Electronic structures mechanical and thermal properties of V–C binary compounds

Abstract

The electronic structure, mechanical and thermal properties of VC, α-V₂C, β-V₂C, V₄C₃, V₆C₅ and V₈C₇ are investigated systematically by the first principles calculation using density functional theory combined with the Debye quasi-harmonic approximation. Formation enthalpy is calculated and used to estimate the stability of the V–C binary compounds. The electronic structures and chemical bonding characteristics are analyzed by the band structures, density of states and Mulliken population analysis. The elastic constants of single crystal, hardness, bulk, shear, Young's modulus and Poisson's ratio of the polycrystalline crystal are obtained and compared with the experimental results. The anisotropic mechanical properties are discussed using the anisotropic index, three-dimensional surface contours and their planar projections on different planes of the bulk and Young's modulus. VC exhibits the largest mechanical modulus because of its strong chemical bonding and α-V₂C reveals the weakest elastic anisotropy. The specific heats at constant pressure and volume and the thermal expansion coefficients of the V–C binary compounds versus T from 0–1000 K are calculated and the largest and smallest thermal expansion coefficients are attributed to α-V₂C and V₄C₃, respectively.

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1. Introduction

Stoichiometric and non-stoichiometric vanadium carbides display a unique combination of thermal and mechanical properties such as high hardness, high melting point, high-temperature strength, corrosion and wear resistance and very efficient electrical and thermal conductivities.^1–3 Owing to this excellent performance, they are widely used in functional coatings, corrosion protection, high-temperature structural materials, microelectronics, catalysts and thermal barriers.⁴ These carbides are also used as advanced enhanced phases in metal matrix composites.⁵ Moreover, vanadium carbides are regarded as one important kind of hard constituents and grain refiners in the high speed steels.

Numerous experimental investigations for vanadium carbides have been published. Liu et al.⁶ synthesized the vanadium carbide (VC) nanoparticles by a novel refluxing-derived precursor and characterized them by X-ray diffraction, Raman spectroscopy, scanning electron microscopy and X-ray photoelectron spectroscopy. Zhang et al.⁷ obtained the vanadium carbides at room temperature using a planetary ball mill and discussed their formation mechanism. They found that the as-produced VC_x (x < 1) which had the same structure as VC transformed into V₈C₇ when annealed at a temperature higher than about 750 °C. Ma et al.⁸ made contribution to disclose good thermal stability and oxidation resistance of VC below 350 °C in air. Jun et al.⁹ performed atomic-scale direct observation of hydrogen-trapping sites in vanadium carbide (VC) precipitation steel with a 3DAP analysis and confirmed the platelets had the chemical composition V₄C₃. Zhong and his co-workers⁵ systematically investigated microstructure and properties of the in situ V₈C₇ particulate-reinforced iron matrix composites. Meanwhile, numerous researches have investigated in detail the electronic structure, the mechanical and thermodynamical properties of vanadium carbides. Ozoliņš and Häglund¹⁰ studied the effective cluster interactions and formation enthalpies of substoichiometric VC_1−x. Joshi and Paliwal¹¹ investigated the structural and bonding properties of VC by an ab initio linear combination of atomic orbitals method. Mechanical properties and electronic structures of VC, V₄C₃ and V₈C₇ were studied and the effect of carbon vacancies was discussed.^12–14 The specific heats of VC was also calculated using different models.^15,16 What's more, structure, vacancy formation, and strength of bcc Fe/V₄C₃ interface were also explored by Kaoru Nakamura and Toshiharu Ohnuma.¹⁷ However, the relationship between mechanical properties, electronic structure and the elastic anisotropy of vanadium carbides are rarely discussed in the literature so far.

In this paper, the first principles calculations are employed to investigate systematically the stability, electronic structure, anisotropic mechanical properties of V–C binary compounds. The thermal properties such as special heats and thermal expansion coefficients of the vanadium carbides are also studied systematically using the Debye quasi-harmonic approximation. The contribution of this work would help to explain the mechanism of some experimental phenomenon and choose the proper vanadium carbides for different purpose.

2. Methods and computational details

In this paper, the crystal structures of vanadium carbides including VC, V₂C, V₄C₃, V₆C₅ are prepared from references and V₈C₇ are adopted though the symmetry of crystal structures and group theory. For V₂C, two polymorph structures are α-V₂C (orthorhombic, Pbcn) and β-V₂C (hexagonal, P6₃/mmc). VC is considered as NaCl-type structure (cubic, Fm3̄m), and the structure of V₄C₃ is cubic rock-salt type which has a C vacancy in the center of VC structure.¹² The crystal structure of V₈C₇ (cubic, P4₃32) is from that the positions of the C vacancies in VC_1−x (Fm3̄m) changed from disorder to order.⁷ What's more, V₆C₅ with hexagonal structure is also investigated in this work.

The calculations are conducted using the Cambridge sequential total energy package (CASTEP) based on the density functional theory (DFT).¹⁸ Norm-conserving pseudopotentials (NCPPs) are used for electron–ion interactions, and a plane wave expansion method is applied for the optimization of crystal structure. The plane wave cut-off energy is chosen to be 500 eV for all phases. The generalized gradient approximation within the functional proposed by Perdew, Burke and Ernzerhof (GGA-PBE) is employed to evaluate the exchange-correlation energy.¹⁹ The Grimme custom method for dispersion-corrected density functional theory (DFT-D) is adopted to improve the calculation accuracy for weak interaction.²⁰ The 3s²3p⁶3d³4s² and 2s²p² are treated as valence electrons configurations for V and C, respectively. The unpaired electrons of V atoms have little impact on stress–strain evaluations, so non-spin-polarized calculations are performed in the work, which can also obtain the reliable mechanical properties. The Brillouin zone is sampled with the Monkhorst–Pack scheme and the k point mesh is selected as 12 × 12 × 12 for all structures.²¹ The Broydene–Fletchere–Goldarbe–Shanno (BFGS) method is performed to optimize the crystal structure until the total energy changes are below 5 × 10⁻⁷ eV per atom and the forces acting on distinct atom are less than 0.02 eV Å⁻¹.

The negative values of cohesive energy and formation enthalpy indicate that the compounds are thermodynamically stable and the two energy parameters can be estimated by the following expressions:

where E_coh(V_xC_y) and ΔH_r(V_xC_y) are the cohesive energy and formation enthalpy of V_xC_y per atom, respectively; E_tot(V_xC_y) is the total energy of V_xC_y phase; E_iso refers to the total energy of a single V or C atom and E_bin is the cohesive energy of crystal of V or C, respectively (Fig. 1).

Fig. 1 The crystal structures of (a) VC, (b) α-V₂C, (c) β-V₂C, (d) V₄C₃, (e) V₆C₅ and (f) V₈C₇.

3. Results and discussion

3.1 Equilibrium lattice constants and stability

The calculated crystal parameters of the V–C binary compounds are given in Table 1. The obtained values in this work are in good agreement with the experimental values and other theoretical values. The calculated results are obtained at 0 K, while the experiment is measured at room temperature. Moreover, the lattice parameters can be underestimated or overestimated when different kind of exchange-correlation functional is used. Therefore, the little deviation can be attributed to the thermodynamic effects on the crystal structure during the experiments and different approximation methods for calculations. It is also evident that the lattice parameter of VC is larger than that of V₄C₃ and a half of that of V₈C₇, which may relate to the existence of C vacancy in V₄C₃ and V₈C₇. The stability of the V–C binary compounds can be judged from their cohesive energies and formation enthalpies. The results calculated within eqn (1) and (2) are tabulated in Table 1. The values of cohesive energies of the V–C binary phases increase in the following sequence: V₆C₅ < V₈C₇ < VC < α-V₂C < V₄C₃ < β-V₂C. The formation enthalpy is used to estimate the relative stability for the vanadium carbides, too.³⁰ The lower the negative values, the more stable the compound. It can be seen that the formation enthalpy of V₆C₅ (−0.541 eV per atom) is the lowest value, indicating the most stable phase V₆C₅ in the V–C binary compounds. On the other hand, β-V₂C has the highest formation enthalpy as −0.277 eV per atom, implying that it is less stable than other vanadium carbides. In addition, the formation enthalpy of V₄C₃ is −0.324 eV per atom, which is higher than that of VC, indicating that the C vacancy in the crystal structure of VC can decrease the stability of the compound. However, the formation enthalpy of V₈C₇ is lower than that of VC, which may be attributed to the different symmetry for the crystal structures of VC and V₈C₇. The results in this work consist with the conclusion by Wang et al.¹⁴

Table 1 Lattice constants, density, volume, cohesive energies (E_coh) and formation enthalpies (ΔH_r) of V–C binary compounds accompanied with the available theoretical (DFT calculations) and experimental values

Species		Lattice constants (Å)			ρ (g cm^−3)	V (Å^3)	Ecoh (eV per atom)	ΔHr (eV per atom)	ICSD
		a	b	c					Ref.
VC	Cal.	4.091	4.091	4.091	6.11	68.47	−9.370	−0.405	This work
	Cal.	4.161	4.161	4.161			−9.563		13
	Cal.	4.157	4.157	4.157			−9.138	−0.494	14
	Cal.	4.165	4.165	4.165		72.25			#22263
	Cal.	4.164	4.164	4.164					22
	Cal.	4.160	4.160	4.160					12
	Exp.	4.163	4.163	4.163					23
α-V2C	Cal.	4.495	5.628	4.929	6.07	124.68	−9.336	−0.466	This work
	Cal.	4.305	6.031	5.080					24
	Exp.	4.567	5.744	5.026		131.85			#9982
β-V2C	Cal.	2.884	2.884	4.797	5.47	34.56	−9.242	−0.277	This work
	Cal.	2.8878	2.8878	4.5743		33.04			#77564
	Cal.	3.045	3.045	4.409					24
V4C3	Cal.	4.044	4.044	4.044	6.02	66.12	−9.248	−0.324	This work
	Cal.	4.109	4.109	4.109					13
	Cal.	4.114	4.114	4.114			−8.988	−0.413	14
	Cal.	4.149	4.149	4.149					12
	Cal.	4.091	4.091	4.091					25
	Exp.	4.160	4.160	4.160					26
V6C5	Cal.	5.005	5.005	14.099	5.96	305.90	−9.480	−0.541	This work
	Cal.	5.09	5.09	14.4		323.09			#71098
V8C7	Cal.	8.181	8.181	8.181	5.96	547.56	−9.468	−0.522	This work
	Cal.	8.29	8.29	8.29			−9.598		13
	Cal.	8.326	8.326	8.326			−9.219	−0.607	14
	Exp.	8.331	8.331	8.331					27
	Exp.	8.334	8.334	8.334					28
	Exp.	8.340	8.340	8.340					29
	Exp.	8.330	8.330	8.330		578.072			#22177

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3.2 Electronic structures

In this paper, the band structures, density of states and Mulliken population analysis are used to analyze the electronic structures and chemical bonding characteristics of V–C binary phases. Fig. 2 shows the band structures of these compounds. It is obvious that all of them are good electrical conductor.³¹ More details about the electronic structures of V–C binary compounds can be acquired by means of total density of states (TDOS) and partial density of states (PDOS) as shown in Fig. 3. All the vanadium carbides are metallic compounds because of their finite TDOS at Fermi level. The value at Fermi level of β-V₂C is higher than other compounds, suggesting that β-V₂C is less stable than other vanadium carbides, which is in consistent with the previous analysis on the formation enthalpy. In the vicinity of Fermi level, the curve shape of total density of states is similar with the shape of V-d states for all the V–C compounds, indicating that the d bands of V atoms dominate the Fermi levels. At the energy range from −10 eV to 4 eV, the C-p bands are all overlapped with the V-d bands, which show the p–d hybridization between C atoms and V atoms. Therefore, strong covalent bonding exists in these compounds, which would be beneficial to their high incompressibility, shear strength and hardness. Moreover, the shapes of C-p states and V-d states are not completely consisting, implying the ionic character for the chemical bonding of the compounds. Additionally, for β-V₂C, the C-s bands are overlapped with the C-p bands in conduction bands, which lead to the covalent bonds between C atoms. Moreover, the conduction bands of β-V₂C extend to 20 eV, which is wider than those of other compounds. For the vanadium carbides with cubic structure, the contribution of C-p orbitals and V-d orbitals to the conduction bands decreases in the following order: VC > V₄C₃ > V₈C₇, and the possible reason is the existence of C vacancy in it.

Fig. 2 Band structures of V–C binary compounds. The dashed line represents the Fermi energy.

Fig. 3 The total density of states (TDOS) and partial density of states (PDOS) for V–C binary compounds. The dashed line represents the Fermi energy.

Population analysis provides more quantitative information for the transferred chargers in the V–C binary compounds, and the results are listed in Table 2. The Mulliken method is applied to calculate the overlap population for a specific kind of bond and the charges carried by distinct atoms. The average bond length and the overlap population are evaluated using the following equations:³²

here L̄(AB) and represent the average bond length and the mean bond population, respectively; N_i is the total number of i bond in the cell and L_i is the bond length of i type. The positive charges carried by V atoms vary from 0.34 to 0.60 electrons, while the negative charges carried by C atoms in α-V₂C have the smallest value −0.68, which is similar with other vanadium carbides. The results clearly indicate that the ionicity of V₄C₃ is slightly stronger than that of other compounds. There are two possible electron transfer paths intra or inter the C and V atoms.³³ The one refers to p–d hybridized covalent bonding between C and V, and the other one is caused by the metallic or weak covalent bonding among V atoms. In the front case, the excessive electrons are transferred from V to C, which is described as 3d4s (V) to 2p (C). The rearrangement of valence electrons intra C atoms is mainly due to the sp¹ hybridization, which leads to the strong covalent bonding between C atoms. Moreover, it is obvious that there are more electrons on C-p band of V₄C₃ and V₈C₇ when compared with VC, which is attributed to the C vacancy in V₄C₃ and V₈C₇ structures. For these carbides, all the V–C bonds show positive populations. β-V₂C owns the largest value of the mean V–C bond population as 0.82 electrons, indicating the stronger covalent interaction between V and C atoms. Meanwhile, all the C–C bonds shows negative populations, and the evaluated population values of the V–V bonds are also negative in these compounds except α-V₂C and V₄C₃, which indicates that the strong repulsion force may exist among them. The smallest V–V bond population value is in β-V₂C, which may make it less stable and deteriorate the mechanical properties. Furthermore, the calculated bond length of C–V is smaller than that of C–C bond and V–V bond. The longest C–V bond is 2.05 Å and the shortest C–V bond is 1.99 Å, which is in VC and α-V₂C, respectively. For the vanadium carbides with cubic structure, the obtained V–C bond population of VC is larger than that of V₄C₃ and V₈C₇. As the population is proportional to the strength of the bonding, the V–C bond in VC is stronger than that of V₄C₃ and V₈C₇. Thus, we deduce that the C vacancy decreases the strength of the V–C bond, which may be harmful for the mechanical properties of compounds. In addition, the total numbers of valence electrons are 68, 120, 30, 64, 294 and 528 for VC, α-V₂C, β-V₂C, V₄C₃, V₆C₅ and V₈C₇, respectively. It shows that the net bonding electrons account for 11.3% of total valence electrons for VC, and the value is 7.6%, 6.5%, 9.4%, 6.3% and 10% for α-V₂C, β-V₂C, V₄C₃, V₆C₅ and V₈C₇, respectively. The results express that chemical bonding in VC is stronger than other carbides. In summary, the chemical bonding of the V–C binary compounds is dominated by the V–C covalent bonds but also possesses the ionic and metallic character, which may lead to high melting point, high mechanical modulus and hardness and good electric conductivity.

Table 2 Population analysis of V–C binary compounds, the calculated average bond length (L̄(AB)) and the mean bond population () are shown

Species	Atom	s	p	d	Total	Charge (e)	Bond	L̄(AB) (Å)	(e)	Net bonding electrons
VC	C	1.45	3.15	0.00	4.60	−0.60	V–C	2.05	0.76	9.12
	V	2.12	6.51	3.78	12.40	0.60	V–V	2.89	−0.09	−0.54
							C–C	2.89	−0.15	−0.90
α-V2C	C	1.43	3.25	0.00	4.68	−0.68	V–C	1.99	0.29	6.96
	V	2.16	6.75	3.75	12.66	0.34	V–V	2.79	0.10	2.32
							C–C	2.83	−0.10	−0.20
β-V2C	C	1.43	3.16	0.00	4.59	−0.59	V–C	2.05	0.82	3.28
	V	2.09	6.54	3.78	12.41	0.59	V–V	2.92	−1.22	−1.22
							C–C	2.40	−0.10	−0.10
V4C3	C	1.45	3.18	0.00	4.63	−0.63	V–C	2.02	0.70	6.27
	V1	2.11	6.51	3.79	12.41	0.59	V–V	2.86	0.04	0.21
	V2	2.11	6.51	3.79	12.57	0.43	C–C	2.86	−0.15	−0.45
V6C5	C1	1.45	3.19	0.00	4.64	−0.64	V–C	2.02	0.30	27.12
	C2	1.44	3.17	0.00	4.61	−0.61	V–V	2.84	−0.10	−5.48
	C3	1.44	3.16	0.00	4.60	−0.60	C–C	2.89	−0.06	−3.06
	V1	2.11	6.58	3.80	12.49	0.51
	V2	2.11	6.57	3.80	12.48	0.52
	V3	2.11	6.57	3.80	12.47	0.53
V8C7	C1	1.45	3.19	0.00	4.64	−0.64	V–C	2.03	0.35	59.28
	C2	1.44	3.16	0.00	4.60	−0.60	V–V	2.83	−0.01	−0.72
	C3	1.45	3.18	0.00	4.63	−0.63	C–C	2.90	−0.04	−5.76
	V1	2.11	6.56	3.80	12.47	0.53
	V2	2.12	6.53	3.79	12.43	0.57

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3.3 Elastic properties

In this part, the independent elastic constants are calculated using the stress-strain method based on the generalized Hook's law as summarized in Table 3.³⁴ The obtained elastic constants of vanadium carbides all satisfy the Born–Huang's mechanical stability criterions,³⁵ implying all the compounds are elastically stable. Because of the symmetry of the crystal, the C₁₁, C₂₂ and C₃₃ of cubic VC are the same and it is evident that the values are the largest among all the compounds, which means that VC is hard to be compressed under the external uniaxial stress along the [100], [010] and [001] directions. α-V₂C has the smallest C₁₁, C₂₂ and C₃₃, indicating that it has low incompressibility compared with other compounds. C₄₄, C₅₅ and C₆₆ represent the shearing strength at (100), (010) and (001) crystal plane, respectively. V₆C₅ exhibits larger C₄₄ and C₅₅ than other compounds, while β-V₂C shows the largest C₆₆ among all the carbides. In addition, for β-V₂C, the values of C₄₄ and C₅₅ are smaller than half of the value of C₆₆. As a result, the shearing strength of β-V₂C at (100) and (010) planes is weaker than (001) plane. The mechanical modulus, such as the bulk modulus (B) and shear modulus (G) of polycrystalline crystal are evaluated within Viogt–Reuss–Hill (VRH) approximation using the elastic constants for single crystal.³⁶ Young's modulus (E) and Poisson's ratio (σ) are estimated by the following expressions:^37,38

E = 9BG/(3B + G) (5)

σ = (3B − 2G)/(6B + 2G) (6)

Table 3 The calculated independent elastic constants (GPa), bulk modulus (GPa), shear modulus (GPa), Yong's modulus (GPa), B/G values, Poisson's ratio (σ), universal anisotropic index (A^U) and percent anisotropy index (A_B and A_G) of V–C binary compounds

Species	C11	C22	C33	C44	C55	C66	C12	C13	C23	B	G	E	B/G	σ	A^U	AB (%)	AG (%)
a Cal. in ref. 25.b Cal. in ref. 13.c Exp. in ref. 39.d Cal. in ref. 24.e Exp. in ref. 40.
VC	748.0			181.8			138.7			341.8	223.9	551.3	1.53	0.23	0.33	0	3.16
	783^a			196^a			131^a			348^a
	615^b			178^b			154^b			304^b	210^b
										390^c	157^c	430^c		0.22^c
α-V2C	452.1	450.3	492.5	122.3	143.0	160.6	206.9	145.5	205.3	278.5	139.5	358.6	2.00	0.29	0.11	0.14	1.05
										244.17^d
β-V2C	549.2		529.3	94.5		194.6	160.0	225.3		316.3	137.4	360.1	2.30	0.31	0.55	0.08	5.18
V4C3	603.7			112.6			149.5			301.6	149.7	385.3	2.01	0.29	0.61	0.02	5.79
	632^a			115^a			151^a			311^a	145^a	378^a		0.298^a
	440^e			136^e			92^e
V6C5	504.9		512.1	228.6		189.6	125.8	154.8		265.7	197.8	475.4	1.34	0.20	0.21	0.05	2.01
V8C7	650.5			179.3			120.2			297.0	209.8	509.4	1.42	0.21	0.19	0	1.82
	619^b			161^b			128^b			292^b	212^b
	527^e			159^e			105^e

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The obtained values are presented in Table 3. VC has the largest B value as 341.8 GPa, indicating that VC is the most difficult one to be compressed under hydrostatic pressure in the V–C binary compounds, which is in consistent with the analysis of elastic constants. Meanwhile, the G and E of VC are also larger than other carbides. Because the intrinsic hardness is proportional to the shear modulus, the high hardness may correspond to VC. Based on the previous discussion of electric structures, the excellent mechanical properties of VC are attributed to the strong covalent bond in it. The Poisson's ratios of α-V₂C, β-V₂C and V₄C₃ are close to 0.3, which means that they have stronger metallic characters than other compounds. The result is in consistent with the analysis of DOS, too. Meanwhile, the ratios of B/G of α-V₂C, β-V₂C and V₄C₃ are all larger than the critical value as 1.75, indicating that they are less brittle than VC, V₆C₅ and V₈C₇.

Fig. 4 reveals the relationship between elastic parameters and the ratio of net bonding electrons accounting for total valence electrons. With the ratio increasing, the variation of B and C₁₁ has the same tendency and the tendency of C₄₄ is also similar with that of G and E. The results indicate that the relationship between C₁₁ and B is simple and clear and C₄₄ has a positive impact on G and E. In Fig. 4(b), we clearly see that the variations of B/G and Poisson's ratio have the same tendency.

Fig. 4 (a) The variations of C₁₁, C₄₄, bulk modulus (B), shear modulus (G) and Young's modulus (E), (b) the Poisson's ratio (σ) and B/G of the V–C binary compounds.

In order to describe the anisotropy of V–C binary compounds, the universal anisotropic index (A^U) and percent anisotropic index (A_B and A_G) proposed by Ranganathan and co-workers are defined by:⁴¹

where B_V, B_R, G_V and G_R are the bulk and shear modulus estimation within Voigt and Reuss approximations, respectively. For isotropic crystal, the values of anisotropic indexes are close to zero and the deviations from zero indicate anisotropy. The calculated results are listed in Table 3. A^U is a better indicator than other index to reflect the anisotropy of the elastic properties. V₄C₃ has the largest value of A^U, indicating that the elastic properties of V₄C₃ have the strongest anisotropy and the elastic modulus of α-V₂C is not strongly dependent on the different orientations. The results are confirmed by the values of A_G. Moreover, the largest value of A_B is only 0.14% in α-V₂C, suggesting that the anisotropy of bulk modulus is very weak for these compounds. However, the three-dimensional (3D) surface contour provides more intuitive and details of the elastic anisotropy. The bulk modulus and Young's modulus are plotted in spherical coordinates as a function of the crystallographic orientation. The 3D representation of bulk modulus and Young's modulus for hexagonal and orthorhombic crystals in any direction is given by:^42,43

Hexagonal crystal:

Orthorhombic crystal:

where S_ij are the elastic compliance constants, and l₁, l₂ and l₃ are the directional cosines. For cubic structure, the following equations are also suitable by assuming S₁₁ = S₃₃, S₁₂ = S₁₃, and S₄₄ = S₆₆. The obtained surface contours of bulk modulus are illustrated in Fig. 5. It is evidence that the bulk modulus of these carbides show weak anisotropy because the surface contours of the vanadium carbides are all close to an ellipsoid. α-V₂C has the relatively strong anisotropy and the result is in good agreement with the values of A_B. From the projections on the (001) and (110) planes, we find that anisotropy of bulk modulus of α-V₂C on (001) plane is stronger than that on (110) plane, while for β-V₂C, the condition is in reverse. In Fig. 6, the Young's modulus of V–C binary compounds show more anisotropic features than the bulk modulus due to the remarkable anisotropic geometry of the surface contour. VC, V₄C₃ and V₈C₇ have the similar surface construction which may relate to the similar crystal structure. Projections on the (001) and (110) planes are deviated from the regular ellipses, indicating the strong anisotropy of Young's modulus for all the compounds. The shapes of planar contours of VC, V₄C₃ and V₈C₇ are also alike and just different in size. Moreover, the planar contours of VC, V₄C₃ and V₈C₇ on the (001) and (110) planes are all polarized and show the maximum Young's modulus along the axes. For β-V₂C, the anisotropy of Young's modulus on (001) plane is weaker than that on (110) plane and the projection on (110) plane is strongly polarized, too. In addition, the Young's modulus of all the V–C binary compounds along [11̄0] direction are larger than that along [100], [010] and [001] directions.

Fig. 5 (a–f) The surface contours of the bulk modulus of V–C binary compounds. (g and h) The (001) and (110) planar projections, respectively.

Fig. 6 (a–f) The surface contours of the Young's modulus of V–C binary compounds. (g and h) The (001) and (110) planar projections, respectively.

Because of the symmetry of the crystal, Young's modulus of the compounds with different crystal structure would have the planar projections on different plane. Besides the projections on (001) and (110) planes, we obtain the planar contours on (010) and (100) planes for α-V₂C and on (100) plane for β-V₂C and V₆C₅. The results are presented in Fig. 7. From Fig. 7(a), we find that the difference between the maximum and minimum modulus of VC on (001) plane is 197.9 GPa, while on (110) plane, the value is 241.3 GPa. We conclude that the deviation of Young's modulus between the theoretical calculation and the experimental measurement may be due to the influence of the strong elastic anisotropy. In Fig. 7(b), the largest Young's modulus for α-V₂C on (010) and (100) planes are all 391.6 GPa and the smallest values are 339.7 GPa and 312.4 GPa on (010) and (100) planes, respectively. From Fig. 7(c), it is clearly to find that the largest and smallest value of Young's modulus for β-V₂C on (100) plane is 444.2 and 289.8 GPa, respectively, while for V₆C₅, the values are 435.3 and 500.2 GPa.

Fig. 7 (a) Planar projections on (001) and (110) plane for VC, (b) planar projections on (010) and (100) plane for α-V₂C, and (c) planar projections on (100) plane for β-V₂C and V₆C₅.

As the general enhanced phase in steel and ceramic hard coating, the hardness of the V–C binary compounds is necessary to be quantitatively investigated. According to Gao's work, there are two important factors that affect the hardness of a covalent material: the number of bonds per unit area and the strength of the bonding. So the equations of calculating the hardness are expressed as follows:⁴⁴

H_v^u(GPa) = 740P^u(v_b^u)^−5/3 (15)

where H_v is the hardness of the compound; H_v^u is the hardness of u type bond; v_b^u is the volume of u type bond; P^u is the Mulliken overlap population of u type bond; d^u refers to the length of u type bond and N^v is the total number of v type bond in the cell, respectively. In this work, the hardness of V–V bonds and C–C bonds are not considered because the hardness of metallic bond is not suitable to evaluate in this model. Moreover, the metallic bond is very soft and the hardness is much lower than of the covalent bond like V–C bond. The values of hardness for the V_xC_y compounds are presented in Table 4. The hardness of V₈C₇ is 36.37 GPa, which is the largest among these compounds, while α-V₂C has the smallest hardness value as 13.76 GPa.

Table 4 The predicted hardness of V–C bonds and V–C binary compounds

Species	Bond	d^u(Å)	P^u	N^u	vb^u	Hv^u (GPa)	Hv (GPa)
a Exp. in ref. 45.b Cal. in ref. 14.
VC	V–C	2.05	0.76	12	5.71	30.87	30.87
							29^a, 28.9^b
α-V2C	V–C	1.98	0.28	8	5.09	13.75	13.76
	V–C	1.99	0.31	8	5.17	14.85
	V–C	2.01	0.28	8	5.33	12.76
β-V2C	V–C	2.05	0.82	4	8.64	16.68	16.68
V4C3	V–C	2.02	0.62	6	7.35	16.52	18.35
	V–C	2.02	0.85	3	7.35	22.65
V6C5	V–C	1.91	0.42	6	2.86	53.86	28.74
	V–C	1.95	0.38	6	3.05	43.94
	V–C	1.96	0.37	6	3.09	41.70
	V–C	2.00	0.32	12	3.29	32.60
	V–C	2.01	0.30	6	3.34	29.81
	V–C	2.02	0.29	6	3.39	28.11
	V–C	2.02	0.30	6	3.39	29.08
	V–C	2.03	0.29	6	3.44	27.42
	V–C	2.04	0.29	6	3.49	26.76
	V–C	2.05	0.28	6	3.54	25.21
	V–C	2.08	0.24	18	3.70	20.10
	V–C	2.09	0.24	6	3.75	19.62
V8C7	V–C	1.91	0.43	24	2.71	60.25	36.37
	V–C	2.00	0.38	24	3.12	42.30
	V–C	2.03	0.36	24	3.26	37.20
	V–C	2.03	0.38	24	3.26	39.26
	V–C	2.06	0.30	24	3.41	28.80
	V–C	2.07	0.32	24	3.45	29.99
	V–C	2.10	0.30	24	3.61	26.16

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3.4 Thermal properties

In order to investigate thermodynamic properties of V–C binary compounds, we estimate the Debye temperature by using the following relations:^46,47here Θ_D is the Debye temperature, h is the Planck's constant, k_B is the Boltzmann constant, N_A is the Avagadro's constant, n is the number of atoms per formula, M is the molecular weight, v_l and v_t are the longitudinal and shear velocities, respectively, ρ is the theoretical density of the compound. The calculated Debye temperature and acoustic velocities are shown in Table 5. Debye temperature represents the strength of covalent bonding in the crystal structure. As can be seen from Table 5, VC has the largest Θ_D value as 977.7 K, indicating the covalent bonds in VC are stronger than other carbides and the conclusion is in coincide with the above discussion about the electric structures. α-V₂C has the lowest Θ_D as 730.5 K, implying the metallic bonds among V atoms in α-V₂C. The largest acoustic velocity is from VC, which attributes to the largest bulk modulus and shear modulus of VC among all the vanadium carbides. Debye temperature is used to estimate the heat capacity (C_p) at low temperature using following eqn (21)–(23):⁴⁸

C_p = γT + βT³ (21)

where γ is directly related to electron density of states, β relates to phonon excitations, k_B is the Boltzmann constant, D_f is the density of states (DOS) value at the Fermi level, R is the molar gas constant and n is the total number of atoms per formula unit. So the heat capacity is consisted of two parts: electron and phonon excitations. The calculated values are summarized in Table 5. Because of the smallest DOS value at the Fermi level, VC has the smallest γ value as 2.54 × 10⁻³ J (mol⁻¹ K⁻²), indicating the weakest metallic character for VC among V–C binary compounds. Because eqn (21) is accurate when , we plot C_p versus T in the 0–70 K temperature range for V–C binary compounds in Fig. 8. When T < 39 K, the values of C_p follow the order VC < α-V₂C < β-V₂C < V₆C₅ < V₄C₃< V₈C₇. This is consistent with the sequences of γ and it indicates that electrons would dominate the heat capacities of compounds. As the temperature exceeds 63 K, phonon excitations make mainly contribution to the heat capacities and the trend of C_p is: VC < β-V₂C < α-V₂C < V₄C₃ < V₆C₅ < V₈C₇.

Table 5 The sound velocity, Debye temperature (Θ_D), DOS value at Fermi surface (D_f), theoretically predicted γ and β values of V–C binary compounds

Species	νl (m s^−1)	νt (m s^−1)	νm (m s^−1)	ΘD (K)	Df (states per eV f.u)	Cp = γT + βT^3
						γ (10^−3, J (mol^−1 K^−2))	β (10^−5, J (mol^−1 K^−4))
VC	10237.2	6053.5	6706.0	977.7	1.08	2.54	0.42
α-V2C	8747.8	4793.9	5344.9	730.5	1.70	4.00	1.49
β-V2C	9556.0	5011.9	5605.5	739.9	2.55	6.00	1.44
V4C3	9124.5	4986.7	5561.0	784.3	5.52	13.00	2.81
V6C5	9425.0	5760.9	6361.1	903.1	4.94	11.63	2.90
V8C7	9837.0	5933.1	6560.1	935.9	7.19	16.93	3.55

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Fig. 8 Low temperature heat capacity of V–C binary compounds plotted in the range of 0–70 K.

The heat capacity at high temperature and thermal expansion coefficients of V–C binary compounds are calculated using the Debye quasi-harmonic approximation. Firstly, the specific heat capacity at constant volume (C_V) as a function of temperature is calculated by:⁴⁹

where D(Θ_D/T) is the Debye integral. The heat capacity at constant pressure (C_P) and the volume thermal expansion coefficient (β) is calculated with:⁵⁰

C_P = C_V + VTB_Tβ² (25)

here γ represents the Grüneisen parameter and B_T is the isothermal bulk modulus determined by⁵¹in which V is the equilibrium volume. The evaluated specific heats from 70–1000 K are illustrated in Fig. 9. The lower bound and upper bound of C_P and C_V for V–C binary compounds are attributed to VC and V₈C₇, respectively. At very low temperature, the two specific heats are basically the same. With the temperature increasing, the C_p values and the C_V values of the carbides show similar temperature dependence and the difference between the two specific heats is due to the volumetric effect. Above the Debye temperature, the specific heat at constant volume approaches to a constant based on Dulong–Petit's law: 3nR, where n is the total number of atoms per formula and R is the gas constant.

Fig. 9 Heat capacity of V–C binary compounds: (a) at constant pressure; (b) at constant volume.

Fig. 10 depicts the calculated thermal expansion coefficients (β) of the six compounds as a function of temperatures. The volume thermal expansion coefficients of V–C binary compounds decrease in the following order: α-V₂C > V₈C₇ > β-V₂C > VC > V₆C₅ > V₄C₃. At far below Debye temperature, the thermal expansion coefficient of α-V₂C, V₈C₇ and β-V₂C increases exponentially due to the changes of specific heats at low temperature. When the temperature is relatively high, the thermal expansion coefficient increases sharply with the growing temperature from 100 K to 300 K. The reason should be the stored thermal energy dominated by the lattice vibrations, which is increased rapidly due to the intensive phonon excitations and the anharmonic effect is greatly enhanced at this stage.⁵² When the temperature goes over 300 K, the thermal expansion coefficients of α-V₂C, V₈C₇ and β-V₂C gradually approach to a linear function with temperature. The propensity of increment becomes very moderate at high temperature. What's more, for α-V₂C, the shape of curve is close to V₈C₇ and β-V₂C, the high-temperature effect on the thermal expansion coefficients is more significant than V₈C₇ and β-V₂C. On the other hand, for VC, V₆C₅ and V₄C₃, the thermal expansion coefficients change slowly with respect to temperature and increased linearly very slightly.

Fig. 10 The calculated thermal expansion coefficients as a function of temperature for V–C binary compounds.

4. Conclusions

The stability, elastic properties, mechanical anisotropy and Debye temperature of V–C binary compounds are extensively studied by first principles calculations. The cohesive energy and formation enthalpy reveal that they are thermodynamically stable. Based on the analysis of electronic structures, the bonding behaviors of vanadium carbides are dominated by the V–C covalent bonds but also possess the ionic and metallic character. VC exhibits the largest bulk, shear and Young's modulus as 341.8, 223.9 and 551.3 GPa, respectively, which may owe to the strong chemical bonding in it. The mechanical anisotropy of V₄C₃ is the strongest and α-V₂C has the weakest anisotropy among all the carbides. Moreover, the hardness of the compounds is evaluated from 13 to 30 GPa, respectively. VC and α-V₂C have the largest and smallest Debye temperature as 977.7 K and 730.5 K, respectively. The thermal properties of vanadium carbides are calculated using the Debye quasi-harmonic approximation. At 0–70 K, the electron and phonon dominate the special heat in turn, respectively. At high temperature, the lower bound and upper bound of C_P and C_V for V–C binary compounds are attributed to VC and V₈C₇, respectively. The volume thermal expansion coefficients of V–C binary compounds vary from 0.17 × 10⁻⁵ K⁻¹ to 2.25 × 10⁻⁵ K⁻¹ at 1000 K. To summarize the systematical theoretical researches of V–C binary compounds in this work, a fundamental physical picture of the properties of the compounds can be established, which is helpful for the experimental research and application of V–C binary compounds in the future.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (nos 51171074 and 51261013).

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