First-principles study on the elastic property of hexagonal alunite

Physica B 407 (2012) 2606–2609

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First-principles study on the elastic property of hexagonal alunite Jianping Long n, Lijun Yang, Xuesong Wei College of Materials and Chemistry & Chemical Engineering, Chengdu University of Technology, Chengdu 610059, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 22 February 2012 Received in revised form 22 March 2012 Accepted 3 April 2012 Available online 10 April 2012

In this paper, we have predicted the structural and elastic characteristics of KAl3(SO4)2(OH)6 compounds through the method of density functional theory within the generalized gradient approximation (GGA) and Local Density Approximation (LDA) using the CASTEP package. The calculated equilibrium lattice parameters, the elastic property, anisotropy factor, Poison’s ratio, Young’s modulus, sound velocities and Debye temperature for KAl3(SO4)2(OH)6 have been calculated and compared with the available experimental data. From these results, this compound behaves as a brittle material and has a good thermal conductivity. & 2012 Elsevier B.V. All rights reserved.

Keywords: Alunite DFT Elastic property Debye temperature

1. Introduction The alunite supergroup [1] consists of three mineral groups that, combined, contain more than 40 mineral species with the general formula MA3(XO4)2(OH)6, wherein M represents cations with a coordination number greater or equal to 9 [2], and A and X represent sites with octahedral and tetrahedral coordination, respectively. For the alunite group, the X in (XO4)2 is dominated by S6 þ [3,4], M is K þ and A is Al3 þ [5]. Substitution of Na for K is common in nature alunites, K4Na for alunite and KoNa for natroalunite [6]. The mineral alunite KAl3(SO4)2(OH)6 occurs in hydrothermal ore deposits [7,8], hot springs [9,10] and sedimentary rocks [11]. It is fairly nondescript, in terms of physical properties of alunite except the hardness H¼ 4. Many studies have been performed to investigate the physical and chemical properties of alunite by theoretical and experimental methods [3–10]. Serna et al. [12] investigated the relationships between the crystallochemical characteristics of alunite–jarosite compounds and their vibrational by Infrared and Raman spectra. Stoffregen et al. [13] measured the rates of alunite–water alkali and isotope exchange by scanning electron microscope. Gaboreau et al. [14] predicted Gibbs free energies of formation for the alunite family. Bayliss et al. [15] studied the thermal decomposition of synthetic and natural alunite by X-ray diffraction (XRD), electron diffraction, and electron microscope methods. Drouet et al. [16] investigated the thermochemistry of jarosite–alunite and natrojarosite– natroalunite solid solutions by XRD, Fourier Transform infrared

n

Corresponding author. Tel.: þ86 18723176181; fax: þ86 23 63962314. E-mail address: [email protected] (J. Long).

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2012.04.004

spectroscopy (FTIR) and electron microprobe analysis. Bishop and Murad [17] investigated the visible and infrared spectral properties of jarosite and alunite. In the present article, we investigate the structural, elastic properties and Debye temperature of alunite as determined from firstprinciples density-functional-theory (DFT) calculations. The data thus obtained can be compared with experimental data and also the single-crystal elastic constants reported by Majzlan et al. [18].

2. Computational methods Our calculations were performed using the CASTEP package [19,20] using the plane-wave pseudopotential method based on density functional theory (DFT) [21]. Pseudoatomic calculations were performed for H: 1s1, O: 2s2 2p4, Na: 2s2 2p6 3s1, Al: 3s2 3p1, S: 3s2 3p4 and K: 3s2 3p6 4s1. For better understanding and comparison, both generalized gradient approximation (GGA) developed by Perdew–Burke–Ernzerhor (PBE) [22] and PBE for solids [23] and local density approximation (LDA) developed by Ceperley–Alder [24] and Perdew–Zunger [25] scheme were employed to treat the exchange-correlation potential. Cut-off energy of plane wave is 410 eV and a 9  9  4 Monkhorst–Pack [26] k-point mesh has been employed in this study to ensure well convergence between the computed structures and energies. The structural parameters optimization were calculated using the Brodyden–Fletcher–Goldfarb–Shanno (BFGS) method [27–29], with the threshold for converged structures: the maximum energy change per atom is 5  10  6 eV/atom, the maximum ˚ the maximum Hellmann–Feynman force per atom is 0.01 eV/A, stress within 0.02 GPa and the maximum displacement of atom is 0.0005 A˚ during the geometry optimization [30].

J. Long et al. / Physica B 407 (2012) 2606–2609

The elastic constants of a material describe its response to an applied stress. The linear elastic constants form a 6  6 symmetric matrix, have 27 different components, as the stress and strain have three tensile and shear components. Hexagonal crystals with R3m symmetry have six nonzero elements in the elastic tensor (C11, C12, C13, C33, C44, and C66), but only five are independent as C66 ¼(C11 C12)/2 [27,28].

3. Results and discussions

Table 1 ˚ c (A), ˚ volume V (A˚ 3), density r (g/ Computed equilibrium lattice parameters a (A), cm3) and zero-pressure Bulk modulus B compared with available experimental data. Method

˚ a (A)

˚ c (A)

V (A˚ 3)

r (g/cm3)

B (GPa)

GGA–PBESOL

7.01981

16.9824

724.735

2.77

68.1

GGA–PBE

7.06795

17.5931

761.130

2.64

49.2

LDA–CA–PZ

6.86327

16.4449

670.848

2.99

89.8

EXP

6.96a 7.020b 6.943c 6.9741d

17.35 17.223 17.227 17.19

727.860 735.050 719.175 724.070

2.83 2.81 2.98 2.83

62.6

The relative errors (%)

0.66e

2.12

8.79

3.1. Geometry and structure properties The crystal structure of KAl3(SO4)2(OH)6 belongs to the space group R3m. The structure of KAl3(SO4)2(OH)6 is made of T–O–T layers, where T is the tetrahedral and O the octahedral layer [31]. The octahedron is distorted being formed by two oxygen atoms, coming from the two adjacent SO4 tetrahedrons and four OH groups. Al lies at the center of symmetry whereas the S atom and the apical O lie on the three-fold axis. The tetrahedron is also distorted and consists of three basal O and an apical one (O1) [32,33]. Fig. 1 shows the crystal structure of KAl3(SO4)2(OH)6. Firstly, our calculated results for the lattice parameters a and c, equilibrium volume V, density r and the zero-pressure Bulk modulus B are listed in Table 1 together with the available experimental data. From Table 1, we can find the calculated results of GGA–PBESOL calculation are in agreement with the experiments, but the values of LDA–CA-PZ and GGA–PBE calculation are different. The differences could be partly attributed to use of both GGA (LDA) approximations, which lead to slightly overestimated (underestimated) results.

2607

1.21

0.09

a

Data from Ref. [34]. Data from Ref. [35]. c Data from Ref. [36]. d Data from Ref. [18]. e The relative errors (%)¼ 1009VGGA-PBESOL  VEXP9/VEXP. b

Table 2 Computed elastic constants Cij (GPa) and compliances Sij (10–12 Pa  1) for KAl3(SO4)2(OH)6 with available experimental. Calculate

Experiment

GGA

C11 C33 C44 C66 C12 C13 C14 S11 S33 S44 S12 S66 S13 S14

LDA

PBESOL

PBE

180.0 86.1 51.4 69.5 41.0 31.3  5.0 6.15 12.9 19.6  1.10 14.49  1.83 0.71

158.6 49. 5 42.9 60.1 38.4 18.7  4.5 6.93 218 23.5  1.46 16.77  2.07 0.88

220.4 107.4 64.1 82.7 55.1 49.4  5.7 5.22 11.2 15. 7  0.87 12.17  2.0 0.54

181.9 66.8 42.8 66.9 48.2 27.1 5.4 6.18 16.6 23.6  1.38 15.11  1.95  0.95

3.2. Elastic properties Elastic constants of solids provide a link between mechanical and dynamical behaviors. Also, they give important information concerning the elastic response of a solid to an external pressure. The accurate calculation of elasticity is essential for understanding the macroscopic mechanical properties and provides key information about the bonding characteristic between adjacent atomic planes and the anisotropic character of the solid. For a hexagonal crystal, the elastic constants must satisfy the well-known Born stability criteria [37–39] at 0 GPa: C 11 40 C 11 C 12 40 C 44 40 ðC 11 þ C 12 ÞC 33 2C 213 4 0

Fig.1. Crystal structure of KAl3(SO4)2(OH)6.

ð1Þ

Obviously, from the calculated values of Cij in Table 2, the above restrictions are all satisfied, implying that KAl3(SO4)2(OH)6 is mechanically stable. As shown in Table 2, we list the elastic constants of KAl3(SO4)2(OH)6 at 0 GPa. It can be seen clearly that GGA–PBESOL results are in agreement with the experimental data.

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J. Long et al. / Physica B 407 (2012) 2606–2609

The elastic properties of crystals have been the most extensively investigated in both the experiment and theoretical. A number of equations for estimating the elastic and physical properties of hexagonal polycrystalline materials are presented in literature [40,41]. The bulk (B) and shear (G) moduli of the polycrystalline can be computed by either the Voigt–Reuss–Hill approximation using the individual elastic constants, Cij. The computations of Voigt (GV) and Reuss shear moduli (GR) and Voigt (BV) and Reuss bulk moduli (BR) for hexagonal lattices can be calculated respectively [30,42,43]: GV ¼

1 1 ð2C 11 þ C 33 C 12 2C 13 Þ þ ð2C 44 þ C 66 Þ 15 5

ð2Þ

BV ¼

2 ðC 11 þC 12 þ 2C 13 þC 33 =2Þ 9

ð3Þ

Table 3 Bulk and shear moduli B and G (all in GPa), Young’s modulus E (all in GPa), Poisson’s ratio s, anisotropic factor A, linear compressibility ratio kc/ka and ratio G/B. Calculate GGA

GR GV GH BR BV BH E

s

1 BR ¼ 2ðS11 þS33 Þ þ 2ðS12 þ 2S13 Þ GR ¼

15 4ð2S11 þ S33 Þ4ðS12 þ 2S13 Þ þ3ð2S44 þ S66 Þ

ð4Þ

S12 ¼

C 33 C 11 2C 233 ðC 11 C 12 Þ½C 33 ðC 11 þ C 12 Þ2C 213  C 213 C 33 C 12 ðC 11 C 12 Þ½C 33 ðC 11 þ C 12 Þ2C 213 

S13 ¼ 

S33 ¼

C 13 C 33 ðC 11 þ C 12 Þ2C 213 C 11 þC 12

C 33 ðC 11 þ C 12 Þ2C 213

ð6Þ

ð7Þ

ð8Þ

ð9Þ

S44 ¼

1 C 44

ð10Þ

S66 ¼

1 C 66

ð11Þ

The elastic moduli of the polycrystalline KAl3(SO4)2(OH)6 aggregates can then be estimated by Hill’s average [45], BH ¼ (1/2)(BV þBR) for bulk modulus and GH ¼(1/2)(GV þGR) for shear modulus. Additionally, Young’s modulus E, Poisson’s ratio s can be computed by the following equations [30], respectively: 9BG 3B þ G

ð12Þ

3B2G s¼ 2ð3B þ GÞ

ð13Þ

E¼

C 44 C 66

PBESOL

PBE

53.38 57.30 55.34 63.68 72.57 68.13 130.65 0.18 2.89 0.81 0.74

42.12 48.58 45.35 40.90 57.58 49.24 104.09 0.15 5.19 0.92 0.71

63.44 68.47 65.95 84.41 95.10 89.75 158.93 0.20 3.05 0.73 0.78

46.3 52.4 49.4 54.5 70.6 62.6 117.3 0.19 4.43 0.78 0.64

0.74 (0.71, 0.78) using the GGA–PBESOL (GGA–PBE, LDA); so this compound shows slight anisotropy. On the other hand, in order to evaluate the elastic anisotropy of KAl3(SO4)2(OH)6, the parameter kc/ka is employed, that expresses the ratio between linear compressibility coefficients [48]. The ratio between linear compressibility coefficients kc/ka for an hexagonal crystals can be expressed as kc/ka ¼(C11 þC12  2C13)/(C33  C13). The information obtained kc/ka 41 demonstrates that the compressibility for KAl3(SO4)2(OH)6 along c axis is larger than along a axis and kc/ ka o1 that the compressibility for KAl3(SO4)2(OH)6 along c axis is smaller than along a axis. The kc/ka of KAl3(SO4)2(OH)6 is 2.89 for GGA–PBESOL and 3.05 for LDA, respectively, which means that the compressibility along the c axis is much greater than that along the a axis. In addition, we have estimated the malleability of materials used Pugh’s criterion [49]. If G/B o0.5 the material will behave in a ductile manner or else the material demonstrates brittleness. The obtained G/B ratios for KAl3(SO4)2(OH)6 is equal to 0.81 (0.92, 0.73) using the GGA–PBESOL (GGA–PBE, LDA). According to this value, this compound behaves as a brittle material. Poisson’s ratio s, is a very important property for industrial applications. It is small (s ¼0.1) for covalent materials and for ionic materials s ¼0.25. The obtained values of Poisson’s ratio (s) are listed in Table 3. The calculated value of s equal to 0.18 is very close to the experimental value.

3.3. Debye temperature (yD) of KAl3(SO4)2(OH)6

The values of Bulk and shear moduli B and G, Young’s modulus E, Poisson’s ratio s, anisotropic factor A, linear compressibility ratio kc/ka and ratio G/B are given in Table 3. Our calculated B, G, E, and s values for KAl3(SO4)2(OH)6 are in agreement with the values reported by Majzlan et al. [18]. Essentially all known crystals are elastically anisotropic. In order to quantify the elastic anisotropy of KAl3(SO4)2(OH)6, the shear anisotropic factor A was calculated as the following [39]: A¼

kc/ka G/B A

LDA

ð5Þ

where the Sij are the elastic compliance constants. For KAl3 (SO4)2(OH)6, we can obtain the relations between Cij and Sij [44]: S11 ¼

Experiment

The Debye temperature (yD) of KAl3(SO4)2(OH)6 can be estimated from the averaged sound velocity, vm, by the following equation [42,46,50]:

yD ¼

ð15Þ

where h is Planck’s constant; k is Boltzmann’s constant; NA is Avogadro’s number; r is the density; M is the molecular weight and n is the number of atoms in a formula unit. "

ð14Þ

For an isotropic crystal, A is equal to1, while any value smaller or lager than 1 indicates anisotropy [46]. The magnitude of the deviation from 1 is a measure of the degree of elastic anisotropy of the crystal [47]. The calculated anisotropic factor of KAl3(SO4)2(OH)6 is equal to

   h 3n NA r 1=3 vm k 4p M

vm ¼

1 2 1 þ 3 3 v3t vl

!#ð1=3Þ ð16Þ

vl and vt are the longitudinal sound velocity and transverse sound velocity, respectively, which can be obtained by the shear

J. Long et al. / Physica B 407 (2012) 2606–2609

Table 4 Longitudinal, transverse, average sound velocity (vl, vt, vm in m/s), and the Debye temperatures (yD in K) for KAl3(SO4)2(OH)6 within LDA, GGA–PBE and GGA– PBESOL levels, respectively. Calculate

Experiment

GGA

vt vl vm

yD (K)

LDA

PBESOL

PBE

4470.78 7159.34 4925.47 697.5

4147.36 6450.67 4554.15 634.44

4695.69 7707.49 5186.68 753.61

modulus (G) and the bulk modulus (B) [51–53]:   B þ ð4=3ÞG 1=2 vl ¼

r

vt ¼

 1=2 G

r

4178.02 6737.55 4606.40 652.48

ð17Þ

ð18Þ

The longitudinal, transverse, average sound velocity and Debye temperature of KAl3(SO4)2(OH)6 are given in Table 4. Our calculation shows that this compound has a relative high Debye temperature value; it is found to be equal to 697.5 K (634.44 K, 753.61 K) using the GGA–PBESOL (GGA–PBE, LDA), we are able to assume that this compound has a good thermal conductivity.

4. Conclusions In this paper, the structural, elastic properties, anisotropy factor, Poison’s ratio, Young’s modulus, sound velocities and Debye temperature of KAl3(SO4)2(OH)6 have been studied within the generalized gradient approximation (GGA) and Local Density Approximation (LDA) method based on DFT. We obtain the structural parameters, elastic properties and bulk modulus at zero pressure which are satisfactory with the available experimental results. From these results, this compound behaves as a brittle material and has a good thermal conductivity.

Acknowledgment The authors are thankful for the partially financial support by the Mineral Resources Chemistry Key Laboratory of Sichuan Higher Education Institutions. References [1] J.L. Jambors, Can. Mineral. 37 (1999) 1323. [2] D.K. Smith, A.C. Roberts, P. Bayliss, F. Liebau, Am. Mineral. 83 (1998) 126.

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