A first-principles study on the mechanical and thermodynamic properties of (Nb_1−xTi_x)C complex carbides based on virtual crystal approximation

Abstract

Effects of component and temperature on mechanical and thermodynamic properties of (Nb_1−xTi_x)C complex carbides were studied. The results show that it is easier to mix a small amount of NbC with TiC to form Ti-rich complex carbides than mixing small amounts of TiC with NbC to form Nb-rich complex carbides. Elastic modulus, hardness, ductility and fracture toughness of the complex carbide can be optimized by controlling the fractions of Nb and Ti in the complex carbide. The electronic property analysis on the complex carbides reveals that strong metal–metal and metal–carbon interactions exist in NbC, while strong metal–carbon interactions but weaker metal–metal interactions in TiC. The metal–metal interactions in the complex carbides attenuated when Nb is replaced by Ti. The bulk modulus and linear thermal expansion coefficient of the complex carbides demonstrate that a small amount of Ti atoms in the complex carbides may effectively alleviate the loss of the bulk modulus at elevated temperatures.

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1. Introduction

Alloying with appropriate elements can effectively tailor or optimize material properties for optical,¹ magnetic,² catalytic^3,4 and mechanical applications.⁵ Studies on how the mechanical properties are affected by alloying with selected elements have generated substantial invaluable information or knowledge for the design of advanced engineering materials with enhanced strength, toughness and fatigue resistance etc.^6,7 As an example, the strength and toughness of high strength low alloyed (HSLA) steels can be readily improved and controlled without sacrificing their workability and weldability by alloying multi-elements such as Nb, Ti, V and Mo etc.,^8–11 which result in complex carbides with adjustable properties, stable multiphase microstructures, and enhanced strength by grain refinement and interphase precipitation.^12–14

Development of carbide-reinforced ferrous matrix composites has also attracted increasing interest.^15–18 Since NbC has its density similar to that of iron, great attempts have made in industry to reinforce cast irons and steels using NbC particles, which can be homogeneously distributed in the ferries matrix. In order to improve such metal–matrix composites, tailoring the properties of NbC through varying its constituents is a promising approach. For instance, partially replacing Nb by Ti or other elements to make complex carbides may lead to improved properties and adjustable density of carbides for reinforcing various metallic matrixes with homogenous particle distribution and maximized performance.

In order to appropriately utilize complex carbides and minimize possible side effects, comprehensive information on properties of the complex carbides is needed. For instance, the thermal stability of carbides, which could influence mechanical properties of the composites or HSLA steels due to coarsening or decomposition of carbide particles during thermomechanical processes. Steels are prone to cleavage fracture initiated by microcracks resulting from the structural inhomogeneity with the presence of coarse carbide.^13,19 Great efforts have been made to increase the high-temperature stability of carbides, to reduce the coarsening rate of carbides and to improve the coherent interface between the carbide and matrix.^19–21 Among various approaches, application of (Nb,Ti)C complex carbides is considered as a promising one to improve the performance of HSLA steels due to their capability of refining grain size and reducing the coarsening rate.^22,23 However, the knowledge about the complex carbide is rather limited. In order to better understand the complex carbide and maximize its benefits to the HSLA steel and carbide-reinforced metallic matrix composites, the information on how the component and temperature affect the properties of (Nb,Ti)C is highly desired.

The first-principles method has demonstrated its high effectiveness for in-depth understanding of advanced engineering material and its power in material design based on well-established computational technique and physical principles including structure-dependent hardness,²⁴ ductility²⁵ and toughness²⁶ etc. These relationships make it possible for using the first-principles method to analyze various properties of materials at atomic/molecular level or nano-scales. As for the variations in the degree of alloy disordering caused by mixing multiple constituent atoms, property analysis could be treated using different approaches, such as the virtual crystal approximation (VCA), coherent potential approximation (CPA), and computational alchemy.²⁷ However, CPA is not feasible for the first-principles energy calculation and the alchemy requires substantial computation time. Thus, VCA is popular for dealing with alloy disordering using the first-principles technique. The VCA method has been demonstrated to be successful in investigating the alloys with not only monovalent atoms but also heterovalent atoms.^28–30 In this study, the first-principles method incorporated with VCA was employed to investigate effects of the component and temperature on properties of (Nb_1−xTi_x)C complex carbide.

2. Methodology

The first-principles calculations based on density functional theory (DFT) were implemented with ABINIT package.^31,32 The norm-conserving-pseudopotentials³³ and Perdew–Burke–Ernzerhof Generalized Gradient Approximation (GGA) of exchange-correlation functional³⁴ were adopted for the calculation. The properties of (Nb_1−xTi_x)C complex carbides which crystallized into NaCl type structure were calculated, where the alloying pseudo-potential that consisted of two types of metal was constructed within a first-principles VCA scheme.²⁷ After convergence tests on the cutoff energy and density of k-points, an energy cutoff of 35 Hartree (1 Hartree = 27.211 eV) and a 16 × 16 × 16 k-point mesh were used to achieve self-consist convergence with the tolerant potential residual V(r) less than 10⁻¹² Hartree. The equilibrium lattice constants and elastic constants were calculated by means of finite displacement methods.^35,36 The elastic moduli of polycrystalline metals were calculated according to the Voight–Reuss–Hill bounds.³⁷ Ideal tensile strengths were calculated by stretching the crystal cell along 〈111〉 direction, and ideal shear strengths were calculated for the easily activated slip system i.e. applying shear deformation along (111)〈21̄1〉 of the (Nb_1−xTi_x)C structure, which is the basic sliding process to form stacking faults in face-centred-cubic (FCC) structure.³⁸ The calculations on ideal tensile strength and ideal shear strength were carried out under a relaxed mode, where except the stress component corresponding to the strain applied to the crystal cell, all other stress components were relaxed to a tolerance within 0.1 GPa and the force on each atom was less than 0.001 eV A⁻¹.

Hardness was calculated using the following relationship,³⁹

H_v = CG^α (1)

where G is shear modulus, C is a coefficient of 0.0194, α is the power law index of 1.337. The relationship between hardness and shear modulus described by eqn (1) works well for brittle materials.

According to S. Ogata et al.,²⁶ the shear strain energy density stored in the unit volume, W_shear, can be calculated as

where τ is the Cauchy stress tensor, J is the deformation gradient, dη is the interval of strain between each deformation steps, Ω₀ and Ω are volumes of cells in ground state and in stressed state, respectively. Tr is the trace of matrix in bracket. The integral is calculated by discretization according to trapezoidal rule. Likewise, the tensile strain energy density stored in the unit volume, W_tensile, is expressed as

Once the shear strain energy density and tensile strain energy density are obtained, the brittleness, β, of complex carbides is evaluated as

The fracture toughness, K_Ic, can be calculated as²⁶

K_Ic = D(β⁻¹)^1.4B(Ω₀/N)^1/6 (5)

where B is bulk modulus, Ω₀ the cell volume in ground state, N the number of atoms in supercell, D is a coefficient of 0.157 for intrinsically brittle materials.

3. Results and discussions

The validation of virtual crystal approximation for treating the alloy disorder was justified by comparing component-dependent elastic constants of the (Nb_1−xTi_x)C complex carbide, which were calculated using two models. One model deals with a cubic supercell of NaCl structure as shown in Fig. 1(a), in which the constituent atoms Nb and Ti are mixed according to the stoichiometry, and individual pseudopotentials of Nb and Ti are used for the calculation. This model can give accurate description of the interaction between atoms but increase the computational cost. Another model can also be used to treat the same lattice system but alloyed pseudopotential is used for metal atoms within VCA scheme. The calculated lattice constants, elastic constants (C₁₁, C₁₂, C₄₄), bulk modulus, shear modulus, Young's modulus and hardness of (Nb_1−xTi_x)C complex carbides are listed in Table I (see ESI†). Fig. 1(b) shows calculated elastic constants. One may see that the elastic constants calculated with the alloy pseudopotential are as good as those calculated using individual pseudopotentials. This consistency demonstrates that the alloyed pseudopotential is accurate to reflect the influence of alloy disordering in (Nb_1−xTi_x)C complex carbides. Moreover, the influence of constituent elements on the elastic constants of the complex carbide can be determined. With an increase in Ti content, the elastic constant C₁₁ gradually decreases. However, some increase in C₄₄ is observed. Since C₄₄ is usually used to evaluate the resistance to shear deformation,⁴⁰ a better component-mediated resistance to shear deformation could be achieved through varying the fractions of components in the complex carbide.

Fig. 1 (a) A NaCl-type crystal structure, in which particles in taupe denote carbon atoms, and particles in light blue denote metal atoms; (b) elastic constants of complex carbides calculated using two models: one was a cubic cell with individual pseudopotentials of Nb atoms and Ti atoms for calculation, and the other was the same cell structure but alloy pseudopotential within the VCA scheme was used for calculation.

3.1. Component-dependent energetic and elastic properties of (Nb_1−xTi_x)C complex carbides

The cohesive or formation energy is a crucial parameter to evaluate the thermodynamic stability of a compound. The formation energy of (Nb_1−xTi_x)C complex carbide was calculated using the following formula:⁴¹where E(Nb_1−xTi_xC) is the total energy of (Nb_1−xTi_x)C carbide, μ^bulk_Nb, μ^bulk_Ti and μ^bulk_C are the chemical potentials of Nb, Ti and C in the form of bulk substance, respectively. x is the atomic fraction of Ti atoms in the complex carbides. As shown by the curve with solid marks in Fig. 2(a), the formation energies of NbC and TiC are −41 kJ mol⁻¹ and −61 kJ mol⁻¹, respectively. When Nb in carbide is gradually replaced by Ti, the formation energy of the complex carbide decreases. The lower formation energy of TiC indicates that TiC is more thermodynamically favourable. With an increase in temperature, the structure stability of (Nb_1−xTi_x)C complex carbide is estimated by the changes in the Gibbs free energy (ΔG)⁴²where ΔH is the mixing enthalpy, E(Nb_1−xTi_xC), E(NbC) and E(TiC) are energies of Nb_1−xTi_xC, NbC and TiC, respectively. ΔS is the entropy, R is the ideal gas constant. The curves with hollow marks in Fig. 2(a) are the component-dependent Gibbs free energies at different temperatures. One may see that the formation of complex carbide (Nb,Ti)C is thermodynamically favourable and this is enhanced by increasing temperature. The calculation of the Gibbs free energies also indicates that phase decomposition of the complex carbide hardly occurs at high temperatures. Another interesting point is that, according to the variations in Gibbs free energy, mixing a small amount of NbC with TiC is easier to form Ti-riched complex carbides than mixing a small amount of TiC with NbC to form Nb-riched complex carbides.

Fig. 2 (a) The formation energies and the temperature dependent mixing Gibbs free energies of complex carbides, (b) the anisotropic ratios of complex carbides, (c) the bulk modulus of complex carbides, (d) the shear modulus of complex carbides, (e) the Young's modulus of complex carbides.

The influence of altering components on the structural anisotropy of (Nb_1−xTi_x)C complex carbides was examined based on the anisotropic ratio,^43–45 which is defined as . This ratio is a comparison between the tetragonal stiffness constant , and the shear stiffness constant, C₄₄. If the ratio A is closer to unit one, the structure is more isotropic, otherwise, the isotropic degree will decrease. Fig. 2(b) presents the anisotropic ratio of (Nb_1−xTi_x)C complex carbides. Although the (Nb_1−xTi_x)C complex carbide is crystallized into the type of NaCl structure, the isotropic degree increases as Nb atoms are gradually replaced by Ti atoms. The higher anisotropic degree of the complex carbides closer to the NbC end could lead to deterioration of the ductile plasticity.

The component-dependent bulk moduli are presented in Fig. 2(c). The calculated bulk moduli of NbC and TiC are 296 GPa and 227 GPa, respectively, which are consistent with experimental measurement.⁴⁶ With an increase in Ti content, the bulk modulus of complex carbide is decreased continuously. Fig. 2(d) illustrates variations in shear modulus of the complex carbide. As shown, with an increase in the Ti content, the shear modulus increases initially and then decreases with a maximal shear modulus of 199 GPa for Nb_0.5Ti_0.5C. A similar trend was also observed for Young's Modulus as illustrated in Fig. 2(e) with the maximal Young's modulus equal to 481 GPa.

3.2. Component dependent ideal strength, ductility, hardness and fracture toughness of (Nb_1−xTi_x)C complex carbides

In order to better understand toughness of the complex carbides, the information on their plasticity is desired. In principle, the plasticity may result from twinning,⁴⁷ phase transformation,⁴⁸ kink and sliding.⁴⁹ Among these processes, the sliding behaviour could be predominant, because the critical resolved shear stress (CRSS) for sliding is strongly correlated to the nucleation and motion of dislocations in solid, and in turn affect the yield process and work hardening.⁵⁰ Thus, the stress barrier or unstable stacking energy needs to be determined for evaluating the plastic behavior of the complex carbides. In this study, the maximal tensile stress, shear stress and relevant properties were calculated on the most likely slip system of (111)〈21̄1〉 in FCC structures. The deformation paths set for the stress barrier calculation are illustrated in Fig. 3(a). The crystal cell in equilibrium was rotated to make the (111) plane parallel to the a–b plane and lattice vector of 〈21̄1〉 point along b axis in the mode I. For ideal tensile strength calculation, the equilibrium cell was stretched along c axis to mode II. For the ideal shear strength calculation, the cell in equilibrium was distorted by rotating c axis around a axis to mode III. The calculated ideal tensile strength (ITS), the ideal shear strength (ISS), the tensile strain energy density, the shear strain energy density, the reciprocal of brittleness (β⁻¹) and the fracture toughness (K_Ic) of (Nb_1−xTi_x)C complex carbides are listed in Table II (see ESI†). Along the two deformation paths, stress–strain curves during deformation are plotted in Fig. 3(b) and (c). It is noticed that the resistance to tensile failure of the complex carbide is enhanced when the carbide is changed from NbC to TiC, while the trend is reversed for the resistance to the shearing failure.

Fig. 3 (a) Schematic illustration of deformation paths for the calculation of ideal tensile and ideal shear strengths. Vector c stands for the direction normal to the (111) plane, vector b stands for the sliding direction of 〈21̄1〉 in (111) plane. For the ideal tensile strength calculation, the deformation gradient, J = dη⋅(c ⊗ c) + I, was applied to the equilibrium cell, resulting in transformation from state I to state II. ⊗ is the tensor product of vector, dη is the interval of strain between each deformation steps, I is the unit matrix for the ideal shear strength calculation. The deformation gradient, J = dη⋅(c ⊗ b) + I, was applied to the equilibrium cell, resulting in transformation from state I to state III. (b) The stress–strain curve for the tensile deformation, (c) the stress–strain curve for shear deformation.

From the observed stress–strain relationship shown in Fig. 3(b) and (c), the component-dependent ideal tensile strength and ideal shear strength are determined and plotted in Fig. 4(a). As shown, the ideal tensile strength increases from 72.5 GPa to 84 GPa when the carbide is changed from NbC to TiC. However, the ideal shear strength decreases from 29 GPa to 25.5 GPa. In general, when a material subjected to deformation, the competition for releasing strain energy between creation of new surface by tensile cleavage and dislocation emission by shear sliding determines the intrinsic fracture behavior i.e. ductile fracture or brittle one. Here, the increase in ideal tensile strength and decrease in ideal shear strength may suggest that TiC is more ductile than NbC. Empirically, such competition was previously evaluated using the ratio of bulk modulus to shear modulus, K/G, or Poisson ratios.⁵¹ However, these parameters are not always effective. Thus, in order to more reliably evaluate the ductility, a parameter termed ‘brittleness’, β, defined by eqn (4) is utilized. The ductile degree of a material could be evaluated by the reciprocal brittleness, β⁻¹. A larger value of β⁻¹ corresponds to higher ductility. Fig. 4(b) presents calculated β⁻¹ for the complex carbides. As demonstrated, TiC has a higher value of 5.21, indicating that it is more ductile than NbC. This is also consistent with the isotropy analysis (see Fig. 2(b)), because the isotropic character is generally associated with higher ductility. In addition, it is noticed that the ductility is continuously improved when Ti content in the complex carbide increases and eventually turns NbC to TiC. It should be pointed out that the word ‘ductile’ is just a comparative description between the complex carbides with different fractions of contained components. For intrinsic ductile materials, the value of β⁻¹ is on the order of 10².²⁶ In the present case, β⁻¹ of the complex carbides is in the range of 1, indicating that they are intrinsically brittle, though TiC is relatively ductile than NbC.

Fig. 4 (a) The ideal tensile strength and ideal shear strength of complex carbides, (b) the ductility parameter of complex carbides, β⁻¹, (c) the hardness of complex carbides, (d) the fracture toughness of complex carbides.

Component-dependent hardness of the complex carbides was calculated using eqn (1) and results of the calculation are presented in Fig. 4(c). As shown, the hardness of complex carbide is somewhat mediated by varying the components in the carbide (Nb_0.5Ti_0.5)C, showing the maximal hardness of 23 GPa. Acchar et al.⁵² reported that hardness of NbC and TiC were around 21 GPa, which are similar to results of our calculation. However, it is also reported that hardness of TiC is higher than that of NbC.⁴⁶ The discrepancy could be attributed to different slip systems activated in NbC and TiC, respectively. Experimentally, the slip system is identified to be (111)〈11̄0〉 in single crystal NbC,⁵³ while that in TiC is (110)〈11̄0〉.⁵⁴ The present calculation shows that the ideal shear strength on slip system of (110)〈11̄0〉 in TiC is 40.5 GPa,²⁶ which is much higher than that of slip system of (111)〈11̄0〉 in NbC around 28.4 GPa. Thus, when the slip system of (110)〈11̄0〉 with the higher shear strength is activated in TiC, it will lead to a higher value of hardness as observed in experiments. Moreover, there is another interesting point worth for further studies why in TiC sliding does not occur preferentially on the slip system of (111)〈11̄0〉 with a lower shear strength of 25.6 GPa, but on slip system of (110)〈11̄0〉 having a higher shear strength of 40.5 GPa.²⁶

Using eqn (5), the component-dependent fracture toughness was calculated and plotted in Fig. 4(d). The calculated fracture toughness ranges from to , which is on the same order of magnitude as the fracture toughness measured by indentation. The experimentally measured fracture toughness of NbC is about ,⁵⁵ and that of TiC is about .⁵⁶ Although the brittle nature of complex carbides revealed by the small fracture toughness, some improvement can still be made by mediating the component fractions. As observed, the maximal fracture toughness of is achieved for (Nb_0.25Ti_0.75)C. Although the improvement in the fracture toughness is slight, it could facilitate the dislocation to move through the precipitated carbides by looping rather than by cutting.⁵⁷

3.3. Electronic properties of (Nb_1−xTi_x)C complex carbides

Fig. 5 presents the band structures and density of states of the complex carbides. Fig. 5(a) illustrates the Brillouin zone, in which the band structures of complex carbides were calculated. The paths indicated by the red lines connect symmetrical points. According to the projected density of states (PDOS), which is plotted in narrow vertical panel next to band structure in Fig. 5(b)–(f), it is noticed that the Bloch orbitals in the complex carbides are hybridized into three types of bands. The bottom one consists of s Bloch orbitals, the middle one consists of p–d Bloch orbitals, and the top one consists of d–d Bloch orbitals. In the band structures, an occupied p–d Bloch orbital is denoted by a red symbol curve. From the location of Fermi energy level (as indicated by the dash line), the types of chemical bonding in the complex carbides can be identified.^58,59 As demonstrated, the Fermi energy level shifts from the d–d band to p–d band when the carbide changes from NbC to TiC. This change indicates that, except the interaction between carbon atom and transition metal atom through p–d band, the interaction between two transition metals through d–d band attenuated with the decrease of Nb concentration in the complex carbide. Until TiC, the Fermi energy level located at a pseudo-gap between d–d band and p–d band, implying that chemical bonding in TiC is dominated by the interaction between carbon atom and transition metal atom. The analysis on the chemical bonding characteristics of the complex carbides is consistent with the experimental observations, where both metal–metal bond and metal–carbon bond are strong in NbC, while metal–carbon bond is strong but metal–metal bond is weak in TiC.⁴⁶ Furthermore, such distinction in chemical bond between TiC and NbC may also be reflected by their physical properties, such as electrical resistivity. For instance, the electrical resistivity of TiC (50 ± 10 μΩ cm) is much higher than that of NbC (35 μΩ cm).⁴⁶

Fig. 5 (a) The illustration of the first Brillouin Zone and the paths along which the band structures were calculated, the band structure and density of state for (b) NbC, (c) (Nb_0.75Ti_0.25)C, (d) (Nb_0.5Ti_0.5)C, (e) (Nb_0.25Ti_0.75)C, and (f) TiC. In the density of state panels; the area hatched in grass-green colour denotes p–d band, and the area hatched in purplish colour denotes d–d band.

3.4. Thermodynamic properties of (Nb_1−xTi_x)C complex carbides

Fig. 6 presents the phonon dispersion curve and phonon density of states for the complex carbides. The calculated phonon dispersion curves of NbC and TiC are in good agreement with experiment results,⁶⁰ which are indicated by blue dots in Fig. 6(a) and (e). A stiffening effect in the acoustic branch, indicated by red arrows, is observed when the carbide changes from NbC to TiC. Considering that the phonon frequency and atomic force constant in metals are influenced by the electron density of states around Fermi level due to the screening effect,^61,62 the lower electron density of states usually leads to poor screening effect and stiffens the phonon frequency. Thus, the stiffening in acoustic branch can be ascribed to the fact that the electronic density of states around Fermi level in TiC is much smaller than that in NbC, as shown in Fig. 5. Moreover, as shown in Fig. 6, the phonon density of states for the acoustic branch extends to higher frequency when the carbide changes from NbC to TiC. As confirmed by thermal conductivities of TiC and NbC,⁴⁶ which are respectively 21 W m⁻¹ K⁻¹ and 14.2 W m⁻¹ K⁻¹, such increase in the phonon frequency may result in higher thermal conductivity.

Fig. 6 Phonon dispersion curves and phonon densities of state for (a) NbC, (b) (Nb_0.75Ti_0.25)C, (c) (Nb_0.5Ti_0.5)C, (d) (Nb_0.25Ti_0.75)C, and (e) TiC, respectively. The blue dots denote the experimental data from ref. 60.

Based on the calculated phonon structures, thermodynamic properties of the complex carbides were calculated using the finite volume method with quasi-harmonic approximation.⁶³ Vibrational free energies are calculated on a set of primitive cells with variable volumes, 0.97V₀, 0.98V₀, 0.99V₀, V₀, 1.01V₀, 1.02V₀, 1.03V₀. Here V₀ is the volume in equilibrium, and the changes in the volume are achieved by homogeneously compressing or expanding the primitive cell. Once the volume dependent vibrational free energies are obtained, the volume of cell and bulk modulus corresponding to the minimal free energy at a given temperature can be obtained by fitting volume dependent free energy according to the third-order Birch–Murnaghan equation of states:⁶³

where E₀ is the total energy of primitive cell in equilibrium, V₀(T) andB₀(T) are volume, bulk modulus of cell at given temperature corresponding to the minimal free energy. B′₀(T) is the derivative of bulk modulus with respect to the pressure. Since the volume is temperature dependent, the linear thermal expansion coefficient was calculated as

The calculated temperature-dependent bulk modulus of complex carbides are plotted in Fig. 7(a). In low temperature region, NbC shows larger bulk modulus, which decreases as the carbide changes from NbC to TiC. With an increase in temperature, the bulk modulus of NbC decreases faster than Ti-contained complex carbides. This is consistent with the high-T behaviors of NbC and TiC.⁶³ Intriguingly, a small amount of Ti atoms can effectively enhance the high-temperature strength of the complex carbides, e.g., Nb_0.75Ti_0.25C. In view of that bulk modulus is function of the valence electron density,⁶⁴ and the valence electron density is influenced by the volume expansion at high temperature, thus, the linear thermal expansion coefficients of the complex carbides were calculated and are plotted in Fig. 7(b). The calculated thermal expansion coefficients are in good agreement with experimental measurements.⁴⁶ NbC shows the maximal thermal expansion coefficient at higher temperatures, which means that the volume expansion of NbC is greater than the other complex carbides. The high volume expansion of NbC could be the reason why its bulk modulus decreased dramatically at elevated temperatures. The other complex carbides have smaller thermal expansion coefficients, compared to NbC, which is beneficial to the thermal shock resistance or the resistance to failure caused by the thermal stress.

Fig. 7 (a) The temperature-dependent bulk modulus of complex carbides, (b) the linear thermal expansion coefficients of complex carbides, the experiment results are cited from ref. 46.

4. Conclusions

In summary, the influences of component and temperature on mechanical and thermodynamic properties of (Nb_1−xTi_x)C complex carbides are studied. The formation of the complex carbides shows different mixing trends; it is easier to mix a small amount of NbC with TiC to form (Nb_1−xTi_x)C complex carbides than mixing a small amount of TiC with NbC. At elevated temperatures, the formed complex carbides show good thermal stability. The component-dependent ideal tensile strength and ideal shear strength, ductility and fracture toughness of complex carbide are calculated. It is observed that the mechanical behavior can be optimized by mediating the component fractions in the complex carbide. The electronic property analysis on the complex carbide reveals that strong metal–carbon and metal–metal interactions exist in NbC, while TiC has strong metal–carbon interaction but weak metal–metal interaction. The attenuated metal–metal interaction in the complex carbides as the Ti fraction increases may explain corresponding decrease in bulk modulus of the complex carbide.

Effects of temperature on the bulk modulus and linear thermal expansion coefficient are analyzed. It is demonstrated that the thermal expansion coefficient of NbC increases faster than that of TiC at elevated temperatures, accompanied with a relatively rapid decrease in bulk modulus. Adding a small amount of Ti atoms to NbC appears to effectively alleviate the loss of bulk modulus at high temperatures.

Acknowledgements

The authors are grateful for the financial support from the Nature Science and Engineering Research Council of Canada and AUTO21.

References

1. R. J. A. Esteves, M. Q. Ho and I. U. Arachchige, Chem. Mater., 2015, 27(5), 1559–1568.
2. S. Shihab, H. Riahi, L. Thevenard, H. J. von Bardeleben, A. Lemaitre and C. Gourdon, Appl. Phys. Lett., 2015, 106(14), 142408.
3. X. F. Zhang, J. J. Guo, P. F. Guan, C. J. Liu, H. Huang, F. H. Xue, X. L. Dong, S. J. Pennycook and M. F. Chisholm, Nat. Commun., 2013, 4, 1924.
4. S. Zafeiratos, S. Piccinin and D. Teschner, Catal. Sci. Technol., 2012, 2(9), 1787–1801.
5. C. Suryanarayana and N. Al-Aqeeli, Prog. Mater. Sci., 2013, 58(4), 383–502.
6. G. S. Cho, K. H. Choe, K. W. Lee and A. Ikenaga, J. Mater. Sci. Technol., 2007, 23(1), 97–101.
7. S. C. Wang and P. W. Kao, J. Mater. Sci., 1993, 28(19), 5169–5175.
8. J. H. Jang, C.-H. Lee, Y.-U. Heo and D.-W. Suh, Acta Mater., 2012, 60(1), 208–217.
9. J. F. Lu, O. Omotoso, J. B. Wiskel, D. G. Ivey and H. Henein, Metall. Mater. Trans. A, 2012, 43(9), 3043–3061.
10. A. Y. Ivanov, R. V. Sulyagin, V. V. Orlov and A. A. Kruglova, Steel in Translation, 2011, 41(7), 611–616.
11. W. B. Morrison, Mater. Sci. Technol., 2009, 25(9), 1066–1073.
12. A. M. Jorge Junior, L. H. Guedes and O. Balancin, J. Mater. Res. Technol., 2012, 1(3), 141–147.
13. R. D. K. Misra, G. C. Weatherly, J. E. Hartmann and A. J. Boucek, Mater. Sci. Technol., 2001, 17(9), 1119–1129.
14. V. Lazic, D. Milosavljevic, S. Aleksandrovic, G. Bogdanovic, P. Marinkovic and B. Nedeljkovic, Tribology in Industry, 2010, 32(2), 11–20.
15. F. Akhtar, Can. Metall. Q., 2014, 53(3), 253–263.
16. M. Sheikhzadeh and S. Sanjabi, Mater. Des., 2012, 39, 366–372.
17. D. H. Bacon, L. Edwards, J. E. Moffatt and M. E. Fitzpatrick, Int. J. Fatigue, 2013, 48, 39–47.
18. A. Chrysanthou, N. W. Davies, J. Li, P. Tsakiropoulos and O. S. Chinyamakobvu, J. Mater. Sci. Lett., 1996, 15(6), 473–474.
19. L. J. Huang, L. Geng and H. X. Peng, Prog. Mater. Sci., 2015, 71, 93–168.
20. V. A. Kukareko, Phys. Met. Metallogr., 2009, 107(1), 96–104.
21. J. Moon and C. Lee, Acta Mater., 2009, 57(7), 2311–2320.
22. G. I. Rees, J. Perdrix, T. Maurickx and H. K. D. H. Bhadeshia, Mater. Sci. Eng., A, 1995, 194(2), 179–186.
23. N. Fujita and H. K. D. H. Bhadeshia, Mater. Sci. Technol., 2001, 17(4), 403–408.
24. F. M. Gao, J. L. He, E. D. Wu, S. M. Liu, D. L. Yu, D. C. Li, S. Y. Zhang and Y. J. Tian, Phys. Rev. Lett., 2003, 91(1), 015502.
25. S. Kamran, K. Y. Chen and L. Chen, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79(2), 024106.
26. S. Ogata and J. Li, J. Appl. Phys., 2009, 106(11), 113534.
27. L. Bellaiche and D. Vanderbilt, Phys. Rev. B: Condens. Matter Mater. Phys., 2000, 61(12), 7877–7882.
28. C. J. Yu and H. Emmerich, J. Phys.: Condens. Matter, 2007, 19(30), 306203.
29. E. Bellotti, F. Bertazzi and M. Goano, J. Appl. Phys., 2007, 101(12), 123706.
30. J. W. Zwanziger, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76(5), 052102.
31. X. Gonze, B. Amadon, P. M. Anglade, J. M. Beuken, F. Bottin, P. Boulanger, F. Bruneval, D. Caliste, R. Caracas, M. Cote, T. Deutsch, L. Genovese, P. Ghosez, M. Giantomassi, S. Goedecker, D. R. Hamann, P. Hermet, F. Jollet, G. Jomard, S. Leroux, M. Mancini, S. Mazevet, M. J. T. Oliveira, G. Onida, Y. Pouillon, T. Rangel, G. M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, M. J. Verstraete, G. Zerah and J. W. Zwanziger, Comput. Phys. Commun., 2009, 180(12), 2582–2615.
32. X. Gonze, G. M. Rignanese, M. Verstraete, J. M. Beuken, Y. Pouillon, R. Caracas, F. Jollet, M. Torrent, G. Zerah, M. Mikami, P. Ghosez, M. Veithen, J. Y. Raty, V. Olevano, F. Bruneval, L. Reining, R. Godby, G. Onida, D. R. Hamann and D. C. Allan, Z. Kristallogr., 2005, 220(5–6), 558–562.
33. M. Fuchs and M. Scheffler, Comput. Phys. Commun., 1999, 119(1), 67–98.
34. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77(18), 3865–3868.
35. O. Beckstein, J. E. Klepeis, G. L. W. Hart and O. Pankratov, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63(13), 134112.
36. L. Fast, J. M. Wills, B. Johansson and O. Eriksson, Phys. Rev. B: Condens. Matter Mater. Phys., 1995, 51(24), 17431–17438.
37. R. Hill, Proc. Phys. Soc., London, Sect. A, 1952, 65(389), 349–355.
38. S. Ogata, J. Li and S. Yip, Science, 2002, 298(5594), 807–811.
39. S. Zhou, X. Zhou and Y. Zhao, J. Appl. Phys., 2008, 104(5), 053508.
40. Š. Masys and V. Jonauskas, J. Chem. Phys., 2013, 139(22), 224705.
41. H. W. Hugosson, O. Eriksson, U. Jansson and B. Johansson, Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63(13), 134108.
42. Y. Zhang, T. T. Zuo, Z. Tang, M. C. Gao, K. A. Dahmen, P. K. Liaw and Z. P. Lu, Prog. Mater. Sci., 2014, 61, 1–93.
43. H. Wang, Z. D. Zhang, R. Q. Wu and L. Z. Sun, Acta Mater., 2013, 61(8), 2919–2925.
44. Y. N. Zhang, R. Q. Wu, H. M. Schurter and A. B. Flatau, J. Appl. Phys., 2010, 108(2), 023513.
45. D. H. Chung and W. R. Buessem, J. Appl. Phys., 1967, 38(5), 2010.
46. H. O. Pierson, Handbook of Refractory Carbides and Nitrides, Noyes Publications, Park Ridge, N.J., 1996.
47. Q. Yu, L. Qi, K. Chen, R. K. Mishra, J. Li and A. M. Minor, Nano Lett., 2012, 12(2), 887–892.
48. S. Pauly, S. Gorantla, G. Wang, U. Kühn and J. Eckert, Nat. Mater., 2010, 9(6), 473–477.
49. J. Li, W. Cai, J. P. Chang and S. Yip, Comput. Mater. Sci., 2003, 187, 359–387.
50. G. M. Hua and D. Y. Li, Phys. Status Solidi B, 2012, 249(8), 1517–1520.
51. S. F. Pugh, Philos. Mag., 1954, 45(367), 823–843.
52. W. Acchar and C. A. Cairo, Mater. Res., 2006, 9, 171–174.
53. G. Morgan and M. H. Lewis, J. Mater. Sci., 1974, 9(3), 349–358.
54. Y. Kumashiro, A. Itoh, T. Kinoshita and M. Sobajima, J. Mater. Sci., 1977, 12(3), 595–601.
55. B. R. Murphy and T. H. Courtney, J. Mater. Res., 1999, 14(11), 4285–4290.
56. C. Maerky, M. O. Guillou, J. L. Henshall and R. M. Hooper, Mater. Sci. Eng., A, 1996, 209(1–2), 329–336.
57. M. C. Zhao, T. Hanamura, H. Qiu and K. Yang, Mater. Trans., 2005, 46(4), 784–789.
58. J. M. An and W. E. Pickett, Phys. Rev. Lett., 2001, 86(19), 4366–4369.
59. S. H. Jhi, S. G. Louie, M. L. Cohen and J. W. Morris, Phys. Rev. Lett., 2001, 87(7), 075503.
60. H. Bilz and W. Kress, Phonon dispersion relations in insulators, Springer-Verlag, Berlin, New York, 1979.
61. K. J. Chang and M. L. Cohen, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 34(7), 4552–4557.
62. B. Fultz, Prog. Mater. Sci., 2010, 55(4), 247–352.
63. S. Iikubo, H. Ohtani and M. Hasebe, Mater. Trans., 2010, 51(3), 574–577.
64. J. J. Gilman, Electronic Basis of the Strength of Materials, Cambridge University Press, 2003.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra22756a

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