimental data [9], are listed in Table 2. The elastic compliance matrix sij of LiAlSi2O6 evaluated directly from Cij are listed in Table 3. It is clear from Table 2 that the single-crystal elastic constants of a-C2/c LiAlSi2O6 are coincided well with the experimental values [9]. According to the lattice dynamical theory by Born and Huang [26], the mechanical stabilities of a, b and g-

Table 2 Calculated elastic constants Cij (in GPa) of LiAlSi2O6. Spodumene

C11

C22

C33

C44

C55

C66

C12

C13

C15

C23

C25

C35

C46

Refs.

a

270.4 280.5 356.9 149.1 211.5

235.1 230.7 277.8

296.4 303.0 314.4 157.0 152.4

96.2 97.83 103.7 49.3 29.4

66.0 68.76 47.9

82.6 87.05 96.8 22.8 49.0

100.2 94.3 116.0 54.1 101.7

74.3 64.6 145.3 64.0 115.3

35.4 27.1 60.6

99.2 91.6 142.2

2.2 1.4 1.2

3.0 4.0 22.6

7.6 10.4 0.6

Present [9] Present Present Present

C2/c P21/c

b g

Table 3 Calculated elastic compliance matrix sij of LiAlSi2O6. Spodu mene

s11

s22

s33

s44

s55

s66

s12

s13

s15

s23

s25

s35

s46

a

0.00487 0.00569 0.00826 0.00899

0.00564 0.00488

0.00400 0.00576 0.00899 0.01594

0.01047 0.00964 0.01960 0.05178

0.01646 0.03462

0.01219 0.01033 0.04701 0.02224

0.00183 0.00089 0.00090 0.00226

0.00059 0.00284 0.00291 0.00530

0.00252 0.00851

0.00144 0.00179

0.00086 0.00015

0.00018 0.00636

0.00096 0.00006

b g

C2/c P21/c

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Table 4 Calculated bulk modulus B (in GPa), shear modulus G (in GPa), Young's modulus E (in GPa), Poisson's ratio n, Pugh's modulus ratio GH/BH, Cauchy pressure C12-C44 (in GPa) and Vickers hardness HV (in GPa) of LiAlSi2O6. Spodumene

BV

BR

BH

GV

GR

GH

E

n

GH/BH

C12-C44

HV

Refs.

a

150.1 146.1 195.1 99.4 131.1

146.8 143.9 188.6 97.1 128.5

148.5 145.0 191.9 98.3 129.8

84.1

78.6

206.4

0.268

0.548

4.0

10.5

86.1 46.0 37.8

59.7 39.1 28.0

81.4 86.0 72.9 42.5 32.9

194.1 111.4 91.0

0.331 0.311 0.383

0.380 0.432 0.253

12.3 4.8 72.3

6.4 5.0 2.3

Present [9] Present Present Present

C2/c P21/c

b g

LiAlSi2O6 in monoclinic, tetragonal and hexagonal structures can be evaluated. It should be noted that, although the value of C35 of aP21/c LiAlSi2O6 is 22.6 GPa, the value of C33C55 e C235 is still larger than zero. Therefore, a, b and g-LiAlSi2O6 are mechanically stable due to their single-crystal elastic constants satisfying the corresponding mechanical stability criterions. It is well known that the single-crystal elastic constants C11 and C33 represent the a- and c-axial resistances to linear compression, respectively. In the case of the considered LiAlSi2O6, the calculated C11 and C33 values are very large among elastic constants, indicating that these LiAlSi2O6 spodumenes are very incompressible under the a- and c-axial uniaxial stress. Meanwhile, it can be found that C11 is larger than C33 for a-P21/c and g-LiAlSi2O6, which represent that the c axis is more compressible than the a axis for these two LiAlSi2O6. For a-C2/c and b-LiAlSi2O6, C33 larger than C11 indicates that they are more incompressible along the c axis than along the a axis. aP21/c LiAlSi2O6 has the largest values of C11 (356.9 GPa) and C33 (314.4 GPa) in these spodumenes, implies that a-P21/c LiAlSi2O6 should be more incompressible than other LiAlSi2O6 and, thus, have a high bulk modulus. Besides, C44 and C66 are the resistances to shear deformation at (100) plane along the [001] and [110] directions, respectively. These two elastic constants are related to the shear modulus. Obviously, the values of both C44 and C66 of aLiAlSi2O6 are larger than those of b and g-LiAlSi2O6, which indicate that two a-LiAlSi2O6 have the larger shear modulus than b and gLiAlSi2O6. 3.3. Polycrystalline elastic moduli Generally, the polycrystalline elastic modulus is more meaningful than the single-crystal constant in practical applications. In our work, we performed the Voigt-Reuss-Hill (VRH) approximation to calculate the polycrystalline elastic properties of LiAlSi2O6, including the bulk (B), shear (G) and Young's moduli (E) and Poisson's ratio (n). The VRH method computes the elastic modulus by an average of two bounds such as the lower bound (Voigt approximation) and the upper bound (Reuss approximation). The average is defined as the Hill approximation. The calculated expressions are as follows [27e29]:

BH ¼

1 ðB þ BV Þ 2 R

(1)

GH ¼

1 ðG þ GV Þ 2 R

(2)

E¼

9GH BH GH þ 3BH

(3)

n¼

3BH  E 6BH

(4)

Here, BH, BR, BV, GH, GR and GV are the bulk modulus and shear modulus in Hill, Voigt and Reuss approximations, respectively. It

should be noted that the computed formulas of B and G in VRH method are related to the monoclinic, tetragonal and hexagonal symmetries. Table 4 lists the calculated elastic moduli and Poisson's ratios of spodumenes. It is observed that the present calculated bulk and shear moduli of a-C2/c LiAlSi2O6 agree well with the experimental results [9]. As is known, bulk modulus can reflect the compressibility of a solid under the hydrostatic pressure. A higher bulk modulus indicates a more incompressible solid. The highest bulk modulus is 191.9 GPa for a-P21/c LiAlSi2O6, which indicates that a-P21/c LiAlSi2O6 shows the most incompressible characteristic. b-LiAlSi2O6 exhibits the most compressibility among these spodumenes due to the lowest bulk modulus (98.3 GPa). Generally speaking, shear modulus is defined as the tolerance to shape change under shear stress. Therefore, a-C2/c LiAlSi2O6 shows the highest shear resistance under shear stress due to the largest shear modulus (81.4 GPa). As we mentioned above, shear modulus is related to the elastic constants C44 and C66. a-LiAlSi2O6 has the large C44 and C66 values. As a result, a-LiAlSi2O6 has the large shear modulus, as listed in Table 4. Moreover, a correlation between shear modulus and formation enthalpy for all considered spodumenes can be found, which is that a larger shear modulus favors a more negative formation enthalpy. This can be confirmed from the facts that the order of formation enthalpy is a-C2/c < a-P21/c < b < g in Table 1, while the order of shear modulus is a-C2/c > a-P21/c > b > g in Table 4. Usually, Young's modulus represents the stiffness of materials, and Poisson's ratio can measure the stability of materials under the shear stress. The material with a large Young's modulus and Poisson's ratio has a high stiffness and a good stability to resist the shear deformation. Therefore, a-C2/c LiAlSi2O6 shows the highest stiffness and g-LiAlSi2O6 has the best stability under shear stress. Theoretically, the intrinsic ductile and brittle behavior of non-

Fig. 3. Correlations between bulk modulus B (shear modulus G) and hardness HV of LiAlSi2O6. a-C and a-P represent a-LiAlSi2O6 with C2/c SG and P21/c SG, respectively.

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Table 5 Calculated elastic anisotropic indexes (AU, Acomp, Ashear, A1, A2 and A3) of LiAlSi2O6. Spodumene

a b g

C2/c P21/c

AU

Acomp (%)

Ashear (%)

A1

A2

A3

0.372 2.246 0.906 1.770

1.111 1.694 1.170 1.002

3.380 18.107 8.108 14.894

0.920 1.090 1.107 0.882

0.793 0.622 0.944 1.861

0.865 0.476 0.780 1.038

metallic LiAlSi2O6 can be indicated by Poisson's ratio n, the ratio of GH to BH and Cauchy pressure (C12-C44) [30e32]. If n is larger than 0.26, GH/BH is smaller than 0.57 and C12-C44 is larger than 0, the material is ductile. Otherwise, the material is brittle. Table 4 also lists the values of n, GH/BH and C12-C44 for LiAlSi2O6. It can be seen in Table 4 that all LiAlSi2O6 have the values of n larger than 0.26, GH/BH smaller than 0.57 and C12-C44 larger than 0, indicating that all LiAlSi2O6 spodumenes show the ductile nature. g-LiAlSi2O6 has the relatively large value of n, which indicates that it is the most ductile spodumene. The smallest GH/BH (0.253) and the largest C12-C44 (72.3 GPa) values for g-LiAlSi2O6 also identify the most ductile nature of g-LiAlSi2O6. Although these spodumenes are nonmetallic, SiO4 tetrahedral chains in a-LiAlSi2O6 (laterally linked by AlO6 octahedrons) array along the c axis, while SiO4 and AlO4 tetrahedrons in b and g-LiAlSi2O6 are parallel to the ab plane. Therefore, the strong covalent bonds in spodumenes are perpendicular to the c axis to form the graphite-like strong-bond layers paralleled the ab plane [17,18]. This is the reason that the non-metallic LiAlSi2O6 spodumenes show the ductile behavior. Recently, a theoretical hardness model is usually used to compute the Vickers hardness of polycrystalline materials based on the bulk and shear moduli according to the following expression [33]:

mechanical durability. The elastic anisotropy of LiAlSi2O6 is described by different ways in this work, such as the anisotropic indexes, the surface constructions of elastic modulus and the projections of elastic modulus. The anisotropic indexes used in this work are the universal elastic anisotropic index (AU), the percent anisotropy (Acomp for the compressible anisotropy and Ashear for the shear anisotropy) and the shear anisotropic factors (A1 for the (100) plane, A2 for the (010) plane and A3 for the (001) plane), and their expressions are as follows [39e41]:

A1 ¼

4C44 C11 þ C33  2C13

(8)


1:137 G HV ¼ 0:92 G0:708 B

A2 ¼

4C55 C22 þ C33  2C23

(9)

A3 ¼

4C66 C11 þ C12  2C12

(10)

AU ¼ 5

Acomp ¼

(5)

Table 4 lists the Vickers hardness of LiAlSi2O6. It can be observed that the hardness for all spodumenes is less than 11 GPa. The largest hardness is 10.5 GPa for a-C2/c LiAlSi2O6, while the smallest hardness (2.3 GPa) is for g-LiAlSi2O6. It should be noted from the theoretical model that the hardness is correlated to the bulk and shear moduli. Thus, we plot the correlations between bulk and shear moduli and hardness in Fig. 3. Many times, the bulk modulus is used as a predictor of hardness according to a large bulk modulus corresponding to a large hardness [34,35]. However, a large bulk modulus does not mean a large hardness from Fig. 3 in this work. For example, a-P21/c LiAlSi2O6 has the largest bulk modulus (86.1 GPa), but the largest hardness is 10.5 GPa for a-C2/c LiAlSi2O6. Such non-conforming correlation between bulk modulus and hardness has been proved by the earlier literatures [36,37]. As a more suitable predictor of hardness, shear modulus considers the parameter of dislocation formation and movement [37,38]. Therefore, the material with a large shear modulus has a high hardness. It can be seen in Fig. 3 that a-C2/c LiAlSi2O6 has the largest shear modulus (81.4 GPa) and the largest hardness (10.5 GPa). However, no or less available experimental and theoretical results of elastic constants and moduli for spodumenes can be found up to now. We hope our present work can offer a helpful guidance for the future works on spodumenes. 3.4. Anisotropy of elastic moduli Generally, the anisotropic elasticity can determine the generation and expansion of microcracks in ceramics, and therefore, it is very important for ceramics to the melioration of the ceramic's

GV BV þ 6 GR BR BV  BR G  GR  100%; Ashear ¼ V  100% BV þ BR GV þ GR

(6)

(7)

Here, GV (BV) and GR (BR) represent the shear (bulk) moduli in Voigt and Reuss approximations, respectively. Cij are the single-crystal elastic constants. Generally, if a solid is isotropic, the values of AU, Acomp, Ashear are zero and the values of A1, A2, A3 are one. Otherwise, the solid is anisotropic. Meanwhile, the large deviations from the standard value (zero or one) represent the high anisotropy in mechanical properties. Table 5 lists the calculated several anisotropic indexes. The values of AU are 0.372, 2.246, 0.906 and 1.770 for a-C2/ c, a-P21/c, b and g-LiAlSi2O6, respectively, indicating that the order of anisotropy in elasticity for spodumenes is a-P21/c > g > b > a-C2/ c. This anisotropic order also can be reflected by Acomp and Ashear. For a-P21/c LiAlSi2O6, the values of Acomp and Ashear are 1.694% and 18.107%, respectively. These two values fora-P21/c LiAlSi2O6 are higher than other spodumenes, which indicates a-P21/c LiAlSi2O6 shows the highest elastic anisotropy. Besides, the deviations of the shear anisotropic factors (A1, A2 and A3) from one are 0.090, 0.378 and 0.524 for a-P21/c LiAlSi2O6, which are larger than for other LiAlSi2O6 spodumenes. This also confirms that the highest anisotropy is for a-P21/c LiAlSi2O6. Based on the data in Table 5, a-C2/c LiAlSi2O6 has the least elastic anisotropy by the similar analysis of AU, Acomp and Ashear, and A1, A2 and A3. To characterize the directional anisotropy in elasticity more clearly, an effective three-dimensional (3D) surface construction of elastic modulus is employed in this work. The expressions of 3D surface constructions of bulk and Young's moduli are as follows [42]: For monoclinic crystal class (a-C2/c and a-P21/c LiAlSi2O6):

H. Shou, Y. Duan / Journal of Alloys and Compounds 756 (2018) 40e49

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Fig. 4. Surface construction of bulk modulus of LiAlSi2O6. (a) a-C2/c LiAlSi2O6, (b) a-P21/c LiAlSi2O6, (c) b-LiAlSi2O6, (d) g-LiAlSi2O6.

1 ¼ ðs11 þ s12 þ s13 Þl21 þ ðs12 þ s22 þ s23 Þl22 þ ðs13 þ s23 þ s33 Þl23 B þ ðs15 þ s25 þ s35 Þl1 l3 (11) 1 ¼ l41 s11 þ 2l21 l22 s12 þ 2l21 l23 s13 þ 2l31 l3 s15 þ l42 s22 þ 2l22 l23 s23 E þ 2l1 l22 l3 s25 þ l43 s33 þ 2l1 l33 s35 þ l22 l23 s44 þ 2l1 l22 l3 s46 þ l21 l23 s55 þ l21 l22 s66 (12) For tetragonal crystal class (b-LiAlSi2O6):

  1 ¼ ðs11 þ s12 þ s13 Þ l21 þ l22 þ ð2s13 þ s33 Þl23 B

(13)

    1 ¼ s11 l41 þ l42 þ ð2s13 þ s44 Þ l21 l23 þ l22 l23 þ s33 l43 þ ð2s12 E þ s66 Þl21 l22 (14) For hexagonal crystal class (g-LiAlSi2O6):

1 ¼ ðs11 þ s12 þ s13 Þ  ðs11 þ s12  s13  s33 Þl23 B

(15)

 2   1 ¼ s11 1  l23 þ s33 l43 þ ð2s13 þ s44 Þl23 1  l23 E

(16)

Here, sij is the usual elastic compliance constant in Table 3, l1, l2 and l3 are the direction cosines. The 3D surface construction is spherical

for the isotropic material, and the deviation of 3D figure from sphere represents the degree of elastic anisotropy. Fig. 4 plots the 3D surface constructions of bulk modulus of LiAlSi2O6. Each graph in the 3D figure denotes the magnitude of bulk modulus along different orientations. It can be seen from Fig. 4 that the bulk modulus for all LiAlSi2O6 shows strong anisotropy along different orientations. a-P21/c LiAlSi2O6 has the most highly anisotropic elasticity, which has been proved by the largest Acomp value (1.694%) in Table 5. Furthermore, the 3D figures for the other LiAlSi2O6 exhibit different shapes and these LiAlSi2O6 have little difference in the Acomp values. Therefore, it is hard to determine the difference in the deviation from sphere from Fig. 4. Because the anisotropy in bulk modulus is not very prominent, we plot the 3D surface constructions of Young's modulus of LiAlSi2O6 in Fig. 5. As can be seen from Fig. 5, there are several distinct anisotropic characteristics. The first is the 3D figure of a-P21/c LiAlSi2O6 has the remarkable anisotropy in Young's modulus. The second is that g-LiAlSi2O6 shows an obviously anisotropic nature in Young's modulus, which is more prominent than in bulk modulus. However, the anisotropic nature of g-LiAlSi2O6 is slightly less than that of a-P21/c LiAlSi2O6. The third is that a-C2/c LiAlSi2O6 has a higher anisotropy in Young's modulus than b-LiAlSi2O6. The order of anisotropy in Young's modulus of LiAlSi2O6 is a-P21/c > g > b > aC2/c, which is coincided well with the evaluated elastic anisotropic indexes. The 3D surface construction of elastic modulus is an effective characterization of the directional elastic anisotropy. However, the more details of the anisotropy in elasticity can be described by the projection of elastic modulus on the different planes. Fig. 6 plots the projections of bulk modulus on the (001), (010) and (100) planes of LiAlSi2O6. As plotted in Fig. 6, the bulk moduli of all LiAlSi2O6 are remarkable anisotropic on the (100) and (010) planes, but show the least anisotropy on the (001) plane. The highest anisotropy in bulk

46

H. Shou, Y. Duan / Journal of Alloys and Compounds 756 (2018) 40e49

Fig. 5. Surface construction of Young's modulus of LiAlSi2O6. a) a-C2/c LiAlSi2O6, (b) a-P21/c LiAlSi2O6, (c) b-LiAlSi2O6, (d) g-LiAlSi2O6.

modulus on the (100) plane is for g-LiAlSi2O6, while that on the (010) plane is for a-P21/c LiAlSi2O6. Whereas, the second highest anisotropic bulk modulus on the (100) plane is for a-P21/c LiAlSi2O6, and that on the (010) plane is for g-LiAlSi2O6. Therefore, a-P21/c LiAlSi2O6 has the highest anisotropy in bulk modulus, which coincides with the largest Acomp value (1.694%). As for a-C2/c and b-LiAlSi2O6, they have the relatively weak anisotropy and the no obvious difference in anisotropy. Fig. 7 plots the projections of Young's modulus on the (001), (010) and (100) planes of LiAlSi2O6. It can be observed that, for

a-P21/c LiAlSi2O6 and g-LiAlSi2O6, on the (100) plane they show the highest anisotropy in Young's modulus, on the (010) plane aP21/c LiAlSi2O6 presents a higher anisotropy than g-LiAlSi2O6, and on the (001) plane they have the weakly anisotropic Young's modulus. For a-C2/c and b-LiAlSi2O6, they are weakly anisotropic on the (100) plane, a-C2/c LiAlSi2O6 exhibits a higher anisotropy than b-LiAlSi2O6 on the (010) plane, and b-LiAlSi2O6 shows a higher anisotropy than a-C2/c LiAlSi2O6 on the (001) plane. Therefore, from the projections of bulk and Young's moduli, a general order of anisotropy in elastic properties can be obtained as

Fig. 6. Projections of bulk moduli at the (001), (010) and (100) crystal planes of LiAlSi2O6.

H. Shou, Y. Duan / Journal of Alloys and Compounds 756 (2018) 40e49

47

Fig. 7. Projections of Young's moduli at the (001), (010) and (100) crystal planes of LiAlSi2O6.

Table 6 The density r (g/cm3), sound velocity (longitudinal nl, transverse nt and average nm) (m/s) and Debye temperature QD (K) of LiAlSi2O6. Spodumene

a b g

C2/c P21/c

r

nl

nt

nm

QD

3.16 3.27 2.37 2.39

9019 9403 8086 8524

5075 4722 4235 3710

5647 5295 4737 4190

783 743 597 529

a-P21/c > g > b > a-C2/c, which agrees well with the calculated anisotropic indexes.

structures can be found in our reported work [46]. The eigenvalues of monoclinic crystals in three principal axes, such as [100], [010] and [001] directions, are expressed as follows [47]: For the [100] direction

3.5. Debye temperatures and anisotropy of sound velocities Debye temperature qD is known to be related to specific heat and melting point, and can be calculated from the sound velocities based on the elastic modulus [43,44]. Table 6 presents the sound velocities and Debye temperatures of LiAlSi2O6. As is known, Debye temperature can characterize the strength of covalent bonds [45]. From Table 6, the order of Debye temperature is a-C2/c > a-P21/ c > b > g. Thus, it can be concluded that the strength of covalent bonds in a-C2/c LiAlSi2O6 is the strongest among these LiAlSi2O6. Besides, Debye temperature is also related to the cohesive energy that a larger Debye temperature corresponds to a more negative cohesive energy. Therefore, a-C2/c LiAlSi2O6 with the largest Debye temperature (783 K in Table 6) has the most negative cohesive energy (7.523 eV/atom in Table 1). Moreover, Debye temperature can indicate the thermal conductivity to some extent, which means that a solid with a larger Debye temperature has a higher thermal conductivity. Accordingly, among these spodumenes, a-C2/c LiAlSi2O6 has the largest Debye temperature, while g-LiAlSi2O6 possesses the smallest Debye temperature, which indicates that the highest thermal conductivity is for a-C2/c LiAlSi2O6 and the lowest one is for g-LiAlSi2O6. As we known, LiAlSi2O6 can crystallize as monoclinic, tetragonal and hexagonal structures. The sound velocities of LiAlSi2O6 should be anisotropic. Therefore, we calculate the anisotropy in sound velocities of LiAlSi2O6 in this work. In tetragonal and hexagonal crystals, there are three pure acoustic modes including one longitudinal mode and two transverse modes. In the monoclinic crystals, the acoustic modes include two mixed modes (a faster mode and a slower mode) and one pure mode. The calculated expressions of sound velocities in acoustic modes for tetragonal and hexagonal

nt ¼

pffiffiffiffiffiffiffiffiffiffiffiffi C66 =r

nþ ¼

n ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  u uðC11 þ C55 Þ þ C 211 þ 4C 215  2C11 C55 þ C 255 2 t 2r

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  u uðC11 þ C55 Þ  C 211 þ 4C 215  2C11 C55 þ C 255 2 t 2r

(17)

For the [010] direction

nl ¼

ntþ

pffiffiffiffiffiffiffiffiffiffiffiffi C22 =r

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  u uðC44 þ C66 Þ þ C 244 þ 4C 246  2C44 C66 þ C 266 2 t ¼ 2r

nt ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  u uðC44 þ C66 Þ  C 244 þ 4C 246  2C44 C66 þ C 266 2 t

For the [001] direction

nt ¼

pffiffiffiffiffiffiffiffiffiffiffiffi C44 =r

2r

(18)

48

H. Shou, Y. Duan / Journal of Alloys and Compounds 756 (2018) 40e49

Table 7 The anisotropic sound velocity (in m/s) of LiAlSi2O6. Spodumene

a b

[100]

C2/c P21/c

g

[010] vþ

v-

vl

vtþ

vt-

vt

vþ

v-

5113 5441 [100] [100]vl

9352 10614

4359 3338

5614 5633

5006 5439

4569 3751

[001]vt2

[100]vt1

[010]vt2

5518 5631 [110] [110]vl

9686 9835

[010]vt1

8625 9217 [001] [001]vl

[001]vt1

7932 4793

3102 9407

4561 3507

8139 7985

3102 3507

3102 3507

7245

4561

[110]vt2 4477

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  u uðC33 þ C55 Þ þ C 233 þ 4C 235  2C33 C55 þ C 255 2 t nþ ¼ 2r

n ¼

[001]

vt

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1  u uðC33 þ C55 Þ  C 233 þ 4C 235  2C33 C55 þ C 255 2 t 2r

kLong min

( " 3 #)13
1
2 3 2 1 1 E 2 2ð2 þ 2nÞ2 þ ¼ n kB n3 3 1n r

kCahill min ¼ (19)

It can be seen in Eqs. 17e19 that in [100] and [001] directions there are two mixed modes (vþ for the faster mode and v- for the slower mode) and one pure transverse (vt), and in the [010] direction there are two pure transverse modes (vtþ and vt-) and one pure longitudinal mode (vl). Table 7 lists the calculated directional sound velocities. Obviously, all LiAlSi2O6 are anisotropic in sound velocities due to their different values in the directions. For the monoclinic a-LiAlSi2O6, the values of vþ (faster mode) are always larger than those of v- (slower mode) in the mixed modes, and the values of v- are the lowest in all considered directions. Moreover, in the [100] and [001] directions, the velocities in the faster mode (vþ) are often larger than those in the transverse and slower modes (vt and v-). In the [010] direction, the velocities in the pure longitudinal mode (vl) are the largest compared to the two pure transverse modes (vtþ and vt-). For the tetragonal b-LiAlSi2O6, the longitudinal sound velocities in three considered directions are larger than the transverse velocities. The reason is that the longitudinal sound velocities are mainly determined by the uniaxial elastic constants, and b-LiAlSi2O6 has the large C11 and C33 values. For the hexagonal g-LiAlSi2O6, the largest velocity is 9407 m/s for the first transverse mode in the [100] direction due to the quite large C11 value (211.5 GPa) among the elastic constants of g-LiAlSi2O6.

3.6. Thermal conductivities The thermal conductivity is crucial to ceramics at high temperatures. As the temperature increases, the thermal conductivity will fall down to an extreme [48]. We have computed the minimum thermal conductivities (kmin) of LiAlSi2O6 by Long's and Cahill's models [49e51] in this work, respectively. The equations are as follows:

kB 2 n3 ðvl þ 2vt Þ 2:48

(20)

(21)

where kB is Boltzmann's constant, n is the number of atoms per unit volume. In Long's model, v is Poisson's ratio, E is Young's modulus, r is the density. In Cahill's model, vl is the longitudinal sound velocity, and vt is the transverse sound velocity. Table 8 lists the minimum thermal conductivities of LiAlSi2O6 computed from the Long's and Cahill's models, respectively. It is obvious that the obtained values of kLong are close to those of kCahill min , which indicates that these two min models are suitable for the calculations of minimum thermal conductivities of LiAlSi2O6. The contribution of phonon spectrum is considered in Cahill's model [52], whereas the elastic anisotropy is taken into account in Long's model [49]. This may be the origin of the difference between the values of kLong and kCahill min . However, as min can be seen in Table 8, a-C2/c LiAlSi2O6 has the largest kmin value, while g-LiAlSi2O6 shows the smallest one. It indicates that a-C2/c LiAlSi2O6 has the best thermal conductivity, which can reduce the thermal stress concentration of LiAlSi2O6 ceramics. As a result, this will be propitious to the synthesis glasses and glass ceramics in Li2O-Al2O3-SiO2 system. Moreover, the kmin values of these LiAlSi2O6 (1.4e2.3 W m1 K1) are lower than those of monazite-type rare-earth phosphates (4e5 W m-1 K-1) [53] and Nano yttria stabilized zirconia (YSZ) (2.7e3.0 W m-1 K-1) [54], which indicates that these LiAlSi2O6 exhibit potential for insulation materials. As is known, the thermal conductivity is mainly originated from the conductivity of lattice vibration (namely lattice thermal conductivity) at the ground-state temperature [55]. At low temperatures, the lattice thermal conductivity is directly proportional to Debye temperature [56]. The order of thermal conductivity for LiAlSi2O6 is a-C2/c > a-P21/c >b > g, and therefore, a-C2/c LiAlSi2O6 has the largest Debye temperature (783 K), while g-LiAlSi2O6 exhibits the smallest Debye temperature (529 K). Cahill's model might be more suitable way to calculate the anisotropic thermal conductivity due to the consideration of three acoustic branches in the model. Therefore, the equation of Cahill's model to investigate the anisotropy in thermal conductivity is as follows [51]:

Table 8 Calculated minimum thermal conductivities kmin (W$m1$K1) of LiAlSi2O6. n(1029)

Spodumene

a b g

C2/c P21/c

1.023 1.059 0.767 0.774

kmin

Long

kCahill min

kmin[100]

kmin[010]

2.153 2.023 1.426 1.406

2.344 2.337 1.663 1.611

2.292 2.215 1.566 1.679

2.343 2.327

kmin[110]

1.636

kmin[001] 2.407 2.393 1.541 1.565

H. Shou, Y. Duan / Journal of Alloys and Compounds 756 (2018) 40e49

kmin

2 k ¼ B n3 ðvl þ vt1 þ vt2 Þ 2:48

(22)

Table 8 lists the directional minimum thermal conductivities of LiAlSi2O6. It can be found that the values of kmin[001] are always higher than kmin[010] and kmin[100] in the monoclinic a-LiAlSi2O6, while the values of kmin[100] are higher than kmin[001] in b-LiAlSi2O6 and g-LiAlSi2O6. The variety of minimum thermal conductivity indicates the different velocities of heat conduction in the different directions. Besides, a-P21/c LiAlSi2O6 has a stronger anisotropy in thermal conductivity than other LiAlSi2O6 due to the larger difference between kmin[100] and kmin[010] or kmin[100]. This is due to that the elastic moduli in a-P21/c LiAlSi2O6 are larger than in other LiAlSi2O6. The thermal conductivity of b-LiAlSi2O6 shows the weakest anisotropy on the crystallographic direction. 4. Conclusions We have systematically investigated the phase stabilities, anisotropic elasticity, sound velocities and thermal conductivities of (a, b, g)-LiAlSi2O6 by using the first-principles calculation. The cohesive energies and formation enthalpies indicate that all LiAlSi2O6 are energetically stable and the order of phase stability is aC2/c > a-P21/c > b > g. Based on the corresponding criterions of mechanical stability, a, b and g-LiAlSi2O6 are mechanically stable. According to Poisson's ratio n, GH/BH and Cauchy pressure (C12-C44), all LiAlSi2O6 have the ductile nature, which is due to that the strong covalent bonds in LiAlSi2O6 are perpendicular to the c axis to form the graphite-like strong-bond layers paralleled the ab plane. Most importantly, the order in elastic anisotropy for LiAlSi2O6 is a-P21/ c > g > b > a-C2/c. Moreover, Debye temperature decreases from aC2/c LiAlSi2O6 to g-LiAlSi2O6 and the sound velocities of all LiAlSi2O6 are anisotropic. Besides, all LiAlSi2O6 exhibit potential for insulation materials and show the anisotropic thermal conductivities, and the order of thermal conductivity for LiAlSi2O6 is found to be a-C2/c > a-P21/c >b > g. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant no. 51761023 and the Reserve Talents Project of Yunnan Province under Grant no.2015HB019. References [1] [2] [3] [4]

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