Theoretical Investigation on the Structural, Elastic and Mechanical Properties

Page 1

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

Theoretical Investigation on the Structural, Elastic and Mechanical Properties of Rh3HxNb1-x(x=0.125, 0.875) 1

M. Manjula1, M. Sundareswari1 1 – Department of Physics, Sathyabama University, Chennai, India DOI 10.2412/mmse.86.89.465 provided by Seo4U.link

Keywords: first-principles theory, density functional theory, electronic properties, mechanical properties, ductility.

ABSTRACT. Electronic, elastic and mechanical properties of Rh3HxNb1-x(x=0.125, 0.87) are investigated from density functional theory using FP-LAPW method within generalized gradient approximations. The lattice parameters and ground state properties are calculated by using optimization method. Shear modulus, Young’s modulus, Poisson’s ratio, G/B ratio and anisotropy factor are calculated using elastic constants C11, C12 and C44. The calculated results are consistent with available theoretical and experimental data. Systematic addition of Hf with Rh3Nb shows that the Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 are ductile. Charge density plots assess the results.

Introduction. L12 intermetallic compounds such as rhodium and iridium based compounds are of great interest in industrial applications [1], [2]. The mechanical and thermal findings on rhodium base alloys are more convenient for high-temperature structural applications than iridium base alloys. Rhodium is most frequently used as an alloying agent in other materials such as platinum and palladium. These alloys are used to make electrodes for aircraft spark plugs, detectors in nuclear reactors, laboratory crucibles and furnace coils. It has higher thermal conductivity, high temperature strength, good oxidation resistances and lower thermal expansion coefficient which are beneficial properties for high temperature applications [3], [4], [5], [6], [7], [8], [9]. The L12 crystal structure offers the possibility of enhanced ductility and workability of these materials. This motivates us to focus our research on rhodium base alloys. Especially, we focus our attention to design new materials with enhanced ductility from existing one. Alloying is one of the effective ways to attain our aim. To our best knowledge no systematic study on Rh3HfxNb1-x ternary alloy system. We have already reported Rh3HfxNb1-x (x= 0.25.0.75) combinations in our previous work [10]. In the present study, a first-principles calculation based on the density-functional theory was carried to investigate the electronic structure and mechanical properties of Rh3HfxNb1-x (x= 0.125.0.875) combinations. The ductile/brittle nature of these compounds is analysed. A number of theoretical and experimental structural have been performed for structural, electronic, elastic and mechanical properties of Rh3Nb. Yamabe et al. investigated the microstructure evolution and high temperature strength of Rh-based alloys [11]. Rajagopalan and Sundareswari reported structural and electronic properties of this compound [12]. Chen et al. [13] investigate elastic and mechanical properties of this compound. The mechanical properties of Rh3Nb are studied by Miura et al. [14]. Some of the thermal properties were measured by Terada et al. [15]. Their strength behaviour was discussed by Yamabe-Mitarai et al. [16] Computational Methods. Our calculations are carried out by means of Full Potential Linearized Augmented Plane wave (FP-LAPW) method implemented in the WIEN2k code [17]. The basis set is obtained by dividing the unit cell into non-overlapping spheres surrounding each atom and creating an interstitial region between the spheres. The exchange and correlation was treated within the generalized gradient approximation by Perdew et al [18]. 10×10×10 k-point mesh is used in the 1

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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

irreducible Brillion Zone. The plane wave expansion is taken as RMT × Kmax = 7.0 and lmax=10. Charge density Fourier expansion are extended up to Gmax=12. The total energies are converged below 0.0001eV and the charges are converged below 0.001mRy. Result and Discussion. The Rh3Nb alloy has a Cu3Au-type structure with space group 221-Pm3m. The Rh and Nb atoms are located at the site (0, 0.5, 0.5) and (0,0,0) respectively. The optimized lattice parameters for Rh3Nb, Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 are presented in Table 1 and calculated lattice constant for Rh3Nb agree very well with experimental and theoretical data [12], [13]. For Rh3Nb, the percentage error between the calculated and the experimental lattice constant is 1.07. The calculated elastic constants (C11, C12 and C44), Shear modulus (G), Young’s modulus (E), Cauchy pressure (C12-C44), G/B ratio, Poisson’s ratio (ν), and anisotropy factor (A) for Rh3HfxNb1-x (x =0, 0.125, 0.875) combinations are reported in Table1 and these values are used to predict ductile/brittle nature of the compounds. From Table 1, one can note that the computed B, G, E and C44 values for Rh3Nb are quantitatively higher than the other two combinations. For cubic system there are three independent elastic constants namely C11, C12 and C44. The mechanical stability conditions for cubic crystal are: C11-C12 >0, C11 >0, C44 >0, C11+2C12 >0. The calculated elastic constants (Table 1) obey the mechanical stability criteria, suggesting that these compounds are mechanically stable. Table 1. The optimized lattice parameter, elastic constants and elastic properties of Rh3HfxNb1-x (x=0, 0.125, 0.875). Parameters

Rh3Nb

Rh3Hf0.125Nb0.875

Rh3Hf0.875Nb0.125

Lattice Constant (a.u.) Present study

aexp =7.2887

aoal= 7.3677

acal= 7.4405

acal =7.3668 Other study a

aexp=7. 289 acal =7.3625

C11

475.48

374.72

322.10

C12

169.58

186.33

174.16

C44

456.58

211.16

102.88

Cauchy Pressure(C12-C44)

-287.00

-24.83

71.28

Bulk Modulus(B), GPa

271.55

249.13

223.47

Shear Modulus(G), GPa

294.81

152.73

90.14

Young’s Modulus(E), GPa

649.42

380.44

238.38

G/B

1.08

0.61

0.40

Poisson’s Ratio(ν)

0.10

0.25

0.32

B/C44

0.594

1.179

2.172

HV

77.87

25.91

10.68

A

2.27

1.62

1.179

Bulk modulus is a measure of average atomic bond strength of materials [19]. It is strongly correlated with cohesive energy or binding energy of atoms in crystals. The large value of shear modulus indicates that the more pronounced directional bonding between atoms [20], [21]. Young’s modulus (E) indicates the stiffness of the material. Higher its value the material will be stiffer. From Table 1, the values of B, G and E reveals that the addition of Hf in Rh3Nb can decrease the atomic bond strength, directional bonding and stiffness of the material. Ductile/brittle nature of the alloy is investigated to analyse the effect on brittleness of Rh3Nb on addition of hafnium. The ductility of the compounds investigated based on Cauchy pressure (C12MMSE Journal. Open Access www.mmse.xyz

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

C44), G/B ratio and Poisson’s ratio (ν). According to Pettifor [22], if C12-C44 is positive, the material exhibits metallic characteristics and it is negative for non-metallic with directional bonding. Rh3Nb and Rh3Hf0.125Nb0.875 are brittle having negative Cauchy pressure (-287GPa & -24.83) and Rh3Hf0.875Nb0.125 is ductile having positive Cauchy pressure (71.28GPa). According to Pugh criterion [23], if G/B < 0.57 the material exhibits ductile behaviour, otherwise, it exhibits brittle behaviour. This ratio for Rh3Hf0.875Nb0.125 (0.39) is less than 0.57 reveals that ductile nature. Poisson’s ratio (ν) [24] explains about the characteristics of the bonding forces. If ν >0.26, the material is ductile; otherwise brittle. The ν value for Rh3Hf0.875Nb0.125 is 0.33 indicates that the ionic contributions to the atomic bonding are dominant for these compounds. Elastic anisotropy factor (A) is an indicator of the degree of anisotropy in the solid structures [25]. For a complete isotropic material A=1, when the value of A is smaller or greater than unity it is a measure of the degree of elastic anisotropy. From Table 1, it can be seen that the Rh3Nb is and anisotropy material due to A>1. With the addition of Hf to Rh3Nb, the degree of anisotropy decreases. The calculated microhardness HV [21] for Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 is 25.91 GPa and 10.68 GPa respectively (Table 1). From the calculations, it is found that the hardness decreases when hafnium is added to the parent Rh3Nb alloy. Along with bulk and shear modulus, the elastic constant C44 is also an important parameter indirectly governing the indentation hardness [26]. Hence, Rh3Hf0.875Nb0.125 identified as less hard material having low hardness and C44 values than Rh3Hf0.125Nb0.875. The results are assessed by plotting charge density plots. Fig. 1 (a-c) shows that the charge density plots of Rh3Nb, Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 alloy combinations respectively. In general, the brittle materials have strong directional characteristic of bonding. From Fig. 1a, one can observe that the charge density contours encloses Nb-Rh-Nb atoms and this can be attributed to the directional covalent nature (brittle). Such directionality is decreased in Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 when Nb is replaced by Hf (Hf-Rh-Nb) shown in Fig. 1b & 1c. In Fig.1c, the electron density contours enclosing Hf & Rh atoms and Hf & Nb atoms are not observed. This makes the directional bonding very weak, it may be attributed to the ductile nature of Rh3Hf0.875Nb0.125. Thus, addition of Hf to Rh3Nb reduces the directional bonding nature present in Rh3Nb, resulting in a transition from brittle to ductile nature in Rh3Hf0.875Nb0.125.

(a)

(b)

(c)

Fig. 1. Charge density plot of (a) Rh3Nb (b) Rh3Hf0.125Nb0.875 and (c) Rh3Hf0.875Nb0.125. Fig. 2 (a-c) represents the DOS curves of Rh3Nb, Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 alloy combinations respectively. From DOS histograms, it is observed that the peaks in the total density of states that lie below the Fermi level are mainly due to the Rh-d states and above the Fermi level are Nb-d and Hf-d states. In Fig. 2a, one can notice a pseudo gap in Rh3Nb and in Fig. 2 (b-c), there is MMSE Journal. Open Access www.mmse.xyz

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

no noticeable pseudo gap in Rh3Hf0.125Nb0.875 and Rh3Hf0.875Nb0.125 combinations. The pseudo gap can directly reflect the strength of covalent bonding.

(a)

(b)

(c)

Fig. 2. DOS histograms of (a) Rh3Nb (b) Rh3Hf0.125Nb0.875and (c) Rh3Hf0.875Nb0.125. The Debye temperature is one of the important parameter closely related to many physical properties. It is a measure of thermal conductivity of materials and it can be related to the strength of covalent bonds [27]. Using the elastic constants, the Debye temperature (đ?œƒD), sound velocities for longitudinal and shear waves (VL and VS) and Debye average velocity (Vm)[28] are calculated and presented in Table 2. From Table 2, it can be found that the Debye temperature value for Rh3Hf0.125Nb0.875 (618 K) and Rh3Hf0.875Nb0.125 (448 K) alloy combinations are decreased. Hence, the strength of covalent bonds decreases in these materials. Table 2. Calculated mass density Ď (gm/cm3), VL and VS (103m/s), Debye average velocity Vm (103m/s), and Debye temperature đ?œƒD (K) for Rh3HfxNb1-x(x=0,0.125,0.875)alloys. Parameters

Rh3Nb

Rh3Hf0.125Nb0.875

Rh3Hf0.875Nb0.125

Ď

45.02

46.21

51.84

VL

3.8422

3.1302

2.5747

VS

2.5589

1.8180

1.3186

Vm

2.7970

2.0174

1.4770

đ?œƒD

857

618

448

Summary. In this work, the structural, elastic and electronic properties of Rh3HfxNb1-x (x =0, 0.125, 0.875) combinations are investigated by means first principles calculations based on DFT with GGA method. The calculated lattice parameters and bulk modulus are consistent with the literature values. Young’s modulus, shear modulus, G/B ratio, Poisson’s ratio and anisotropy factor have been calculated and discussed. Also in Rh3Nb, a transition from brittle nature to ductile nature is observed when Hf is added. Rh3Hf0.875Nb0.125 shows ductile nature having highest Cauchy pressure and Poisson’ ratio and lowest shear modulus, Young’s modulus and G/B ratio. Charge density plots reveal decrease in directional bonding nature in Rh3Nb when Hf is added. The sound velocities and Debye temperatures of the alloys have been calculated. References [1] Y. Yamabe-Mitarai, Y. Koizumi, H. Murakami, Y. Ro, T. Maruko, H. Harada, Scr. Metall. Mater. Vol. 35 (1996) 211. [2] Y. Yamabe-Mitarai, Y. Ro, T. Maruko, H. Harada, Metall. Mater.Trans. A, Vol. 29 (1998) 537. MMSE Journal. Open Access www.mmse.xyz

Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

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Mechanics, Materials Science & Engineering, July 2017 – ISSN 2412-5954

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