Introduction

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. So they are widely used in functional coatings, corrosion protection, high-temperature structural materials, microelectronics and catalysts. Vanadium form the stoichiometric carbides such as VC and the non-stoichiometric cubic monocarbide VCy which are all with B1 (NaCl) structure. A feature peculiar to the structure of non-stoichiometric VCy is the intrusion of carbon atoms in the octahedral interstices of the metallic sites and the carbon atoms may occupy only a fraction of the interstitial sites and the rests are filled with structural vacancies. The carbon-atom vacancies arranging randomly on the fcc carbon sublattice is called disordered phases, while the ordered carbon vacancies take on a periodic arrangement. The disordered state of non-stoichiometric carbides is in thermodynamic equilibrium only at a high temperature, whereas the ordered state is in thermodynamic equilibrium at a temperature below 1300 K1. The disorder–order phase transition in the vanadium carbides and a cubic ordered V8C7 phase can arise in the VCy carbide over the range VC0.86–VC0.882,3. The low-temperature phase V8C7 does not have an fcc structure, it crystallizes in the space group P4332 with the vanadium atoms occupying positions slightly off the ideal fcc positions and the carbon atoms having an ordered arrangement on the interstitial sites4. The crystal structures of VC and V8C7 are presented in Fig. 1. The existence of ordered carbon vacancies can affect the stability, mechanical, thermal and electronic properties of vanadium carbides. The influence of the order-disorder transformation on the electrical resistivity of single crystals of vanadium carbide was studied by Shacklette and Williams4. They found that the electrical resistivity of the disordered vanadium carbide is significantly higher than that of ordered phase and the temperature of order–disorder transformation is 1124 ± 15 °C for V8C7. Emmons and Williams studied the thermodynamics of δ-VC1−x order-disorder transformation using powder samples5. Lipatnikov’s research results show that ordered structure produces much higher hardness and electrical conductivity6. They also investigated the disorder–order phase transformations in the region of homogeneity of a non-stoichiometric cubic vanadium carbide, VCy (0.66 < y < 0.88) and constructed the equilibrium phase diagram of the V–C system which allows for the formation of ordered phases in a non-stoichiometric vanadium carbide1. Xing et al. reported a new type vanadium carbide V5C3 and predicted its mechanical properties7. In our previous work, we calculated the mechanical and thermal properties of the ordered vanadium carbides system including VC, V2C, V4C3, V6C5 and V8C78. But the difference for the thermodynamic and mechanical properties of ordered vanadium carbides have not been reported so for. This work contains two aspects. One is the difference for relatively thermodynamic stability and mechanical properties for the ordered stoichiometric VC and non-stoichiometric V8C7 at high temperature are obtained and compared with the theoretical and experimental results. The other one is a simple way is proposed to characterize the anisotropy of lattice thermal conductivity based on the Cahill’s model.

Figure 1
figure 1

Crystal structure of VC and V8C7.

(a) 2 × 2 × 2 supercell of VC with 32 vanadium atoms and 32 carbon atoms; (b) Unit cell of V8C7 with 32 vanadium atoms and 28 carbon atoms.

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Results and Discussion

Crystal structure

Figure 2 exhibits the carbon vacancy site in V8C7 which is not an fcc structure. Details for the structure of VC and V8C7 can be seen in Table S1. By comparing the simulated XRD of VC and V8C7 with the experimental results of V8C7 and VC1−x, we find our simulated results are in agreement with the experimental results9,10. And from the XRD pattern shown in Fig. 3, it is difficult to distinguish the ordered VC, V8C7 and disordered VC1−x. So we investigate the difference of the mechanical and thermal properties of them under different temperature.

Figure 2
figure 2

The crystal structure of VC and the ordered carbon vacancy site in V8C7.

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Figure 3
figure 3

The simulated XRD of VC and V8C7.

The results are compared with the experimental results of V8C7 and VC1−x9,10.

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Thermodynamic properties

The calculated results of thermal properties from Helmholtz free energy at elevated temperatures are showing in Fig. 4. From Fig. 4(b), we can see that the linear expansion coefficient of V8C7 is larger than VC during the temperature from 0 K to 1000 K, which leads to the Lattice constant of V8C7 increases more rapidly than the 2 × 2 × 2 supercell of VC with the temperature increasing (Fig. 4(c)).

Figure 4
figure 4

(a) The curve of the volume dependence of total energy; (b) The linear expansion coefficients of VC and V8C7 at different temperature; (c) Lattice constant of V8C7 and 2 × 2 × 2 supercell of VC as a function of temperature.

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Moreover, the obtained thermodynamic parameters, such as entropy (S) and Gibbs free energy (G), of VC and ordered V8C7 as a function of temperature exhibit in Fig. 5, which are in good agreement with the experimental results11. From Fig. 5(a), the entropy of V8C7 is larger than the entropy of VC, indicating that the V8C7 is thermodynamic more stable than VC at different temperature. The result can be confirmed by the Gibbs free energy of VC and V8C7 as shown in Fig. 5(b). The calculated Gibbs free energy of V8C7 is smaller than that of VC, which also suggests that the V8C7 is more stable at high temperature and easier to form than VC. What’s more, there is a little error between the experimental and theoretical data of entropy and Gibbs energy of VC, which can attribute to the experimental conditions and magnetic contribution to the entropy and Gibbs energy.

Figure 5
figure 5

(a) The entropy (S) of VC and V8C7 as a function of temperature; (b) The Gibbs free energy (G) of VC and V8C7 as a function of temperature companied with the experimental data11.

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The further analysis for the relatively thermodynamic stability of VC and V8C7 can be seen in Fig. 6. The Gibbs free energy of V32C32 is larger than the system of V32C28 and graphite, which shows that the V32C32 tends to decompose into V32C28 and graphite from the room temperature to 1000 K. And the result supports that the introduction of carbon vacancies in the B1 crystal structure and different symmetry increase the stability of carbides12. This founding explains the phenomenon that vanadium carbides phases in the high-vanadium iron and steel are not the stoichiometric compounds VC, but always the non-stoichiometric VC1−x such as V8C7 and V4C313.

Figure 6
figure 6

The Gibbs free energy of V32C32 and the V32C28 + graphite system as a function of temperature.

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Elastic properties at finite temperature

The obtained isothermal and isentropic elastic constants of VC and ordered V8C7 as a function of temperature from 0 K to 1000 K are displayed in Fig. 7(a,b). Different decreased amplitudes of and values with the increase of temperature could be found for VC and V8C7. It can be seen clearly that the elastic constants of VC decrease slightly and approach zero slope with the increase of temperature. The reduction in elastic constants values is more pronounced for V8C7 than VC and that such a decrease of C11 with temperature is more apparent. The value of is larger than for V8C7 and the deviation of our calculated results to experimental data for of V8C7 is about 5%.

Figure 7
figure 7

Temperature dependent elastic constants, mechanical modulus and intrinsic hardness of VC and V8C7.

(a) Elastic constants of VC; (b) Elastic constants of V8C7; (c) Bulk modulus of VC and V8C7; (d) Shear modulus of VC and V8C7; (e) Young’s modulus of VC and V8C7; (f) Intrinsic hardness of VC and V8C7.

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Using the elastic constants, the aggregate properties of polycrystalline such as the isothermal and isentropic bulk modulus (BT and BS), shear modulus (GT and GS) and Young’s modulus (ET and ES) of VC and V8C7 are calculated by VRH approach at different temperature. Moreover, the bulk modulus can also be obtained by the thermal equation of states (EOS)14:

where γ is the Grüneisen parameter, α is the thermal expansion coefficient and V is the equilibrium volume. The results are shown in Fig. 7(c–e). It can be found that the modulus values are larger for VC than those for V8C7, which is in according with the previous results. Apparently, the mechanical modulus all decease when temperature increases and an almost linear decrease could be discerned above 400 K. The response of mechanical properties of polycrystalline to the temperature is quite similar to that of elastic constants of single crystal. The isentropic modulus is larger than the isothermal one for VC and V8C7. And the decrease of mechanical modulus of V8C7 is more obvious than VC with the temperature increasing. In addition, the intrinsic hardness (HV) at different temperature is calculated by Chen’s and Tian’s models15,16,17,18 and the results are shown in Fig. 8(f). The intrinsic hardness obtained by Tian’s model is higher than that by Chen’s model for VC, but this is opposite for V8C7. All the two models describe the hardness with the elastic parameters of materials. But there is an extra intercept term (−3) in Chen’s model compared with Tian’s model. Indeed, proper hardness model should be selected according to the chemical bonding characteristics of the crystal. The predicted hardness of V8C7 is higher than VC but reduce pronouncedly at high temperature. It is known that a material would be less elastic heat resistive if its elastic properties are more sensitive to temperature. So these results indicate that the elastic heat resistant property of VC is superior to those of V8C7. Our theoretical results deviate from other calculated and experimental values of VC and V8C7 by less than 3% for BS and GS. The slight discrepancy may be attributed to the ignorance of the anharmonic effect and electron and maganetic contributionon on the kinetic energy. However, the deviation of our calculated results to experimental data for ES and HV of VC is larger than 10%, which may be caused by the relative strong anisotropy of Young’s modulus. The anisotropy of Young’s modulus for VC and V8C7 has been demonstrated and investigated in our previous work8. Another important reason is that the single crystal of vanadium carbide in the high vanadium iron and steel grows dependent on the orientations and the morphology of them is always dendritic as reported in ref. 19. So the characterized results of Young’s modulus and hardness are related to the location of the vanadium carbide20,21.

Figure 8
figure 8

The planner projections of the anisotropy of Young’s modulus of VC and V8C7 on (001) and (110) planes at different temperatures.

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The anisotropy of the Young’s modulus for VC and V8C7 can be evaluated by calculating the Young’s modulus as a function of crystallographic orientation. The expressions for Young’s modulus in the three principal directions [100], [010] and [001] for cubic crystal is given as22:

Moreover, in [1 0] and [1 1 1] directions, they are calculated by:

Where Sij represents the elastic compliances matrices, which can be calculated directly from . The temperature-dependent ES is plotted in polar coordinates and projected onto the plane as shown in Fig. 8. For VC and V8C7, the curve on (001) plane with maximum values along [100] direction and minimum values along the [110] direction could be observed, while on (110) plane, the maximum values and minimum values could be found along [001] and [10] directions, respectively. Furthermore, our calculated anisotropy of Young’s modulus at 0 K and 1000 K are all a little weaker than other theoretical and experimental results.

Minimum lattice thermal conductivity

This section we will discuss the anisotropy of lattice thermal conductivity for VC and V8C7 based on Cahill’s model23:

where n is the density of number of atoms per volume and vl, vt1 and vt2 are the longitudinal and transverse modes, respectively. Cahill’s model is suitable for studying the anisotropy of thermal conductivity, because the model already treats the total thermal conductivity as the summation of three acoustic branches. Detailed method for calculating the longitudinal and transverse modes can be found in ref. 24. The results are plotted for VC and V8C7 on the (001) and (110) crystallographic planes in Fig. 9. One finds that the anisotropy in the calculated thermal conductivities kl due to vl is stronger than that of vt1 and vt2. The shapes of these planar contours are related to the anisotropy in elasticity. At (001) crystallographic plane the most contribution to total thermal conductivities is from kl, while at (110) crystallographic plane it is kt1. The most intriguing conclusion obtained from these plots is that the total lattice thermal conductivity (k) of VC is almost isotropic. And the thermal conductivity of VC is a little larger than V8C7, which means that the carbon vacancy or defect can decrease the intrinsic lattice thermal conductivity.

Figure 9
figure 9

Anisotropy in thermal conductivity of VC and V8C7 at (001) and (110) crystallographic planes.

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It is known that the more strict treatment of the anisotropy in lattice thermal conductivity of single-crystalline involves the computation of phonon dispersions and applying the relaxation time approximation for phonon scattering. And another way to reveal the anisotropy in thermal conductivity is to evaluate the anisotropy in the Grüneisen constant. In this paper, we provide a simpler way to characterize the anisotropy of lattice thermal conductivity.

Conclusions

The intrusion of ordered carbon vacancies in non-stoichiometric vanadium carbides can significantly affect their stability, mechanical, thermal and electronic properties. The relatively thermodynamic stability, mechanical properties at high temperature and minimum lattice thermal conductivity of VC and ordered V8C7 are investigated by first-principle calculations combined with the quasi-harmonic approximation. V8C7 is more thermodynamically stable at high temperature and easier to form than VC according to the entropy and Gibbs free energy. The isothermal and isentropic elastic constants, mechanical modulus and intrinsic hardness are calculated and we found that the elastic properties of V8C7 are more sensitive to temperature than VC, which meaning that the elastic heat resistant properties of VC is superior to those of V8C7. The anisotropic Young’s modulus of VC and V8C7 are evaluated by calculating the Young’s modulus as a function of crystallographic orientation. Finally, the minimum lattice thermal conductivity of VC and V8C7 is predicted and a simple way to characterize the anisotropy of lattice thermal conductivity is proposed based on the Cahill’s model.

Methods

First-principle calculations details

In this work, all the calculations are performed based on the density functional perturbation theory (DFPT) as implemented in the CASTEP code25,26. A generalized gradient approximation (GGA) approach in the form of Perdew–Burke–Ernzerhof (PBE) was used to calculate the exchange and correlation functional27. The interactions between the ionic cores and the valence electrons were indicated by Ultrasoft pseudo-potentials (UPPs). A plane wave expansion method was applied for the optimization of the crystal structure. A special k-point sampling in the first irreducible Brillouin zone was confirmed by the Monkhorst–Pack scheme and the k point mesh was selected as 10 × 10 × 10. A kinetic energy cut-off value of 400.0 eV was used for the plane wave expansion in reciprocal space. So the selected values are suitable for the chosen system. The Broyden–Fletcher–Goldfarb–Shannon (BFGS) method was applied to relax the whole structure based on total energy minimization. The total energy changes during the optimization processes were finally converged to 1 × 10−6 eV and the forces per atom were reduced to 0.05 eV·Å−1. The second-order elastic constants (C11, C22 and C44) at the equilibrium volumes are calculated using an efficient stress–strain method, which was consisted of applying several different types of Lagrangian strains on crystals and calculated Cauchy stress tensor for each strain after optimizing the internal degrees of freedom28. Then the Voigt–Reuss–Hill approach was used to calculate the bulk (B), shear (G) and Young’s modulus (E) for polycrystalline crystals29.

Calculations of the thermodynamic properties

The thermodynamic properties of VC and V8C7 can be calculated by means of the quasi-harmonic approximation and thermal electronic excitation. For a system with the given volume (V) and temperature (T), the Helmholtz free energy F (V, T) is obtained through combining vibrational Fvib (V, T) and electronic Fel (V, T) free energy as follows30,31,32.

where Estat(V) is the static energy at 0 K and volume V. g (ω, V) is the phonon density of states at phonon frequency ω and volume V. Moreover, the entropy (S (V, T)) is calculated by the vibrational and thermal electronic contribution (Sel (V, T))33,34.

Finally, the temperature dependent equilibrium volume V (T) and Gibbs free energy G (T) are obtained by minimization of the following expression35 using the thermal equation of states (EOS)14:

From the results of equilibrium volume V (T), the volume coefficient of thermal expansion (CTE) αV can be calculated by the following equation36,37:

For the cubic crystal, the linear expansion coefficient (αl) and volumetric coefficient (αV) are related by αV = 3αl.

Calculation of the temperature-dependent mechanical properties

The temperature-dependent isothermal elastic properties can be estimated by a quasistatic approach, which is assumed that the change of elastic properties is mainly caused by volume change and the anharmonic effect as well as the contributions due to the kinetic energy and the fluctuation of microscopic stress tensors are ignored38,39. The procedure for calculating the temperature-dependent isothermal elastic constants () can be summarized in three steps as follows: (i) calculating the static elastic constants at 0 K as a function of volume Cij (V) using the stress-strain energy method; (ii) predicting the volume change as a function of temperature at ambient pressure V (T, 0) using the quasiharmonic approximation40; (iii) then the temperature dependence of isothermal elastic constants can be derived as Cij (T) = Cij (T(V)).

Most of the experimental data of elastic stiffness constants are usually reported as isentropic elastic stiffness constants, therefore the calculated isothermal elastic stiffness constants () must be converted to the isentropic ones () with their thermodynamic relationship given by Davies41. For cubic crystals, the thermodynamic relations can be simplified as42:

where CV and CP are heat capacities at constant volume and pressure, respectively. Details about the estimations of CV and CP from Helmholtz energy and the results can be found in our previous work8.

Additional Information

How to cite this article: Chong, X. Y. et al. The effects of ordered carbon vacancies on stability and thermo-mechanical properties of V8C7 compared with VC. Sci. Rep. 6, 34007; doi: 10.1038/srep34007 (2016).