Anisotropies in elastic properties and thermal conductivities of trigonal TM2C (TM = V, Nb, Ta) carbides

Solid State Sciences 98 (2019) 106027

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Anisotropies in elastic properties and thermal conductivities of trigonal TM2C (TM ¼ V, Nb, Ta) carbides Longke Bao, Deyi Qu, Zhuangzhuang Kong, 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

Keywords: First-principles calculations Carbides Anisotropy Elastic properties Thermal conductivity

In this work, first-principles calculations based on density functional theory were employed to explore anisot­ ropies in elastic moduli, sound velocities and thermal conductivities of trigonal TM2C (TM ¼ V, Nb, Ta) carbides. Elastic moduli were obtained according to elastic constants, while Poisson’s ratio, sound velocity and Debye temperature were calculated based on elastic moduli and constants. The anisotropy in elastic modulus was investigated using several elastic anisotropy indexes and three-dimensional surface construction and its planar projection. Moreover, the minimum thermal conductivity kmin was evaluated by Clarke’s model and Cahill’s model. Meanwhile, the anisotropy in kmin was also discussed. The results indicated that TM2C carbides are mechanically stable and ductile, and the intrinsic ductility is in a sequence of Nb2C > V2C > Ta2C. Both the se­ quences of anisotropy in elastic modulus and in kmin are V2C > Nb2C > Ta2C.

1. Introduction Transition metal carbides (TMCs) are familiar ceramics that have large elastic modulus, high melting point and chemical resistance. Therefore, they are extensively utilized in structural materials working at corrosive environment, high temperature and heavy load. Among these, the V–C, Nb–C and Ta–C systems are of interest owing to that there are a number of phases with different carbon content, and varied orders of vacancies in the carbon sublattice [1–3]. From TM-C (TM ¼ V, Nb, Ta) phase diagrams, TM2Cs can be widely found in a component range of stoichiometry related to their nominal chemical formulas [4–6]. With regard to crystal structure of TM2C (TM ¼ V, Nb, Ta), there are three main controversies that are hexagonal structure, trigonal structure and orthorhombic structure. Generally, trigonal systems are often clas­ sified as hexagonal systems due to that there is a crystal lattice satisfying not only the symmetry of trigonal systems, but also the symmetry of hexagonal systems. Virtually, for these TM2C (TM ¼ V, Nb, Ta) carbides, the space group (SG) of hexagonal structure is 194-P63/mmc, while that of trigonal structure is 164-P-3m1. Moreover, for V2C, the orthorhombic structure with SG of Pbcn was considered [1]; for Nb2C, the ortho­ rhombic structures with SGs of Pnma and Pbcn were also discussed [2]. The phase transitions between trigonal and other structures in TM2C (TM ¼ V, Nb, Ta) carbides have been reported. The orthorhombic V2C

undergoes a phase transition to trigonal V2C at 1023 K [7]. Ortho­ rhombic Nb2C transforms to trigonal Nb2C at temperature range of 1473–1523 K, and trigonal Nb2C has a phase transition to hexagonal Nb2C at 2773 K [7,8]. In Ta–C system, trigonal Ta2C has a phase tran­ sition to hexagonal Ta2C at temperature ranging from 2308 to 2433 K [9]. Besides, the hexagonal TM2Cs (TM ¼ V, Nb, Ta) behave interesting magnetic properties that Ta2C and Nb2C showed superconducting at 3.26 and 9.18 K, respectively, and V2C did not become superconducting down to 1.2 K [10]. However, trigonal TM2Cs (TM ¼ V, Nb, Ta) have been indicated that they are nonmagnetic [11]. At present, most of investigations of TM2C (TM ¼ V, Nb, Ta) carbides focused on orthorhombic and hexagonal structures, especially on their phase stability, electronic, mechanical and thermal properties. The crystal structures of vacancy-ordered and faulted TM2C (TM ¼ V, Nb, Ta) carbides were predicted using first-principles investigations of phase stability, and the results showed that orthorhombic V2C, orthorhombic Nb2C and trigonal Ta2C are the lowest energy structure in corresponding TM2C (TM ¼ V, Nb, Ta) carbides [12,13]. The theoretical investigations of orthorhombic V2C and hexagonal V2C indicated that orthorhombic V2C is more stable than hexagonal V2C, while hexagonal V2C has more anisotropic elastic modulus than orthorhombic V2C [14]. Using density functional theory (DFT), the electronic structure and formation energy of TM2C (TM are VB-VIB transition metals) were calculated [15]. Here, hexagonal V2C and hexagonal Nb2C, respectively, are the most

* Corresponding author. E-mail addresses: [email protected] (Z. Kong), [email protected] (Y. Duan). https://doi.org/10.1016/j.solidstatesciences.2019.106027 Received 8 April 2019; Received in revised form 23 August 2019; Accepted 4 October 2019 Available online 5 October 2019 1293-2558/© 2019 Elsevier Masson SAS. All rights reserved.

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Fig. 1. Crystal structures of TM2C (TM ¼ V, Nb, Ta) carbides: (a) Top view, (b) Side view.

stable phases in V2C and Nb2C carbides. But in another report ortho­ rhombic Nb2C showed the lowest formation energy and hexagonal Nb2C had the highest formation energy, indicating that orthorhombic Nb2C is the most stable phase [16]. The calculated formation enthalpy and mechanical properties of orthorhombic V2C presented that ortho­ rhombic V2C is a stable phase, and bulk, shear and Young’s moduli are 242, 121 and 311 GPa, respectively [1]. Orthorhombic Nb2C possessed bulk, shear and Young’s moduli of 226, 121 and 307 GPa, respectively [2]. As for trigonal trigonal TM2C (TM ¼ V, Nb, Ta) carbides, there are few reports related to theoretical calculations, which can only be found in a few literatures. The calculated formation enthalpies of trigonal TM2C (TM ¼ V, Nb, Ta) carbides were 0.42, 0.50 and 0.60 eV/ atom, respectively [12]. Moreover, bulk moduli of trigonal TM2C (TM ¼ V, Nb, Ta) carbides were computed and were 206.5, 211.93 and 246.45 GPa, respectively [15]. So far, no more information about me­ chanical properties of trigonal TM2C (TM ¼ V, Nb, Ta) carbides can be discovered. Therefore, the current investigation of these carbides is fragmented and unsystematic. In this work, we performed the first-principles calculations to explore systematically the phase stability, polycrystalline elastic modulus, thermal conductivity and their anisot­ ropies, which can contribute to high-temperature synthesis and appli­ cation of TM2C (TM ¼ V, Nb, Ta) carbides.

Table 1 Experimental and theoretic lattice parameters and formation enthalpies ΔH of TM2C (TM ¼ V, Nb, Ta) carbides. TM2C V2C

Nb2C

Ta2C

Lattice parameters (Å)

V (Å3)

a

c

2.897 2.881 2.9043 2.893 2.900

4.539 4.547 4.5793 4.532 4.524

32.989 32.68 33.451 32.849

3.151 3.120 3.148 3.140

5.000 4.957 5.007 4.973

42.980 41.789 42.968

3.133 3.1030 3.125 3.130

5.028 4.9378 4.971 4.933

42.730 41.174 42.041

ΔH (eV/atom) 0.472 0.451 0.470 0.42 0.509 0.489 0.508 0.50 0.615 0.601 0.661 0.60

Refs. Present [20] [21] [11] [15] [12] Present [22] [11] [15] [12] Present [21] [11] [15] [12]

are three atoms that two TM atoms occupy 2c site (0.3333, 0.6667, 0.25) and one C atom locates at 1a site (0, 0, 0). After geometry optimization, the symmetries of all these TM2C carbides still follow trigonal structure. It indicates that the setting computational parameters and firstprinciples calculations can be used to calculate physical properties of TM2C carbides. Table 1 shows the optimized structural parameters of these carbides, along with the previous theoretical and experimental data [11,15, 20–22]. The calculated structural parameters in this work are slightly larger than the reported values. The overestimation of structural pa­ rameters is the intrinsic nature of GGA approximation. However, the calculated structural parameters coincide with the already reported values due to the small average deviation between the present and the previous data of less than 1.5%, which also indicate that the computa­ tional parameters employed in this work can be used to compute elastic and thermal properties. For ceramics, the phase stability is important for establishing the heating and sintering process. Formation enthalpy ΔH represents the difference in energy released and absorbed after the reaction of the substance. The formation enthalpy ΔH is negative and smaller, indi­ cating that the reaction process is easier and the phase is more stable. However, cohesive energy represents the energy required to decompose a crystal into infinitely far neutral free atoms at absolute zero. As is known, the phase stability can be better reflected by formation enthalpy than by cohesive energy. Therefore, in this work, the formation enthalpy ΔH is applied to evaluate the phase stability of trigonal TM2C (TM ¼ V, Nb, Ta) carbides by the following equation:

2. Computational methods To explore the structural properties, anisotropy in elastic moduli, sound velocities and thermal conductivities of trigonal TM2C (TM ¼ V, Nb, Ta) carbides, the first-principles calculations based on density functional the­ ory (DFT) [17] were employed using CASTEP code [18] with the ultrasoft pseudo-potentials (USPPs). The generalized gradient approximation (GGA) within the Perdew-Burke-Ernzerhof (PBE) scheme [19] was applied to represent exchange-correlation energy. The valence electrons in present work were V 3s23p63d34s2, Nb 4s24p64d45s1, Ta 5s25p65d36s2 and C 2s22p2, respectively. Geometry optimization of crystal structure was performed using Broyden-Fletcher-Goldfarb-Shannon (BFGS) method. After conver­ gence tests, k points and cutoff energy were 40 � 40 � 22 and 500 eV, respectively. During the first-principles calculations, the total energy convergence, the maximum force and maximum stress were set as 5 � 10 6 eV/atom, 0.01 eV/Å and 0.02 GPa, respectively. Besides, when elastic constants were calculated, the number of steps for each strain and maximum strain amplitude were 0.003 and 7, respectively. 3. Results and discussion 3.1. Structural properties and phase stability Fig. 1 plots crystal structures of TM2C (TM ¼ V, Nb, Ta) carbides. As shown in Fig. 1, TM2C carbides crystallize as trigonal structure with the space group of P-3m1 in this work. In the trigonal TM2C carbides, there

ΔH ¼ ½ETM2C

2

2ETM

EC �=3

(1)

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It can be seen from Table 2 that, for TM2C (TM ¼ V, Nb, Ta) carbides, values of C11 are larger than those of C12. And Cij’s satisfying the me­ chanical stability criterion indicates that TM2C (TM ¼ V, Nb, Ta) car­ bides are mechanically stable. Elastic constants of trigonal TM2C (TM ¼ V, Nb, Ta) carbides are anisotropic due to C11 ¼ C22 6¼ C33, which will lead to the anisotropies in elastic moduli of these carbides. Generally, C11, C22 and C33 reflect the resistances to linear compression along x-, y- and z-axes, respectively. It can be observed that C33 is slightly larger than C11, indicating that trigonal TM2C (TM ¼ V, Nb, Ta) carbides show more compressible along x- and y-axes than along the z-axis under uniaxial stress. Besides, C44 and C66 correspond to the [001] and [110] directional resistances to shear stress in (100) plane, respectively. C12 represents the [110] directional resistance to shear stress in (110) plane. From Table 2, C44, C66 and C12 are not equal to each other, which indicate their anisotropies in shear moduli. It should be noted that the difference between C11 and C33 for trigonal TM2C (TM ¼ V, Nb, Ta) carbides is generally smaller than that between C12, C44 and C66, and therefore in these carbides, bulk modulus should show the anisotropy less than shear modulus.

Table 2 Calculated elastic constants Cij (in GPa) of TM2C (TM ¼ V, Nb, Ta) carbides. TM2C

C11

C33

C44

C66

C12

C13

C14

V2C Nb2C Ta2C

434.5 400.8 464.5

446.2 417.6 492.1

46.6 57.3 124.8

115.9 129.1 152.4

126.1 140.2 159.5

97.8 115.9 139.7

40.5 43.6 44.4

Here, ETM2C represents the total energy of TM2C, ETM means the energy of one TM atom in the body-centered cubic TM crystal (TM ¼ V, Nb, Ta). EC is the energy of a C atom in graphite. Generally, a compound with a negative formation enthalpy is stable, and the more negative formation enthalpy represents the better phase stability. The calculated formation enthalpies of trigonal TM2C (TM ¼ V, Nb, Ta) carbides, together with the previous theoretical ones [11,12,15], are listed in Table 1. From Table 1, the present calculated formation enthalpies agree well with the theoretical data. Moreover, these trigonal TM2C carbides have negative formation enthalpies, indicating that they are stable. In all of these carbides (orthorhombic, hexagonal and trigonal TM2C), the formation energy is negative, and the difference in formation energy between carbides is very small, so carbides can be present in the material [16]. Besides, Ta2C behaves the most negative formation enthalpy ( 0.598 eV/atom), while V2C possesses the least negative formation enthalpy ( 0.455 eV/atom), revealing that Ta2C is the most energeti­ cally stable and V2C is the least energetically stable. Thus, the phase stability of trigonal TM2C (TM ¼ V, Nb, Ta) carbides is in a sequence of Ta2C > Nb2C > V2C. As is known, phonon spectrums calculation is a strict measure for structural stability [23]. It should be noted that, in our work, we have mainly investigated the elastic properties and thermal conductivities of trigonal TM2C (TM ¼ V, Nb, Ta). However, the phonon calculation will be considered in our next work.

3.3. Polycrystalline elastic moduli According to the single-crystal elastic constants, the polycrystalline elastic moduli, such as bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio v, can be evaluated by Voigt-Reuss-Hill approximations [25–27]. For trigonal TM2C (TM ¼ V, Nb, Ta) car­ bides, the Hill bound of bulk (shear) modulus BH (GH) can be calculated from the Voigt and Reuss bounds (BV (GV) and BR (GR)) from the following expressions [28]:

�

BR ¼ ½ðC11 þ C12 C33

3.2. Single-crystal elastic constants

GV ¼ ð7C11

��� � �� GR ¼ 15=2 2C11 þ 2C12 þ 4C13 þ C33 C33 C11 þ C12 ��� � �� 2 2C14 1 þ6C44 C44 C11 C12

(3)

BV ¼ 2ðC11 þ C12 þ C33 =2 þ 2C13 Þ=9

� 2C213 þ 3C11

2C213 Þ�=ðC11 þ C12 þ 2C33

5C12 þ 12C44 þ 2C33

4C13

�

(4) (5)

4C13 Þ=30

2C12

Generally, single-crystal elastic constants are related to chemical bonds, specific heat, and thermal expansion. In this work, the stressstrain approach based on the generalized Hook’s low, which is applying a small strain to the equilibrium lattice and computing the resultant change in its total energy in the CASTEP code [24], was used to investigate single-crystal elastic constants of trigonal TM2C (TM ¼ V, Nb, Ta) carbides, and the results are listed in Table 2. Table 3 presents the elastic compliance constants Sij obtained directly from elastic con­ stants Cij. According to Born-Huang’s lattice dynamics theory, the criterion of mechanical stability for trigonal crystal can be defined by elastic con­ stants as follows: � � C11 jC12 j > 0; C11 þ C12 C33 > 2C213 ; C11 C12 C44 > 2C214 (2)

(6)

BH ¼ ðBV þ BR Þ=2

(7)

GH ¼ ðGV þ GR Þ=2

(8)

Accordingly, Young’s modulus E and Poisson’s ratio v can be ob­ tained from BH and GH: (9)

E ¼ 9BH GH =ð3BH þ GH Þ v ¼ ð3BH

(10)

2GH Þ=ð6BH þ 2GH Þ

Table 4 presents the calculated elastic moduli of trigonal TM2C (TM ¼ V, Nb, Ta) carbides, together with the previous bulk moduli [15].

Table 3 Calculated elastic compliance matrix Sij of TM2C (TM ¼ V, Nb, Ta) carbides. TM2C

S11

S33

S44

V2C Nb2C Ta2C

0.00307 0.00362 0.00275

0.00243 0.00272 0.00233

0.02781 0.02341 0.00894

S12

3

0.00114 0.00153 0.00091

S13 0.00042 0.00058 0.00053

S14 0.00365 0.00392 0.00130

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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 ν, Pugh’s modulus ratio BH/GH and Cauchy pressure (C13–C44 and C12–C66) (in GPa) of TM2C (TM ¼ V, Nb, Ta) carbides. Here, bulk modulus is taken to 0.01 GPa for calculating percent compressible anisotropy. TM2C

BV

BR

BH

GV

GR

GH

E

ν

BH/GH

C13–C44

C12–C66

V2C

217.62

217.56

115.7

64.0

89.9

237.0

0.318

2.421

51.2

10.2

Nb2C

218.13

218.12

105.5

67.6

86.5

229.2

0.325

2.521

58.6

11.1

Ta2C

255.43

255.42

217.59 206.50 [15] 218.13 211.93 [15] 255.43 246.45 [15]

145.8

130.7

138.2

351.3

0.271

1.848

14.9

7.1

It is obvious that the calculated bulk moduli agree well with the reported data, which also verify the high validity of computational parameters used in this work. Bulk modulus is defined as the ratio of the increase of infinitesimal pressure to the decrease of the volume, and it is a measure of how resistant to compression. The crystal with a higher bulk modulus behaves a less compressibility. The different B values of these trigonal carbides indicate their different compressibility. Ta2C possesses the highest bulk modulus of 255.5 GPa, revealing the least compressibility for Ta2C. As for V2C and Nb2C, their bulk moduli are close to each other, which illustrate that they have the similar compressibility. However, bulk modulus of Nb2C is slightly higher than that of V2C. Thus, V2C has the greatest compressibility among these trigonal TM2C (TM ¼ V, Nb, Ta) carbides. Besides, bulk modulus can to some extent reflect the strength of chemical bonds in crystal, and a higher bulk modulus cor­ responds to stronger chemical bonds. Accordingly, due to the largest bulk modulus for Ta2C, Ta–C bonds in Ta2C may be the strongest than Nb–C bonds in Nb2C and V–C bonds in V2C. Of course, this result needs to be further verified by electronic structures. Shear modulus is the ratio of shear stress to the shear strain, and it describes the response of material to shear stress. A larger shear modulus corresponds to a smaller deformation of a material under the shear stress. From Table 4, Ta2C has the largest shear modulus (138.2 GPa), while Nb2C possesses the smallest one (86.5 GPa), indicating that the deformation of Ta2C is the smallest and that of Nb2C is greatest under the same shear stress. Moreover, shear modulus is concerned with elastic constants C44 and C66. Generally, the larger C44 and C66 correspond to a larger shear modulus. C44 and C66 of Ta2C are 124.6 and 152.4 GPa, respectively, which are larger than those of V2C and Nb2C. Therefore, Ta2C has the largest shear modulus. Meanwhile, Nb2C has a slightly larger C44 and C66 than V2C, however, C11 of V2C is much larger than that of Nb2C. As a result, V2C has a larger shear modulus than Nb2C. Young’s modulus is a measure of the stiffness of a solid. A solid with a larger Young’s modulus is stiffer. Thus, the stiffness of trigonal TM2C (TM ¼ V, Nb, Ta) carbides is in a sequence of Ta2C > V2C > Nb2C. Poisson’s ratio v can characterize the stability of material in the linear

elasticity regime of a shear deformation. For a stable material, its Poisson’s ratio varies in the range from 1.0 to 0.5. Poisson’s ratios of V2C, Nb2C and Ta2C are 0.318, 0.325 and 0.271, respectively, which fall within the range of 1.0–0.5. Thus, these trigonal TM2C (TM ¼ V, Nb, Ta) carbides are stable in the linear elasticity regime of a shear deformation. For non-metallic trigonal TM2C (TM ¼ V, Nb, Ta) carbides, their intrinsic ductility and brittleness can be generally evaluated by ν [29], BH/GH [30] and Cauchy pressure (C13–C44 and C12–C66). If a carbide has ν > 0.26, BH/GH > 1.75, and C13–C44 > 0 and C12–C66 > 0, it is ductile. Otherwise, it is brittle. Table 4 also presents the calculated ν, BH/GH and C13–C44 for trigonal TM2C (TM ¼ V, Nb, Ta) carbides. Fig. 2 shows variations of ν, BH/GH, C13–C44 and C12–C66. Form Fig. 2, ν, BH/GH, C13–C44 and C12–C66 for trigonal TM2C (TM ¼ V, Nb, Ta) carbides satisfy the ductile conditions, indicating that these carbides are ductile. Moreover, the curves of ν, BH/GH, C13–C44 and C12–C66 show the similar trend that, from V2C to Ta2C, the curves increase first and then decrease. Nb2C possesses the largest ν, BH/GH, C13–C44 and C12–C66, while Ta2C has the smallest ones. Therefore, Nb2C is more ductile than V2C and Ta2C, and the intrinsic ductility of TM2C (TM ¼ V, Nb, Ta) carbides is in a sequence of Nb2C > V2C > Ta2C. The ductile nature of these carbides indicates that these carbides can be used as reinforcing phases in TM-based alloys, which will not affect the plasticity of alloys, but also increase the strength of alloys. 3.4. Anisotropy of elastic moduli For ceramics, the anisotropies in elastic and thermal properties can induce microcracks to reduce the mechanical durability. Thus, several anisotropic indexes such as the universal elastic anisotropic index (AU) [31], percent anisotropy in compressibility and shear (Acomp and Ashear) [32], and the shear anisotropic factors on the (100), (010), and (001) planes (A1, A2 and A3) [33], were used to characterize the elastic an­ isotropies of TM2C (TM ¼ V, Nb, Ta) carbides by the following equations: AU ¼ 5GV =GR þ BV =BR Acomp ¼ BV

(11)

6

BR Þ=ðBV þ BR Þ;

Ashear ¼ GV

GR Þ=ðGV þ GR

�

(12)

A1 ¼ 4C44 =ðC11 þ C33

2C13 Þ

(13)

A2 ¼ 4C55 =ðC22 þ C33

2C23 Þ

(14)

A3 ¼ 4C66 =ðC11 þ C22

2C12 Þ

(15) U

For an elastically isotropic solid, A ¼ Acomp ¼ Ashear ¼ 0 and Table 5 Calculated elastic anisotropic indexes (AU, Acomp, Ashear, A1 and A3) of TM2C (TM ¼ V, Nb, Ta).

Fig. 2. Variations of v and BH/GH (a), and C13–C44 and C12–C66 (b) of TM2C (TM ¼ V, Nb, Ta). 4

TM2C

AU

Acomp ( � 10 3)

Ashear

A1

A3

V2C Nb2C Ta2C

4.034 2.803 0.575

0.133 0.031 0.025

0.287 0.219 0.054

0.2721 0.3907 0.7393

0.9877 0.9908 0.9993

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Fig. 3. Variations of elastic anisotropic indexes of AU, Ashear and Acomp (a) and 1-A1, 1-A3 (b) of TM2C (TM ¼ V, Nb, Ta).

Fig. 4. Surface constructions ((a), (b) and (c)) and plane projections ((d), (e) and (f)) of bulk, shear and Young’s moduli of V2C.

A1 ¼ A2 ¼ A3 ¼ 1, while the larger values of AU, Acomp, Ashear and 1-Ai (i¼1, 2, 3) represent a more elastic anisotropy. In trigonal crystals, due to C11 ¼ C22, C13 ¼ C23 and C44 ¼ C55, A1 is equivalent to A2. Table 5 lists the values of AU, Acomp, Ashear, A1 and A3 obtained. Fig. 3 shows varia­ tions of AU, Acomp, Ashear, 1-A1 and 1-A3 of trigonal TM2C (TM ¼ V, Nb, Ta) carbides. From AU listed in Tables 5, V2C possesses the largest AU value (4.034) and Ta2C has the lowest one (0.575), indicating that V2C is the most anisotropic in elasticity and Ta2C is the least elastically anisotropic. This result can be verified by their Acomp and Ashear. It is obvious that the curves of AU, Acomp and Ashear in Fig. 3(a) show a downward trend,

which mean that V2C has the largest AU, Acomp and Ashear, while Ta2C possesses the lowest ones. The difference in variation curves of AU, Acomp and Ashear may be originated from the fact that only one kind of modulus is considered in Acomp and Ashear, and both bulk modulus and shear modulus are introduced in AU. Thus, AU can be used to characterize elastic anisotropy better than Acomp and Ashear. According to AU, the order of anisotropy in elastic properties of trigonal TM2C (TM ¼ V, Nb, Ta) carbides is in a sequence of V2C > Nb2C > Ta2C. Besides, we used the difference between shear anisotropic factors (A1 and A3) and 1, such as 1-A1 and 1-A3, to describe the shear anisotropy of these carbides. Larger 1-A1 and 1-A3 reflect a higher anisotropy. It is 5

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Fig. 5. Surface constructions ((a), (b) and (c)) and plane projections ((d), (e) and (f)) of bulk, shear and Young’s moduli of Nb2C.

evident that variations of 1-A1 and 1-A3 are similar in Fig. 3(b). Values of 1-A1 and 1-A3 of V2C are the largest (0.7279 and 0.0123, respectively), indicating that V2C shows the highest shear anisotropy in the three principal planes. While values of 1-A1 and 1-A3 of Ta2C are the lowest (0.2607 and 0.0007, respectively), suggesting the lowest shear

anisotropy of Ta2C in the three principal planes. This conclusion is coincided with that obtained from AU, Acomp and Ashear. From the elastic anisotropic indexes AU, Acomp, Ashear, A1 and A3, trigonal TM2C (TM ¼ V, Nb, Ta) carbides present anisotropy in elastic modulus along different crystallographic directions. Thus, the

Fig. 6. Surface constructions ((a), (b) and (c)) and plane projections ((d), (e) and (f)) of bulk, shear and Young’s moduli of Ta2C. 6

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Table 6 Calculated directional elastic moduli in three principal directions (in GPa) of TM2C (TM ¼ V, Nb, Ta). TM2C

B

G

E

[100]

[010]

[001]

[100]

[010]

[001]

[100]

[010]

[001]

V2C Nb2C Ta2C

662.3 662.3 763.4

662.3 662.3 763.4

628.9 641.0 787.4

55.2 59.3 123.0

55.2 59.3 123.0

33.9 38.9 90.4

325.7 276.2 363.6

325.7 276.2 363.6

415.5 347.6 429.2

directional elastic anisotropies of these carbides also were discussed by three-dimensional (3D) surface constructions of B, G and E. For trigonal crystals, 3D surface constructions of B, Gt (torsion shear modulus) and E can be obtained by the reciprocals of B, Gt and E [34,35]: � � S11 þ S12 S13 S33 l23 B 1 ¼ S11 þ S12 þ S13 (16) Gt 1 ¼ S11 3l21 þ 3l22

2l41

2l42

þ S44 =2Þðl21 þ l22 þ 2l23 4l22 l23 � 4S13 l23 l21 þ l23 þ 2S14 2l32 l3 E

1

¼ S11 1

� � 4l21 l22 þ 2S33 l23 1 l23 � � S12 l21 þ l22 4l21 l23 � 2 2 5l1 2l3 l2 l1 l3

� � l23 þ S33 l43 þ 2S13 þ S44 l23 1

� l23 þ 2S14 l2 l3 3l21

Table 7 The density ρ, sound velocity (longitudinal νl, transverse νt and average νm), acoustic Grüneisen parameter γa and Debye temperature ΘD of TM2C (TM ¼ V, Nb, Ta).

(17)

l22

�

TM2C

V2C

Nb2C

Ta2C

ρ (g/cm3) νl (m/s) νt (m/s) νm (m/s)

5.73 7674 3961 4435 1.893 594

7.64 6606 3365 3770 1.938 462

14.53 5501 3084 3432 1.605 422

γa ΘD (K)

(18)

respectively. Therefore, the deviations of B[100]/B[001] ratios from 1 for V2C, Nb2C and Ta2C are 0.053, 0.033 and 0.030, respectively, indicating that the anisotropy in bulk modulus is in a sequence of V2C > Nb2C > Ta2C. This agrees well with the result from Acomp. More­ over, deviations of G[100]/G[001] ratios from 1 for V2C, Nb2C and Ta2C are 0.628, 0.524 and 0.361, respectively, which indicate the sequence of anisotropy in shear modulus be V2C > Nb2C > Ta2C. This result of shear modulus is in a good agreement with that from Ashear, 1-A1 and 1-A3. Besides, deviations of E[100]/E[001] ratios from 1 for V2C, Nb2C and Ta2C are 0.216, 0.205 and 0.153, respectively. It reveals that V2C shows the highest anisotropy in Young’s modulus while Ta2C possesses the lowest one, which agrees well with the result from AU.

Here, Sij are the elastic compliance constants shown in Table 3. li (i¼1, is the direction cosine. As is known, an elastically isotropic solid has a spherical 3D surface construction of elastic modulus. Otherwise, the non-spherical 3D surface construction is for an elastically anisotropic solid. Besides, the more the deviation of 3D surface construction from sphere is, the higher the degree of elastic anisotropy is. Fig. 4 plots 3D surface constructions and planar projections of bulk, shear and Young’s moduli of trigonal V2C. From Fig. 4 it can be seen that 3D graph of bulk modulus for V2C shows a sphere-like shape, while 3D graphs of shear and Young’s moduli present obvious deviations from the sphere, indicating that bulk modulus of V2C is approximately isotropic, and shear and Young’s moduli are anisotropic. The projection of elastic modulus confirms the difference in anisotropy between bulk modulus and shear (Young’s) modulus. Projections of bulk modulus on the (100) and (001) planes are close to circle, while those of shear and Young’s moduli on the (100) plane are distinctly different from on the (001) plane. The (001) planar projections of shear and Young’s moduli are circle, and the (100) planar projections are irregular. Fig. 5 shows 3D surface constructions of bulk, shear and Young’s moduli of trigonal Nb2C. Bulk modulus of Nb2C is similar to that of V2C: a near-spherical 3D graph and approximate circular projection of bulk modulus can be observed in Fig. 5(a) and (d), respectively. Shear and Young’s moduli of Nb2C are anisotropic due to the no-spherical 3D graphs (Fig. 5(b) and (c)) and non-circular projections (Fig. 5(e) and (f)). Besides, 3D graphs of shear and Young’s moduli of Nb2C have the smaller deviations from sphere than those of V2C, indicating that Nb2C possesses a lower anisotropy in elastic modulus than V2C. Fig. 6 shows 3D surface constructions of bulk, shear and Young’s moduli of trigonal Ta2C. Bulk modulus of Ta2C behaves the similar characteristic as V2C and Nb2C, but shear and Young’s moduli show the different nature that lower anisotropy than Nb2C can be found in Fig. 6. Therefore, the sequence of anisotropy in elastic modulus is V2C > Nb2C > Ta2C. Planar projections can provide more details of elastic anisotropy. Table 6 lists the directional elastic moduli of TM2C (TM ¼ V, Nb, Ta) in the [100], [010] and [001] directions based on Figs. 4–6. Due to the symmetry of trigonal structure, the [100] directional elastic modulus is equivalent to the [010] directional one. It can be observed from Table 6 that elastic moduli in the [001] direction are different from those in the [100] direction, indicating the anisotropy in elastic modulus. Moreover, the ratio of elastic modulus in the [100] and [001] directions was employed to determine the magnitude of elastic modulus anisotropy. The larger the ratio deviates from 1, the higher the elastic anisotropy is. B[100]/B[001] ratios of V2C, Nb2C and Ta2C are 1.053, 1.033 and 0.970, 2, 3)

3.5. Debye temperatures and anisotropy of sound velocities Debye temperature θD is an essential thermal parameter of solids, which can be evaluated from sound velocities based on bulk modulus and shear modulus [36,37]: � vm ¼ ½ð2=v3t þ 1=v31 Þ=3

1=3

;

vl ¼ ½ðB þ 4G=3Þ=ρ

i1=2 ;

vt ¼

�1=2 � G=ρ (19)

θD ¼ ðh=kB Þ½ð3n=4πÞðNAρ =MÞ�1=3 vm

(20)

where vl, vt and vm are the longitudinal, transverse and mean sound velocities, respectively. h, kB and NA are Planck’s, Boltzmann’s and Avogadro’s constants, respectively. n is the total atomic number in each TM2C unit cell. ρ is the density and M is the molecular weight. Table 7 presents calculated vl, vt, vm, and θD of TM2C (TM ¼ V, Nb, Ta) carbides. From Table 7, it is obvious that TM2C carbide with a lower density has larger sound velocities and a higher Debye temperature. For example, the density of V2C is 5.73 g/cm3, which is lower than Nb2C and Ta2C. Values of vl, vt, vm of V2C are 7674, 3961 and 4435 m/s, respec­ tively, which are larger than those of Nb2C and Ta2C, and Debye tem­ perature of V2C (594 K) is also higher than Nb2C(462 K) and Ta2C (422 K). Moreover, Debye temperature is connected with chemical bond strength. The higher the Debye temperature is, the stronger the chemical bond strength is. Therefore, Debye temperature of V2C is higher than those of o Nb2C and Ta2C, which indicate that V2C behaves the stronger chemical bond strength than other TM2C carbides. The chemical bond strength in Ta2C is the weakest. Besides, Debye temperature is also related to thermal conductivity. The higher the Debye temperature is, the larger the thermal conductivity is. Thus, V2C has the larger thermal 7

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Solid State Sciences 98 (2019) 106027

Table 8 The anisotropic sound velocity (in m/s) of TM2C (TM ¼ V, Nb, Ta).

�1=2 h �1=2 h i � i � C12 Þ=2ρ�1=2 ; 010 vt1 ¼ C11 =ρ ; 001 vt2 ¼ C44 =ρ ; �1=2 h i h i � 1=2 ½001�vl ¼ ðC33 =ρÞ ; 100 vt1 ¼ 010 vt2 ¼ C44 =ρ

½100�vl ¼ ½ðC11

TM2C

[100]

[001]

[100]vl

[010]vt1

[001]vt2

[001]vl

[100]vt1

[010]vt2

V2C Nb2C Ta2C

5188 4130 3240

8708 7243 5648

2852 2739 2931

8824 7393 5826

2852 2739 2931

2852 2739 2931

(22) Here, vt1 and vt2 refer to the first and second modes of transverse sound velocities, respectively. Table 8 lists the obtained directional sound velocities of TM2C carbides. From Table 8, due to the largest C33 values, trigonal TM2C carbides have the largest longitudinal sound ve­ locities in the [001] direction ([001]vl). While the [100] direction has the largest first mode of transverse sound velocity ([010]vt1) owing to C11 larger than C44. The different directional sound velocity of TM2C carbides indicate that sound velocity of TM2C carbides is anisotropic.

conductivities than Nb2C and Ta2C. The anharmonicity of interatomic interactions of solids can be described by acoustic Grüneisen constant γ a. γa is associated with ther­ mal expansion and conduction, and elastic properties at different tem­ perature [38], and it can be calculated from yl and vt [39]: � γa ¼ 3=2Þð3v21 4v2t Þ=ðv21 þ 2v2t (21)

3.6. Thermal conductivities

The calculated γ a were also listed in Table 7. Ta2C possesses the lowest γa (1.605), while Nb2C has the highest one, indicating that effect of external conditions (such as temperature and pressure) on lattice dynamics of Ta2C is the least and that of Nb2C is the greatest. Elastic modulus of TM2C (TM ¼ V, Nb, Ta) carbides is anisotropic. Meanwhile, sound velocities are calculated from elastic modulus. Therefore, the anisotropy of sound velocities should be considered here. For trigonal crystals, [100] and [001] directions are propagating modes of sound velocities, and the calculation formulas are as follows:

For solids, as temperature increases, the thermal conductivity will decrease to a limitation [40]. Therefore, the minimum thermal con­ ductivity should be taken into account for ceramics utilized at high temperatures. In this work, Clarke’s model [41,42] and Cahill’ model [43] were employed to calculate the minimum thermal conductivities kmin of TM2C carbides, and the formulas are as follows: Clarke’s model: kmin ¼ 0:87kB M a 2=3 E1=2 ρ1=6

(23)

Ma ¼ ½M=ðmNlÞ� Table 9 Calculated thermal conductivities kmin (W⋅m TM2C

⋅K 1) of TM2C (TM ¼ V, Nb, Ta).

Clarke model Ma(10

V2C Nb2C Ta2C

1

0.631 1.095 2.070

22

)

Cahill model kmin

kmin[100]

kmin[001]

n(1028)

kmin

kmin[100]

kmin[001]

1.561 1.114 1.005

2.056 1.411 1.110

1.729 1.223 1.022

9.09 6.98 7.02

1.755 1.258 1.106

1.885 1.331 1.120

1.635 1.214 1.080

Fig. 7. Surface constructions ((a), (b) and (c)) and plane projections ((d), (e) and (f)) of minimum thermal conductivities of V2C, Nb2C and Ta2C. 8

Solid State Sciences 98 (2019) 106027

L. Bao et al.

Here, kB is Boltzmann’s constant, Ma is the mean atomic mass in each TM2C unit cell, E refers to Young’s modulus, ρ represents the density, M denotes the molar mass, m indicates the total number of atom in each unit cell, NA means Avogadro’s constant. Cahill’s model: � � (24) kmin ¼ kB =2:48 n2=3 vl þ 2vt

from 1, the higher the anisotropy in kmin is. kmin[100]/kmin[001] ratios of V2C, Nb2C and Ta2C are 1.189, 1.54 and 1.086, respectively. Therefore, the deviations of kmin[100]/kmin[001] ratios from 1 for V2C, Nb2C and Ta2C are 0.189, 0.154 and 0.086, respectively, indicating that the anisotropy in kmin is in a sequence of V2C > Nb2C > Ta2C. 4. Conclusions

where n represents density of the number of atom per unit volume. vl is the longitudinal sound velocity and vt is transverse sound velocity. The calculated Ma and kmin in Clarke’s model, and n and kmin in Cahill’s model were listed in Table 9. It can be observed from Table 9 that V2C has the highest kmin, while Ta2C possesses the lowest one. These just reflect that V2C has the highest Debye temperature (594 K) and Ta2C behaves the lowest Debye temperature (422 K). Therefore, the sequence of kmin is the same as that of θD, which is due to that, at the ground-state conditions, the lattice thermal conductivity plays the leading role in the thermal conductivity, and thus the lower Debye temperature corre­ sponds to a smaller lattice thermal conductivity indicates [44]. Besides, kmin values calculated by Cahill’s model are slightly larger than those by Clarke’s model. This originates from that the phonon spectrum is much accounted in Cahill’s model, while the optical phonon is ignored in Clarke’s model. Moreover, the calculated minimum thermal conductivities of these TM2C carbides are in range of 1.0–1.8 W m 1 K 1. The new promising ceramic thermal barrier coatings Ln2Zr2O7 and Ln2SrAl2O7 have kmin of 1.2–1.4 W m 1 K 1 [45] and 1.49–1.60 W m 1 K 1 [46], respectively. These TM2C carbides have comparable minimum thermal conductivities to Ln2Zr2O7 and Ln2SrAl2O7. Thus, these TM2C carbides can be used as potential thermal insulating materials. Since there are anisotropies in vl and vt, and kmin in Cahill’s model is related to vl and vt, Cahill model expressed in three acoustic branches (vl, vt1 and vt2) was used to describe the anisotropy in kmin as follows: � � (25) kmin ¼ kB =2:48 n2=3 vl þ vt1 þ vt2

To summarize, anisotropies in elastic moduli, sound velocities and thermal conductivities of trigonal TM2C (TM ¼ V, Nb, Ta) carbides were explored using first-principles calculations based on density functional theory (DFT). The calculated structural parameters and formation en­ thalpies coincide with the previous reported values, and the sequence of phase stability is Ta2C > Nb2C > V2C. According to the mechanical sta­ bility criterion, trigonal TM2C (TM ¼ V, Nb, Ta) carbides are mechani­ cally stable. In the light of ν, BH/GH and Cauchy pressure, TM2C carbides are ductile and the intrinsic ductility is in a sequence of Nb2C > V2C > Ta2C. The sequence of anisotropy in elastic modulus is V2C > Nb2C > Ta2C based on elastic anisotropic indexes (AU, Acomp, Ashear, A1 and A3), 3D surface constructions and planar projections of elastic modulus. Moreover, sound velocity of TM2C carbides also is anisotropic. Besides, the calculated minimum thermal conductivities kmin indicate that these TM2C carbides can be used as potential thermal insulating materials, and the anisotropy in kmin is in a sequence of V2C > Nb2C > Ta2C. Acknowledgments This work was supported by the Yunnan Ten Thousand Talents Plan Young & Elite Talents Project under Grant no.YNWR-QNBJ-2018-044, the National Natural Science Foundation of China under Grant no. 51761023, and the Reserve Talents Project of Yunnan Province under Grant no.2015HB019. References

Table 9 also listed the calculated minimum thermal conductivities in [100] and [001] directions by Cahill’s model. It is obvious that mini­ mum thermal conductivities in the [100] direction differ from those in the [001] direction, which indicates that the minimum thermal con­ ductivities of TM2C are anisotropic. Besides, kmin[100] is always slightly higher than kmin[001], indicating the faster heat transfer rate in the [100] direction. This is attributed to the atomic arrangement that big TM atoms are parallel to the [001] direction in trigonal TM2C, which lead to that the phonon free path in the [001] direction is smaller than that in the [100] direction, and thus, kmin[001] is smaller than kmin[100]. According to Clarke’s model, the anisotropy of minimum thermal conductivity can be described by the anisotropy of Young’s modulus. Therefore, 3D surface constructions and their planar projections of minimum thermal conductivities of TM2C carbides were plotted in Fig. 7 to investigate the anisotropy of minimum thermal conductivity. One can see that TM2C carbides have a strong anisotropy in minimum thermal conductivity. The degree of deviation of 3D graphs from sphere de­ creases from V2C to Ta2C, indicating that the sequence of anisotropy in kmin is V2C > Nb2C > Ta2C. The planar projections can offer more details about the directional dependence of kmin for TM2C. Fig. 7(d), (e) and (f) show that kmin of TM2C on the (100) plane has stronger anisotropy than that on the (001) plane. The projections on the (001) plane are circular, while those on the (100) plane are four-petal shaped. V2C shows the strongest anisotropy in kmin on the (100) plane, and followed by Nb2C and finally by Ta2C. Table 9 lists the directional kmin of TM2C carbides in the [100] and [001] directions from Clarke’s model according to Fig. 7. It can be found from Table 9 that kmin in the [001] direction differs from those in the [100] direction, which means that kmin of TM2C is anisotropic. More­ over, the ratio of kmin in the [100] and [001] directions was used to evaluate the degree of anisotropy in kmin. The larger the ratio deviates

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