Chemical bonding characterization, expansivity and compressibility of RECrO4

Journal of Alloys and Compounds 582 (2014) 151–156

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Chemical bonding characterization, expansivity and compressibility of RECrO4 Huaiyong Li a,⇑, Hyeon Mi Noh b, Byung Kee Moon b, Byung Chun Choi b, Jung Hyun Jeong b,⇑ a b

Department of Materials Science and Engineering, Liaocheng University, Liaocheng 252059, PR China Department of Physics, Pukyong National University, Busan 608-737, Republic of Korea

a r t i c l e

i n f o

Article history: Received 9 June 2013 Received in revised form 1 August 2013 Accepted 1 August 2013 Available online 13 August 2013 Keywords: Rare earth compounds Crystal binding Thermal expansion Bulk modulus

a b s t r a c t Theoretical researches were performed on zircon-type RECrO4 (RE = rare earth elements) compounds by using dielectric chemical bond theory of complex crystals. The characterization of the chemical bonding, the expansivity and compressibility of the compounds were studied. The results revealed that both RE–O and Cr–O bonds were ionically dominated, and the ionicity fraction decreased gradually with the decreasing of the RE–O bond length. Cr–O bonds had a low linear thermal expansion coefficients (LTEC) and high bulk modulus than RE–O bonds. While the LTEC and bulk modulus of the compounds were mainly determined by RE–O bonds because they had a large bond volume. When RE varied from Pr to Lu, the LTEC decreased linearly from 6.00 to 5.71 106/K and the bulk modulus increased from 117.9 to 132.2 GPa. YCrO4 in zircon-phase had high lattice energy than YCrO4 in scheelite-phase, the bulk moduli of YCrO4 in zircon- and scheelite-phase were determined to be 135 GPa and 153 GPa, respectively, which agreed well with the experimental values. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction ABO4. compounds crystallizing in the zircon structure are important materials in both theoretical and technological aspects. Many of them have been explored intensively due to their interesting crystal structural, optical, magnetic and mechanical properties. While as one branch of the zircon family, rare earth chromates with the general of RECrO4 with RE = rare earths have been seldom studied until recently. The RECrO4 compounds with RE standing for Pr, Nd, Sm–Lu, Y have been synthesized by Schwarz and at the same time their cell parameters have been reported [1–4]. Buisson et al. has refined the coordinations of the atoms in the crystals except TmCrO4 [5]. After that the structural characteristic and magnetic property of these compounds have attracted many interests [6–17]. Most recently, the structural phase transition of some RECrO4 compounds from the zircon structure to scheelite structure has been observed and been investigated both experimentally and theoretically [17–23]. For instance, the high pressure Raman spectra indicates that the onset pressures are 1.3, 4.1, and 3.0 GPa for NdCrO4 [19], DyCrO4 [19] and YCrO4 [21], respectively. A theoretical calculation gives out a similar phase transition onset pressure of 2.9 GPa for YCrO4 [22]. The behavior of DrCrO4 and YCrO4 in both zircon- and scheelite-phase under pressure has also been investi-

⇑ Corresponding authors. Tel.: +82 15966290630; fax: +82 51 629 5549. E-mail addresses: [email protected] (H. Li), [email protected] (J.H. Jeong). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.08.006

gated [24,19]. The bulk moduli of YCrO4 in zircon- and scheelitephase are determined to be 136 GPa and 151 GPa, respectively [24]. On the other side, the magnetic behavior of RECrO4 is also attractive due to the unusually high chemical valence of chromium (Cr+5). It is well known that the intrinsic physical properties of inorganic crystals are determined by the constituent elements, the spatial distribution of the atoms and the interactions among them. The last one is referred to as chemical bond, which plays a significant role in the mechanical properties such as hardness, thermal expansion and compressibility. Therefore getting knowledge of the chemical bond property will be helpful to deep our understanding on the physical properties of a crystal under high pressure and temperature. And further more, chromium (Cr) in the RECrO4 series takes the unusual high valence state, namely +5, to our knowledge, the chemical bond property of Cr5+–O has not been explored. In this paper, we report a theoretical investigation on the zircon-type rare earth chromates by using dielectric chemical bond theory of complex crystal. A brief description of the theoretical method is given out in the following part first. By using this method the covalency fraction of RE3+–O and Cr5+–O bonds are calculated. On this basis the bonding characterization of RECrO4 compounds, such as lattice energy, LTEC and the bulk modulus of the chemical bonds are calculated. The expansivity and compressibility of the chemical bonds as well as the compounds are discussed from the viewpoint of bonding. The properties of YCrO4 in zircon- and scheelite-structure are compared, and the structural stability and compressibility are discussed.

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 2 2 l 2 Elg ¼ ðC l Þ þ ðEh Þ

2. Theoretical method 2.1. Chemical bond parameters

ð4Þ

l 2:48

l

Eh ¼ 39:74ðd Þ For simple binary crystals, such as NaCl-, ZnS-, and CaF2-type, containing only one species of chemical bond, the properties of the chemical bond in a crystal can be easily obtained from the theory suggested by Phillips, [25] and the related physical properties can be predicted While for complex crystals containing multiple types of chemical bonds, such as RECrO4 compounds, the situation is different. RECrO4 compounds with RE = Pr, Nd, Sm–Lu, Y crystallize in the zircon phase, which belongs to a tetragonal structure with space group I41/amd and contains four formula units for a total of 24 atoms. In this structure, the rare earth elements, Cr and O atoms occupy 4a (0, 3/4, 1/8), 4b (0, 1/4, 3/8) and 16h (0, y, z) sites respectively according to the Wyckoff notation. The rare earth atoms are coordinated by eight oxygen atoms having two kinds of bond lengths, the Cr atoms coordinated by four oxygen atoms, and the O atoms are threefold coordinated, forming a plane. The tetragonal structure is constructed by a framework ofREO8 dodecahedra with the CrO4 tetrahedra embed in it. Since there are two polyhedron and three different species of chemical bonds, it is a challenge to getting knowledge of the properties of the chemical bonds. The dielectric chemical bond theory of complex crystals provides an efficient method to study the properties of the chemical bonds in complex crystals, and to predict the related physical properties of a crystal from bonding standpoint. Given the chemical formula of a complex crystal is AmBn (A stands for a cation and B for an anion), and A has i types of crystallographic sites, and the number of A on site i is ai. In order to distinguish different crystallographic sites, we write the chemical formula into a crystal formula as A1a1 A2a2 . . . Aiai B1b1 B2b2 . . . Bjbj , Aiai or Bjbj represents the same element in different sites, and the subscripts ai and bj represent the number of the corresponding element. By using this method chemical bonds having different bond lengths and symmetries can be distinguished. According to the chemical bond theory of complex crystals, when the structural parameters of a complex crystal are known, the crystal formula of a complex crystal can be written as a linear combination of a series of sub-formulas, which correspond to a series of fictitious binary crystals containing every individual chemical bond. The decomposition of the crystal formula into sub-formulas can be achieved by the following equations:

A1a1 A2a2

   Aiai B1b1 B2b2

   Bjbj

X i j X ¼ Ami Bnj ¼ ðAm Bn Þl

mi ¼

i

NC ðB  A Þ  ai NC ðAi Þ

i

;

nj ¼

NC ðA  B Þ  bj NC ðBj Þ

ð2Þ

!2

l

l

fc ¼

Cl Elg ! l 2 Eh Elg

ð6Þ mPn

l

where d ¼ 2r 0 is the bond length of a chemical bond, in Å, l expðks r l0 Þ is the Thomas–Fermi screening factor. Z lA and Z lB are the effective valence electron numbers of A and B ions, respectively, bl is a structural correction factor. 2.2. Lattice energy, LTEC and bulk modulus Lattice energy is one of the most important quantities in elucidating the structure, character, and behavior of ionic crystals. Therefore a great amount of effort has been devoted to the calculation of lattice energy. Liu et al. [33] proposed a method to estimate the lattice energy of ionic crystals based on the chemical bond theory. For a fictitious binary crystal ðAm Bn Þl , its lattice energy U(mn)l l contains two parts, namely ionic part UðmnÞi and covalent part l UðmnÞc , and the total lattice energy of a complex crystals can be obtained from summarizing the contributions of the decomposed binary crystals by using the following equations: l

UmnÞl ¼ UðmnÞi þ UðmnÞlc U¼

X UðmnÞl

ð7Þ ð8Þ

l

The lattice energy density ul of a binary crystal is defined as

uðmnÞl ¼

UðmnÞl l NAV nl v b

ð9Þ

where NAV is the Avogadro constant, nl is the number of the cheml ical bond in one formula unit, v b is the chemical bond volume of the chemical bond. On this basis, it has been revealed that the LTECs and bulk moduli of binary crystals have a significantly linear relationship with the lattice energies or lattice energy densities. The LTEC of a complex crystal can be estimated from the following expressions,

a¼

X l F mn almn

ð106 =KÞ

ð10Þ

l

almn ¼ 3:1685 þ 0:8376vlmn ð106 =KÞ

j

N C ðBj  Ai Þ is the number of Bj ions in the coordination sphere of an Ai ion, and N C ðAi Þ is the nearest coordination number of the Ai ion. ðAm Bn Þl is the sub-formula. Since the fictitious binary crystal contains only one species of chemical bond, its properties can be easily obtained. For the chemical bond in a fictitious binary crystal, the fractions of ionicity fil and of covalence fcl can be evaluated from the average band gap Elg as well as its heteropolar part Cl and homopolar part Elh [26–32]:

fi ¼

l

nPm

l

i;j j

ð1Þ

 l l h l n l i 1 l C l ¼ 14:4b exp ks r 0 Z A  Z B l m r0 i  l l hm l l l 1 l Z  ZB l C ¼ 14:4b exp ks r 0 n A r0

ð5Þ

ð3Þ

ð11Þ

l

the parameter vmn links the lattice energy and the LTEC of a binary crystal together, which can be obtained from the crystal structural data, F lmn is the fraction of one type of chemical bond in the complex crystal. The detailed theoretical method can be found in Ref. [33]. The bulk moduli B(mn)l of a certain type of binary crystals, on the other side, have a linear relationship with the lattice energy densities u(mn)l [34–36],

BðmnÞl ¼ dl þ

uðmnÞl bðmnÞl

ð12Þ

here dl is a constant, the proportion factor b(mn)l depends on the chemical valence ZA and the coordination number NC(A) of the cation. The bulk modulus of a complex crystal can be calculated from

Bm ¼

1

jm

ð13Þ

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H. Li et al. / Journal of Alloys and Compounds 582 (2014) 151–156

Fig. 1. The crystal structure of YCrO4 in: (A) zircon and (B) scheelite phases. The biggest blue spheres are Y, the medium green ones are Cr and the smallest red ones are O. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

jm ¼

1 X l 1 X Vl V jl ¼ Vm l V m l BðmnÞl

3. Results and discussion

ð14Þ

3.1. Bonding characteristic, expansivity and compressibility of RECrO4 (RE = rare earths) in zircon structure

where Vl is the volume of a species of chemical bond in one formula unit. The applicability of this method on complex ionic and covalent crystals has been confirmed in Refs. [37,38]. And therefore we will use this method to estimate the properties of the chemical bonds in RECrO4 and their expansivity and compressibility.

RECrO4 compounds with RE = Pr, Nd, Sm–Lu, Y crystallize in the zircon phase at ambient conditions. The structure of YCrO4 is shown in Fig. 1(A) as an illustration. By meanings of Eqs. (1) and (2), the crystal formula of RECrO4 compounds can be decomposed into three sub-formulas:

Table 1 Chemical bonding characterization, lattice energy, LTEC and bulk modulus of RECrO4. The bond length values were obtained from Ref. [5]. Bond length dl in Å, lattice energy Ul and U in kJ/mol, LTEC al and a in 106/K, bulk modulus Blm and Bm in GPa, and the volume of one molecule Vm in Å3. Crystal

Bond

d l;

fc

Ul

al

Blm

Vm

U

a

Bm

PrCrO4

Pr–O1 Pr–O2 Cr–O

2.394 2.518 1.718

0.1555 0.1523 0.3558

2823 2710 22958

8.53 9.02 0.49

115.98 95.88 482.71

86.67

28491

6.01

117.7

NdCrO4

Nd–O1 Nd–O2 Cr–O

2.383 2.508 1.711

0.1559 0.1526 0.3562

2833 2719 23022

8.48 8.98 0.48

118.05 97.36 490.08

85.55

28574

5.98

119.6

SmCrO4

Sm–O1 Sm–O2 Cr–O

2.364 2.489 1.698

0.1565 0.1531 0.3571

2852 2736 23143

8.41 8.90 0.46

121.72 100.24 504.03

83.54

28730

5.93

123.2

EuCrO4

Eu–O1 Eu–O2 Cr–O

2.355 2.479 1.691

0.1567 0.1534 0.3575

2860 2745 23208

8.38 8.86 0.45

123.48 101.76 511.63

82.56

28814

5.90

125.0

GdCrO4

Gd–O1 Gd–O2 Cr–O

2.349 2.473 1.687

0.1570 0.1535 0.3578

2866 2750 23246

8.35 8.84 0.45

124.70 102.72 516.17

81.93

28862

5.88

126.3

TbCrO4

Tb–O1 Tb–O2 Cr–O

2.338 2.464 1.680

0.1573 0.1538 0.3583

2877 2758 23312

8.31 8.80 0.44

126.97 104.19 524.21

80.89

28947

5.85

128.2

DyCrO4

Dy–O1 Dy–O2 Cr–O

2.329 2.455 1.674

0.1576 0.1540 0.3586

2886 2766 23369

8.27 8.77 0.43

128.77 105.59 530.84

80.03

29021

5.82

130.0

HoCrO4

Ho–O1 Ho–O2 Cr–O

2.322 2.448 1.668

0.1578 0.1542 0.3590

2893 2773 23426

8.25 8.74 0.42

130.30 106.77 538.01

79.27

29092

5.80

131.5

ErCrO4

Er–O1 Er–O2 Cr–O

2.313 2.438 1.662

0.1581 0.1544 0.3594

2902 2782 23484

8.21 8.70 0.41

132.19 108.41 544.98

78.36

29167

5.77

133.5

YbCrO4

Yb–O1 Yb–O2 Cr–O

2.302 2.431 1.655

0.1585 0.1547 0.3600

2913 2788 23550

8.17 8.67 0.40

134.68 109.67 553.76

77.41

29252

5.75

135.4

LuCrO4

Lu–O1 Lu–O2 Cr–O

2.292 2.423 1.649

0.1589 0.1549 0.3604

2923 2796 23608

8.13 8.64 0.39

136.92 111.05 561.16

76.54

29327

5.72

137.3

l

154

RECrO4 ¼ RE1=2 O14=3 þ RE1=2 O24=3 þ Cr1 O4=3

H. Li et al. / Journal of Alloys and Compounds 582 (2014) 151–156

ð15Þ

On the basis of the sub-formula equation, the chemical bond parameters, lattice energies, LTECs and bulk moduli of the chemical bonds in RECrO4 are calculated by using the method shown in Section 2. From the results shown in Table 1, one can find that both RE–O bonds and Cr–O bonds are ionically dominated. The ionicity fractions of RE–O bonds are apparently higher than those of the Cr– O bonds. The ionicity fractions are ca. 83% for RE–O bonds, and ca. 63% for Cr–O bonds. On the other side, when the rare earth goes from Pr to Lu, the ionic radius decreases, as a result, the ionicity fraction decreases gradually with the decreasing of the RE–O bond length. At the same time, the ionicity fraction of the Cr–O bonds varies similarly as theRE–O bonds. This phenomenon can be easily understood from the polarizability of the cations. The chemical valence of Cr in RECrO4 compounds is +5 and the ionic radius is 0.345 Å [39], while the chemical valence of RE is +3 and the ionic radii (coordination number = VIII) are in a range of 0.977–1.126 Å [39]. Cr has a higher chemical valence and smaller ionic radius than RE, therefore Cr has a higher polarizing ability than RE. As a consequence, Cr–O bonds have a higher covalency fraction than RE–O bonds. The lattice energies of the binary crystals are plotted as a function of the ionic radii of rare earths in Fig. 2. It is observed that the lattice energies of RE–O bonds are far less than those of the Cr–O bonds. This is due to the fact that Cr has a higher valence charge than RE and Cr–O bonds have a shorter bond length than RE–O bonds, the interaction potential between Cr and O are larger than RE and O. It can be also observed that with the decreasing of the bond length, the lattice energies of both RE–O and Cr–O bonds decrease, as a result, the total lattice energy of RECrO4 decreases. From the viewpoint of energy, some quantitative conclusion might be drawn on the deformation behavior of RE–O and Cr–O bonds. Since the lattice energies of RE–O bonds are less than those of the Cr–O bonds, it is easy to understand that RE–O bonds are softer than Cr–O bonds, therefore RE–O bonds are easier to deform than Cr–O bonds when the crystal are heated or under pressure. This conclusion is confirmed by the calculated LTEC and bulk modulus of the bonds. The calculated LTECs and bulk moduli of the compounds are also listed in Table 1. As expected that the expansivity of Cr–O bonds is in a range of 0.39–0.49 106/K, which is much smaller than that of RE–O bonds (the expansivity is in a range of 8.13– 10.11 106/K). On the other side, the Cr–O bonds have a high bulk modulus of ca. 500 GPa, while the RE–O bonds are much less

Fig. 2. Lattice energies of the RECrO4 (RE = rare earths) compounds and of the decomposed binary sub-formula as a function of the ionic radii of RE.

harder than Cr–O bonds, and only have a bulk modulus of ca. 100 GPa. A similar investigation has also performed on the BaY2O4 and SrY2O4 compounds [40]. We notice that the bulk modulus of the Y–O bonds in those compounds have a higher value of ca. 200 GPa[40]. It is obvious that there is great inconsistency. We would like to ascribe this result to the different coordination number of Y. In BaY2O4 and SrY2O4 compounds, Y has a coordination number of 6, while in YCrO4 the coordination number of Y is 8. This naturally leads to the fact that the population of electrons on Y–O bonds in YCrO4 is less than that in BaY2O4 and SrY2O4. The LTECs and bulk moduli of the compounds are also estimated, and plotted as a function of the ionic radii of the rare earths [39], as well as the lattice energy density in Figs. 3 and 4, respectively. It is revealed that the LTECs and bulk moduli varies systematically with the ionic radii of the rare earths and with the lattice energy density. The LTECs decrease from 6.01  106/K of PrCrO4 to 5.72  106/K of LuCrO4, and the bulk moduli increase from 117.7 to 137.3 GPa. The decrease of the rare earth ionic radii is correlated with the reduction of the RE–O and Cr–O bonds (as indicated in Table 1), leading to a densification of the zircon structure. 3.2. Comparison of zircon- and scheelite-type YCrO4 At ambient conditions, RECrO4 compounds (RE = Pr, Nd, Sm– Lu, Y) are crystallized in the zircon structure, while it is found

Fig. 3. The linear thermal expansion coefficient and bulk modulus of RECrO4 (RE = rare earths) as a function of the ionic radii of RE.

Fig. 4. Linear thermal expansion coefficient and bulk modulus of RECrO4 (RE = rare earths) as a function of lattice energy density.

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H. Li et al. / Journal of Alloys and Compounds 582 (2014) 151–156

Table 2 Chemical bonding characterization, lattice energy, LTEC and bulk modulus of YCrO4 in zircon (–Z) and the scheelite (–S) phases. Bond length dl in Å, lattice energy Ul and U in kJ/ mol, LTEC al and a in 106/K, bulk modulus Blm and Bm in GPa, and the volume of one molecule Vm in Å3. Crystal a

YCrO4–Z a = 7.1073 c = 6.2486 YCrO4–Sa a = 5.0030 c = 11.2636 YCrO4–Zb a = 7.2099 c = 6.3152 YCrO4–Sb a = 5.0753 c = 11.3307 a b c

Bond

dl

fc

Ul

al

Blm

Vm

U

a

Bm

Y–O1 Y–O2 Cr–O Y–O1 Y–O2 Cr–O Y–O1 Y–O2 Cr–O Y–O1 Y–O2 Cr–O

2.290 2.434 1.700 2.318 2.414 1.711 2.330 2.453 1.719 2.357 2.423 1.734

0.1670 0.1625 0.3708 0.1798 0.1769 0.3940 0.1653 0.1616 0.3690 0.1782 0.1763 0.3921

2930 2790 23051 2910 2817 22825 2890 2772 22879 2871 2808 22618

9.40 10.03 0.48 9.49 9.91 0.51 9.58 10.12 0.50 9.66 9.95 0.55

136.0 108.1 495.3 146.6 125.8 541.1 127.02 104.64 474.20 137.22 123.65 513.71

78.91

28772

6.64

135.3 136c

70.48

28551

6.64

153.1 151c

82.07

28541

6.73

129.2 121.6b

72.97

28297

6.72

147.5 141.3b

l

Experimental structural parameters reported in Ref. [21]. First principles calculated values reported in Ref. [22]. Experimental values reported in Ref. [24].

recently that there exists a structural phase transition from zircon-type to scheelite-type of YCrO4, NdCrO4 and DyCrO4. As shown in Fig. 1(B), the coordination environment of Y, Cr and O in scheelite phase is similar to that in zircon phase. REO8 units are connected with each other along the a-axis direction to form a framework. The CrO4 tetrahedra populate alternately at the caxis direction. Therefore the scheelite phase shares the same sub-formula with the zircon phase (see Eq. (15)). On this basis, we perform a calculation on YCrO4 in both zircon- and scheelite-structure. The lattice parameters determined from experimental refinement and first principle theoretical calculation are used as input data for comparison. The results shown in Table 2 indicate that the zircon phase has higher lattice energy, while a lower bulk modulus. These results are consistent with the experimental phenomenon as well as the results obtained from first principle theoretical calculation. As mentioned above, YCrO4 is crystallized in the zircon phase at ambient conditions, and Long et al. [21] have reported that there exists a phase transition from the zircon phase to scheelite under a pressure above 3.0 GPa. On the other side, Li et al. [22] have performed a first principle calculation on YCrO4. The calculation reveals that YCrO4 in zircon phase has lower total energy than in scheelite at ambient condition, and it also predicts a phase transition at 2.9 GPa. Our calculation reveals that the zircon phase has higher lattice energy than the scheelite phase, which means the zircon phase is more stable than the scheelite phase at ambient conditions. This is consistent with the results obtained by Long et al. and Li et al. [21–23] Although YCrO4 in zircon phase has high lattice energy, its bulk modulus is lower than YCrO4 in scheelite phase. These results are not contradiction according to our theory because the bulk modulus of a crystal depends on the lattice energy density instead of lattice energy. The lattice energy decreases when YCrO4 undergoes a structure transition from zircon to scheelite, while at the same time, the cell volume reduces greatly from 78.91 Å3 to 70.48 Å3. Due to this reason the lattice energy density increases, which results in a higher bulk modulus of the scheelite phase. The predicted bulk modulus increases from 135.3 GPa of zircon-type to 153.1 GPa of scheelite-type, which agree well with the experimental results of 136 GPa and 151 GPa, respectively. We also perform a calculation with the theoretical cell parameters, which are larger than the experimental ones. As a consequence, the predicted bulk moduli of the theoretical cells have smaller values than the experimental ones.

4. Conclusion Rare earth chromates RECrO4 (RE = Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Yb, Lu, Y) within zircon structure have been investigated systematically by using dielectric chemical bond theory of complex crystals. The results reveal that the chemical bonds in RECrO4 compounds are ionically dominated and the ionicity of the RE–O bonds is higher than that of the Cr–O bonds, and the ionicity fraction decreases gradually with the decreasing of the RE–O bond length. Cr– O bonds in the compounds have a low LTEC and high bulk modulus than RE–O bonds, which indicates that CrO4 tetrahedra are more rigid than REO8 polyhedra. The LTEC and bulk modulus of the compounds vary linearly with the ionic radii of the rare earth and with the lattice energy density. The calculated lattice energy of YCrO4 in zircon-phase is higher than that in scheelite-phase, which is consistent with the fact that the zircon phase is more stable at ambient conditions. The calculated bulk moduli agree well with the experimental values. Acknowledgements This work was supported by a Research Grant of Pukyong National University (2013-0282). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

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