First-principles calculations of the elastic properties of ZrC and ZrN

Journal of Alloys and Compounds 377 (2004) 221–224

First-principles calculations of the elastic properties of ZrC and ZrN Dayong Cheng∗ , Shaoqing Wang, Hengqiang Ye Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China Received 22 April 2002; received in revised form 5 October 2003; accepted 13 January 2004

Abstract The elastic properties of ZrC and ZrN were calculated employing all-electron full-potential linearized augmented plane-wave (FLAPW) method. The errors between the calculated and experimental lattice constants are within 2%. The elastic properties of polycrystalline ZrC and ZrN were derived. The calculated results indicated the Young’s modulus of ZrC is much higher than that of Zr-based single phase glassy alloys. By comparing the elastic properties of ZrC and ZrN, it is suggested that the mechanical properties of Zr-based glass alloys will be improved if ZrN as the disperse phase is introduced to the glassy alloys. © 2004 Elsevier B.V. All rights reserved. Keywords: Amorphous materials; Elasticity; ZrC; ZrN

1. Introduction Zr-based glassy alloys have high glass-forming ability and high thermal stability against crystallization. The recent improvement [1–3] of mechanical properties of glassy alloys allows us use them for fundamental investigations and practical use. One of the best glass formers is known to be the multicomponent Zr–Al–transition metals (TM) [4,5] alloy system. The critical diameter reaching 30 mm has been achieved for Zr55 Al10 Ni5 Cu30 alloy prepared by a suction casting method. With the aim of improving the mechanical properties, amorphous composite materials containing second phases (ceramic, ductile metals and so on) have been investigated up to date. In 1980, the first report [6] in this category has indicated an increase of yield stress and a reduction in the degree of localization in the deformation mode. Recently, Kato et al. [7] found that bulk glassy Zr–Al–Ni–Cu composite materials containing ZrC particles exhibit higher fracture strength and larger plastic strain as compared with the Zr–Al–Ni–Cu single phase. The Young’s modulus E increases with increasing volume fraction Vf of ZrC, i.e., 101 GPa at 0% Vf , 121 GPa at 15% Vf . The fracture strength and plastic strain also increase with an increase of Vf , e.g., 1836 MPa and ∼0% at 0% Vf , and 2060 MPa and 4.5% at 15% Vf . From these experimental results, it is seen that the mechanical properties of the glassy alloys are influenced dramatically by ZrC. So it is necessary to study the mechan-

ical properties of ZrC, which is helpful to explain why the ZrC particles can enhance the mechanical properties of the glassy alloys. This will also help us improve the mechanical properties of the glassy alloys further by experiments. In addition, it is known that TiC and TiN have the same crystal structure and similar physical properties, for example high melting points, ultrahardness. ZrC and ZrN also have the same crystal structure. It is known that Ti and Zr are in the same group in the Periodic Table. It is thought that ZrC and ZrN should also possess the similar physical properties, so it is possible to improve the mechanical properties of the glassy alloys if ZrN as second phase is introduced in the multicomponent Zr–Al–TM alloy system. It is why the mechanical properties of ZrN are also studied. It is difficult to measure the mechanical properties of ZrC and ZrN from experiments for the size of their particles is too small. So studying their mechanical properties by first-principles is a feasible method. 2. Computational details The elastic constants are defined by means of a Taylor expansion of the total energy of the system with respect to a small strain of the lattice. The Taylor expansion of the total energy at the equilibrium volume V0 can be written as V0
E(V, ε) = E(V0 , 0) − P(V0 )V + Cij εi εj + O[ε3i ]. 2 i,j

∗

Corresponding author.

0925-8388/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2004.01.058

(1)

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D. Cheng et al. / Journal of Alloys and Compounds 377 (2004) 221–224

where E is the total energy, Cij are the elastic constants, V0 is the volume of the undistorted lattice, P(V0 ) is the pressure of the undistorted lattice at volume V0 , V is the change in volume of the lattice due to the strain, and O[ε3i ] indicates that the neglected terms in the polynomial expansion are cubic and a higher power of the εi . There are 21 independent elastic constants Cij in Eq. (1). Symmetry reduces this number to three (C11 , C12 , and C44 ) for the cubic lattices. At any volume V, the bulk modulus B is related to the elastic constants by B = (C11 + 2C12 )/3. The single-crystal shear modulus for the {1 1 0} plane along the [1 1¯ 0] direction can be simply given by G{1 0 0} = C44 and G{1 1 0} = C = (C11 − C12 )/2, respectively. The relationship between E(V, ε) and ε is determined by first-principles electronic structure calculation. The elastic constants are then obtained from the curvature of the total energy versus strain curves. A set of three independent calculations has been performed for each alloy for the following strain conditions: (1) Uniform hydrostatic pressure to determine the equilibrium lattice constants a0 and the bulk modulus B. In this case, the strain elements are ε1 = ε2 = ε3 = ε and ε4 = ε5 = ε6 = 0. (2) Uni-axial strain to determine C11 , with strain elements ε3 = ε and ε1 = ε2 = ε4 = ε5 = ε6 = 0. (3) Pure shear strain to determine C44 , with strain elements ε6 = ε and ε1 = ε2 = ε3 = ε4 = ε5 = 0. The volume is allowed to change in the calculation. We calculate these total energies under each of the above-mentioned conditions by using the all-electron full-potential linearized augmented plane-wave (FLAPW) method [8–10]. The FLAPW method is based on the density-functional-theory (DFT) theory [11,12]. In the FLAPW method, no shape approximation is made to the potential and the charge density. The DFT equations, incorporated with the generalized gradient approximation (GGA) for the exchange-correlation functional [13], are solved self-consistently for each lattice geometry condition mentioned above. The total energy changes slightly when the crystal is under small strain. Therefore, very accurate total energy calculation is required. The representative points in the Brillouin Zone were chosen according to the special points scheme [14]. The number of k points was increased until the total energy was converged to less than ∼0.1 mRy.

3. Results and discussion At first, the equilibrium volumes of ZrC and ZrN were calculated, and then the elastic constants of ZrC and ZrN at their equilibrium volumes were calculated. Fig. 1 represented the calculated total energy E for different ε values for each of the three conditions for ZrC. The bulk modulus B, C11 and C44 can be routinely obtained from these calculated results. The calculated results are presented in Table 1.

Fig. 1. Calculated total energy as a function of volume (a), strain of tetragonal deformation (b), and strain of orthorhombic deformation (c) for ZrC.

The experimental lattice constants of ZrC and ZrN are 0.472 and 0.463 nm respectively. The errors of equilibrium lattice constants between the calculated and experimental values are within 2%. The experimental data of elastic properties

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Table 1 Calculated lattice constants, bulk moduli and elastic constants C11 , C12 , and C44 Compounds

Lattice constant (Å)

B (×1011 N m−2 )

C11 (×1011 N m−2 )

C12 (×1011 N m−2 )

C44 (×1011 N m−2 )

ZrC ZrN

4.689 4.564

2.32 2.46

2.72 3.04

1.34 1.14

4.29 5.11

Table 2 Calculated shear modulus G, Young’s modulus E and Poisson’s ratio ␯ Compounds

GV (×1011 N m−2 )

GR (×1011 N m−2 )

EV (×1011 N m−2 )

ER (×1011 N m−2 )

νV

νR

ZrC ZrN

2.23 2.61

2.04 2.51

5.06 5.79

4.73 5.61

0.14 0.11

0.16 0.11

The experimental values of TiC and TiN are collected. The shear moduli of TiC and TiN are 1.90 [19] and 1.96 [20]. The Young’s moduli of TiC and TiN are 4.42 [19] and 4.75 [20].

of ZrC and ZrN aren’t found. But from previous calculated results compared with experiments [15], it is evaluated that the errors of the calculated bulk moduli of ZrC and ZrN are within 30%, the errors of the calculated C11 and C44 will be larger than bulk modulus because the lower symmetry in C11 and C44 calculations lead to low precision in the calculations. Since the same convergence criterion was adopted in all the calculations, the results can qualitatively reflect the relative elastic properties of ZrC and ZrN. From Table 1, it is seen that the bulk modulus, the elastic constant C11 and C44 increase from ZrC to ZrN, that is, these constants increase with increasing valence electron number. The variation of these constants from ZrC to ZrN consists with the variation from TiC to TiN [16]. The average size of ZrC particles dispersed in the cast amorphous phase is 3–4 ␮m. For a ZrC particle, it is the single-crystal. But the collection of plenty of ZrC particles dispersed in an amorphous phase will perform the mechanical behaviors of the polycrystalline materials for the orientation of each ZrC particle is different. So the elastic moduli for polycrystalline single-phase materials are derived by averaging the anisotropic single-crystal elastic properties over all possible orientations of crystallites. Upper and lower bounds are formed for the shear modulus G of isotropic materials according to Voigt [17] and Reuss and Angew [18]. They are GV = 15 (C11 − C12 + 3C44 )

(2)

which is derived assuming uniform strain throughout the sample and GR =

5(C11 − C12 )C44 4C44 + 3(C11 − C12 )

(3)

which is derived assuming uniform stress. From the averaged shear modulus G and bulk modulus B, the averaged Young’s modulus E and the averaged Poisson’s ratio ν are derived from the relations E=

9GB 3B + G

ν=

3B − 2G 2(3B + G)

(4)

The calculated elastic moduli for polycrystals, based on the first-principles results for single crystals, are collected in Table 2. The Young’s modulus E of matrix phase, Zr–Al–TM alloys, is only 101 GPa, from Table 2, it is seen that the Young’s modulus of polycrystals of ZrC is much higher than the matrix’s. It was confirmed that the absence of any crystalline phase and the achievement of truly bonding state at the interface between amorphous and ZrC phases from transmission electron microscopic (TEM) observation [21]. It means that the improvement of mechanical properties of the matrix phase depends on the mechanical properties of ZrC particles, not on the chemical interaction between the matrix and disperse phase. The calculated results of ZrC and ZrN tell us that their mechanical properties are similar. So if the ZrN as the disperse phase is introduced to the matrix, the mechanical properties of the matrix should be also improved. From Tables 1 and 2, it is seen that the calculated elastic constants, shear modulus G and Young’s modulus E of ZrN are all higher than that of ZrC, which predicts that the Young’s modulus of bulk glassy materials will be higher with the same volume fraction ZrN particles than with ZrC particles.

4. Conclusions The elastic properties of ZrC and ZrN were studied by the FLAPW method. The calculated results indicated the Young’s modulus E of ZrC is much higher than the single-phase glassy alloys, which is thought to be the important reason of the improvement of the mechanical properties of glassy alloys containing ZrC particles. It is predicted that if ZrN as the disperse phase is introduced to the alloys the mechanical properties of the matrix should be also improved.

Acknowledgements We acknowledge financial support of this work by the Special Funds for the Major State Basic Research Projects

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of China under grant number G2000067104 and by the National Pandeng Research Program of China (grant no. 95-Yu-41).

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