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Ab initio investigation of mechanical and thermodynamic properties of vanadium-nitride
Materials Chemistry and Physics 228 (2019) 237–243
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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Ab initio investigation of mechanical and thermodynamic properties of vanadium-nitride
T
Bahram Abedi Ravana, Mahdi Faghihnasirib,∗, Homayoun Jafaric a
Department of Basic Sciences, Shahid Sattari Aeronautical University of Science and Technology, Tehran, Iran Computational Materials Science Laboratory, Nano Research and Training Center, NRTC, Iran c Department of Physics, Iran University of Science and Technology, Tehran, Iran b
H I GH L IG H T S
and thermodynamic properties of vanadium nitride have been studied. • Mechanical ratio between 2.35 and 2.52 is predicted. • Ductility • Expansion coefficient reaches to 2.63 × 10 1/K at room temperature. −5
A R T I C LE I N FO
A B S T R A C T
Keywords: Ab-initio Mechanical properties Bulk modulus by temperature Vanadium nitride
Density functional theory-based calculations in combination with the continuum approach are used to study the mechanical properties of vanadium nitride (VN) and the method of Debye-Grüneisen is used to investigate its thermodynamic characteristics. Elastic constants of the structure C11 = 580.97, C12 = 205.02 and C44 = 108.92 GPa are obtained. Ductility ratio between 2.35 and 2.52 is calculated which predicts the VN compound to be malleable and structurally stable and it is categorized among tough coating materials. Also, thermodynamic behavior, bulk modulus, specific heat capacity in constant pressure and temperature expansion coefficient of VN are rigorously investigated. Results indicate that as the temperature increases, the bulk modulus decreases. At room temperature the bulk modulus reaches 294.53 GPa, at 400 K it is 284.24 GPa and finally reduces to 221.25 GPa at 1000 K. Hence, compared to the ground state calculations the bulk modulus obtained here is fairly consistent with the experimental data.
1. Introduction Adding nitrogen to a crystalline lattice made of transition metals, creates special nitride compounds which have received considerable research interests in the past decades [1]. The nitride compounds such as ZrN [2], HfN [3], TiN [4], ScN [5], VN [6] and CrN [7] are usually crystallized in the rock-salt (NaCl-type) structure. Among these nitride compounds the vanadium nitride (VN) has received great attention because of its unique electronic properties and great thermal and electrical conductivity and also its promising catalytic features [8]. VN is a potential material of choice for fabricating the electrodes of supercapacitors [9]. The “metal-metal” and “metal-nitrogen” bonds in this category of nitrogen compounds causes the materials to have better chemical stability and interesting ceramic properties [10,11]. These compounds possess special physical and chemical features which make them
∗
desirable for many practical applications. In recent years, scientists have made great efforts towards discovery of tough materials in order to use them as thin protection layers over different surfaces. Coating of hard materials on surfaces which are prone to damage or on cutting tools protects them against corrosion or abrasion and consequently increases their lifetime [12,13]. Hard materials are susceptible to tensions beyond their harmonic region. This low tensile strength leads to their cracking under heavy tensile loads. For hard coating applications the coating material is preferred to be malleable, a property that is directly proportional to the material's bulk modulus and inversely proportional to its shear modulus. Being hard and structurally strong alongside formability puts a material into the high-toughness category of materials [12]. Many studies have been carried out investigating mechanical properties and toughness of the VN compound but still, compared to other nitride compounds, in some cases controversies can be seen in the
Corresponding author. E-mail address: [email protected] (M. Faghihnasiri).
https://doi.org/10.1016/j.matchemphys.2019.02.082 Received 22 November 2018; Received in revised form 17 February 2019; Accepted 19 February 2019 Available online 21 February 2019 0254-0584/ © 2019 Elsevier B.V. All rights reserved.
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reported results. For instance, the elastic constant C11 of VN is experimentally measured to be 533 GPa [14], while theoretical calculations have predicted this parameter to be between 591 and 689 GPa [15,16]. Additionally, very scarce data is available regarding thermodynamic properties of this material. Hence, in this study using the most accurate computational approaches we have calculated the mechanical properties and thermodynamic characteristics of VN. The electronic and mechanical calculations are conducted based on the density functional theory (DFT) and the thermodynamic properties are investigated using the continuum theory and the Debye-Grüneisen approach.
Table 1 Lattice constant (a) of the cubic VN obtained in this work alongside previous theoretical and experimental results. Method
a (Å)
Ref.
PBEsol Exp. acoustic Exp. HP-XRD LDA LDA GGA GGA
4.08 4.13 4.13 4.05 4.06 4.10 4.12
This work [14] [20] [21] [22] [23] [21]
2. Computational details First-principle study by DFT calculations were carried out with the QUANTUM ESPRESSO package [17]. The exchange-correlation term is described by adopting generalized gradient approximation (GGA) using Perdew-Burke-Ernzerhof revised for solids (PBEsol) functional [18]. This approximation is appreciably accurate for investigating solid lattice parameters and crystalline structures. To reduce the required wave functions for describing the electrons near the atomic cores, projectoraugmented wave (PAW) pseudopotentials are applied to replace the core electron wave functions. Electronic configurations which have been used for atoms V and N were [Ar] 3 d3 4s2 and [He] 2s2 2p3, respectively. The kinetic energy cutoff for the plane wave basis set was chosen to be 700 eV which is computationally enough for this structure and a Brillouin zone sampling using Monkhorst-Pack [19] k-mesh of 6 × 6 × 6 is employed. Atomic positions were relaxed until the energy differences were converged within 10−6 eV and the maximum Hellmann–Feynman force on any atom was below 10−5 eV/Å.
Fig. 2. Calculated phonon-dispersion curves for VN.
3. Results and discussion Unit cell of the cubic VN containing 8 atoms, 4 vanadium and 4 nitrogen atoms, can be seen in Fig. 1. At first, lattice vectors and atomic positions of VN are optimized. Lattice vectors obtained using the PBEsol approximation are reported in Table 1 and the calculated results are compared with the similar structures. Furthermore, the stability of VN has been investigated at its equilibrium state and can be seen in phonon dispersion graphs shown in Fig. 2. Since, any negative modes have not been observed in this graph, they elucidate the dynamically stability properties of our structure. Also, we have investigated the electronic band structure and density of states (DOS) of the cubic VN, which, as shown in Fig. 3, illustrates that the structure at its ground state has no band gap and hence no interesting electronic properties. Inspecting the electronic band structure and DOS of the structurally optimized cubic VN reveals that not only the material is metal but also there exist lots of states both on and around the Fermi level. In the next step, to calculate the elastic constants of the cubic VN we used the method explained in our previous works [24–27], as brought
Fig. 3. Electronic band structure (left) along high-symmetry points in reciprocal space and DOS (right) of the cubic VN.
in Supporting Information (SI). According to this method the unit cell is subjected to a set of deformations based on the crystal structure so that elastic constants can be calculated. Due to symmetry of the cubic structures there are only three independent elastic constants C11, C12 and C44 which are calculated using the following cubic, orthorhombic, and monoclinic deformation tensors proposed by Jamal et al. [28].
Dcubic
⎡η 0 ⎡η 0 0 ⎤ 0 −η = ⎢ 0 η 0 ⎥, Dortho = ⎢ ⎢ ⎥ ⎢ ⎢0 0 ⎣0 0 η ⎦ ⎣
0 ⎤ ⎡0 η 0 ⎥ η 0 , Dmono = ⎢ ⎢ η2 ⎥ ⎢0 0 ⎥ 1 − η2 ⎦ ⎣
0 ⎤ 0 ⎥ η2 ⎥ ⎥ 1 − η2 ⎦
Each deformation mode contains −10% to 20% strains which are applied on the structure. The three nonzero second order elastic constants of this structure are C11, C12 and C44. Based on these elastic constants, the bulk modulus and shear modulus can be directly calculated, Eqs. S3 and S4. Consequently, Young's modulus (E) and Poisson's ratio (ν) can be calculated using the bulk and shear moduli, Eqs. S5 and S6. Utilizing elastic constants, many mechanical and thermodynamic parameters can be computed. Elastic constants can also help us obtain Young's, bulk and shear moduli and the Poisson ratio. Moreover, speed of the sound and Debye's temperature inside the crystal can also be calculated.
Fig. 1. Schematic view of cubic VN atomic configuration and its unit cell (black solid line). 238
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Fig. 4. Energy-strain plots for three different types of deformations. (a) Cubic, (b) Ortho and (c) Mono.
elastic moduli and Poisson's ratio obtained from three approaches of Voigt, Reuss and Hill are very close to each other. Bulk modulus of the cubic VN is the same in the three approaches (330.33 GPa), Young's modulus faces 22 GPa differences, shear modulus has around 9.59 GPa variations and alterations the Poisson ratio is of order 0.01. Also in Table 2 we can observe a clear discrepancy between the computed mechanical parameters and the experimental data reported in the literature. However, most reported values for the bulk modulus are within the range of 299–333 GPa and we see that, in comparison to the LDA, the GGA approximation is more accurate and gives 303 GPa which is closer to the experimental data. The bulk modulus obtained in this work using PBEsol exchange-correlation term and PAW approximation is 330.33 GPa. We believe the inconsistency with the experimental data comes from the fact that the PBEsol approximation has calculated the ground state lattice constant of the cubic VN equal to 4.08 Å while the
As previously mentioned, in order to calculate second order elastic constants, three different deformations will be applied on the structure. In Fig. 4 it can be seen that no phase transition and no phase shifting occurs under the applied strains. Fig. 4 alongside Eqs. S1 and S2 can help us to calculate elastic constants of the structure, as presented in Table 2 in which for the cubic VN they are C11 = 580.97, C12 = 205.02 and C44 = 108.92 GPa. For mechanical stability of the structure the elastic stiffness constants matrix should satisfy the Born stability criteria. The Born stability criteria for cubic structures are as follows: C11 > 0, C11-C12 > 0, C11+2C12 > 0, C44 > 0.
(1)
It can be seen that the elastic constants of Table 2 satisfy the Born conditions. In the next step, using Eqs. S3-S6, mechanical moduli and Poisson's ratio are calculated (Table 2). Results show the fact that all the
Table 2 Second order elastic constants (Cij in GPa), bulk (B), Young's (E) and shear (G) moduli (in GPa). Shear, compressional and average sound velocities (m/s), Debye temperature (θD in K), ductility (B/G) and Poisson's ratio (ν) of the cubic VN. This work Voigt C11 C12 C44 B E G Vt Vl Vm θD B/G ν
580.97 205.02 108.92 330.33 370.26 140.54 4704 9030 6244 1043 2.35 0.31
Exp. Reuss
Hill
[14]
GGA [20]
[21]
[29]
[16]
[30]
[15]
333.16
621 172 126 321
591 159 137 303
628.70 144.63 147.41 305.98
165
164
185.26
689.69 138.69 125.57 320.08 437.80 172 5287.05 94447.30 5875.86 848.44 1.86
533 135 133 330.33 347 130.95 4541 8917 1012 1012 2.52 0.32
330.33 358.17 135.75 4623 8974 6152 1028 2.43 0.31
299.74
239
LDA [22]
382.51
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Fig. 5. 3D curves of (a) Young and (b) Shear moduli, (c) Poisson ratio and (d) linear compressibility of the cubic VN.
experimental value is reported to be 4.13 Å which led to a 0.05 Å underestimation in the lattice constant calculation. Young's modulus obtained in this work (∼370.26 GPa) for the cubic VN is greater than its bulk modulus. This result indicates that the cubic VN under uniaxial strains is harder and shows higher stiffness or in specific directions it shows greater mechanical toughness. To better analyze this phenomenon, the three dimensional (3D) Young's modulus, shear modulus, Poisson's ratio and linear compressibility graphs are plotted in Fig. 5 (using the ELATE software [29]). We can observe that Young's modulus is anisotropic and has greater values along directions parallel to the V-N bonds. Maximum and minimum values of Young's modulus in the graph are 474.01 and 294.4 GPa, respectively (Table 3) which explain the crystal's anisotropic behavior. Maximum Young's
modulus of 437.80 GPa, calculated using the GGA approximation, is reported by Chauhan and coworkers [14]. In Fig. 5 we observe that the shear modulus and Poisson's ratio of the cubic VN are also, as stated in Table 3, anisotropic where we read a maximum and minimum shear modulus of 187.9 and 108.92 GPa, respectively. The maximum and minimum values for Poisson's ratio are respectively 0.49 and 0.17 whereas it can be seen that the linear compressibility (defined by 1/B) is completely isotropic. Furthermore, parameters such as speed of sound and Debye's temperature are presented in Table 2 which correspond to the chemical bonds of the structure and are calculated using the second order elastic constants. Debye's temperature is directly proportional to the elastic
Table 3 Minimum and maximum values of Young's modulus, linear compressibility, shear modulus and Poisson's ratio. Young's modulus (GPa)
Linear compressibility (TPa−1)
Shear modulus (GPa)
Poisson's ratio
Emin
Emax
βmin
βmax
Gmin
Gmax
νmin
νmax
294.4
474.01
1.0091
1.0091
108.92
187.97
0.17896
0.49288
240
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strains, PK2 strain curves are calculated using Eq. S(11). True stress is computed using the QUANTUM ESPRESSO package under DFT framework. Employing DFT alongside the PBEsol approximation, the PK2 curves are calculated and the results are depicted in Fig. 6. These curves are plotted to display the symmetric deformation matrix Dcubic applied on VN. In these diagrams, the maxima of the curves reveal the ultimate stress which a material can withstand under the associated strain, and the corresponding strain is the ultimate strain. The ultimate strain of a pristine material is always larger than the most durable strain of the same material in realistic conditions where there exist thermal vibrations or crystal imperfections. Hence, a material is meta-stable beyond its ultimate strain and will be destroyed under crystalline defects, vacancies or thermal vibration. Below the ultimate strain is called “elastic region” and above that is “plastic region”. Within the elastic region, the structure can return to its initial state after eliminating the external stress but in the plastic region the structure cannot reverse. The ultimate strain and the corresponding stress for the cubic VN are 6.5% and 0.27 eV/Å3, respectively. Compressive strain increases the stress more than the tensile strain, as is obvious in the PK2 curves in Fig. 6 where, for instance, we can see the amount of stress for a −5% strain is twice as much as the stress for a 5% strain. This behavior can be explained based on the Leonard-Jones potential which describes the interaction energy between molecules and atoms. According to formula S12, this potential is formed based on repulsion component of order 12 and attraction component of order 6. According to this equation, variation of Leonard-Jones potential is of order 12 in short distances causing more intense interactions between atoms in close vicinity to each other. It should be mentioned that compressive strain more than 10% is not accessible due to the sharp stress variation and, furthermore, the structure would not be stable under these stresses. For this reason, we have not investigated the strains more than −10%. In addition to what is already mentioned, as Pugh and coworkers [31] have reported the ratio of bulk to shear modulus (B/G) illustrates the ductility or brittleness of materials. If B/G > 1.75 then the material is said to be ductile but if this ratio is less than 1.75 then the material is categorized as brittle or fragile. For the cubic VN the B/G ratio calculated using the PBEsol approximation is 2.35 (Table 2) which proves the ductility of this structure. Thermodynamic properties. By computing the Equation of State (EoS) of a material, one can investigate its behavior at high pressures. Having pressure versus volume variations and applying the BirchMurnaghan EoS we can calculate the bulk modulus and its pressure derivative B’. To calculate the EoS and other thermodynamic parameters of cubic VN, we employed the Gibbs2 code [32,33]. This code utilizes the fourth order Birch-Murnaghan equation and Debye–Grüneisen model to calculate the thermodynamic properties up to 1000 K, which is lower than the calculated Debye temperature (1043 K). Thus, using static energy or the ground state energy, and nonequilibrium Gibbs function defined in Eq. S(13) we can, with the help of the Debye–Grüneisen model, compute free energy at any temperature. At the first step, using the thermodynamic calculations we investigate variations of the bulk modulus as a function of temperature. Results are displayed in Fig. 7 where we see that as the temperature increases the bulk modulus decreases. Here, the bulk modulus is based on the third-order Birch-Murnaghan isothermal equation of state [34] which is slightly different from the one computed in the previous section due to the different approaches employed. However, the bulk modulus calculated using this method at zero temperature is 301.68 GPa which increases gradually with temperature. At room temperature the bulk modulus is about 294.53 K. It reduces as the temperature increases until at 400 K it becomes 284.24 GPa and finally reduces to 221.25 GPa at 1000 K. As it is obvious from the slope of the curve in Fig. 7, variations near
Fig. 6. Stress-strain curves for the cubic VN along two axial (Σa and Σb) and one biaxial (Σc) deformations. The elastic-plastic transition point is indicated by red line separator between white-yellow regions. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 7. Temperature dependence of the bulk modulus of the cubic VN.
Fig. 8. Specific heat capacity obtained in this work for constant volume (Cv) and constant pressure (Cp) of cubic VN against temperature.
constants, heat capacity and melting point. It can be calculated using low temperature elastic constants. Debye's temperature, Vt and Vl , which are shear and compressional velocities and average sound velocity (Vm ) can be obtained via Eqs. S7-S10. The calculated results are given in Table 2. Debye's temperature of the cubic VN structure is obtained about 1012–1043 K, which is comparable with theoretical values 848.88 [14] and 838 K [30] and the experimental amount of 772 K [31]. Since for temperatures lower than Debye's temperature it is possible to calculate thermodynamic properties of crystalline structures, using the Debye–Grüneisen approach, all the thermodynamic properties of cubic VN are presented further. Within the continuum approach, the second Piola-Kirchhoff (PK2) can be used to investigate the effect of large strains on structure's behavior. In order to investigate the structure's behavior under large 241
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Fig. 9. Thermal expansion (α ) and volume (strain) variation curves against temperatures up to the Debye temperature for cubic VN.
ductility, studying the PK2 plots revealed that VN has considerable elasticity of about 8%. Then, using the elastic constants and mechanical parameters, we calculated the Debye temperature of VN. Since thermodynamic behavior of materials at temperatures lower than Debye's temperature can be investigated using the Debye–Grüneisen approach, we used this method to investigate temperature dependence of the bulk modulus, specific heat capacity and thermal expansion coefficient of the cubic VN. The problem was that the experimentally measured value for the bulk modulus of VN is 299.74 GPa, but the theoretical calculations have in cases by far overestimated it. Our calculations indicated that the bulk modulus decreases with temperature. It was found out that at room temperature the bulk modulus was about 294.53 GPa which is fairly consistent with the experimentally measured value of 299.74 GPa. Furthermore, according to our calculations as the temperature was raised to 400 K the bulk modulus plummeted to 284.24 GPa and finally it dropped to 221.25 GPa at 1000 K. At last, we observed that as the temperature increases the material undergoes a thermal expansion which is equivalent to the applied strain such that at room temperature the material experiences a 19% strain.
the ground state temperature are very moderate and at about 150 K the slope begins to decline more steeply. For temperatures higher than approximately 400 K we observe a linear reduction behavior in the bulk modulus. Specific heat coefficients at constant volume (CV ) and constant pressure (Cp ) are also plotted in Fig. 8. Specific heat illustrates vibrational properties of the structure. CV and Cp of VN at atmospheric pressure and room temperature are 133.43 and 135.95 J mol−1 K−1, respectively. The relationship between CV and Cp is defined as S14. In Fig. 8 we can clearly see that at low temperatures CV and Cp are very close to each other and the two curves are fairly overlapping; however, as the temperature increases the difference between these two coefficients gradually rises or they split up by increasing the expansion coefficient. This is the main reason that at temperatures higher than 300 K these two curves separate from each other such that Cp becomes greater than CV . Thermal expansion curve in terms of temperature is depicted in Fig. 9. It is easy to see that as the temperature increases the expansion coefficient rises up such that at room temperature it becomes 2.63 × 10−5 1/K. It continues going up until at 400 K it reaches to 3.19 × 10−5 1/K and finally hits 5.13 × 10−5 1/K at 1000 K. Using the obtained volume expansion coefficient defined as Eq. S (15), length variation and the strain, Eq. S(16), due to the temperature increase can be calculated for the VN structure. In Fig. 9 we observe that by raising the temperature to room temperature VN undergoes a 19.9% strain with respect to its equilibrium state. This strain increases to 23.3% at 400 K and eventually at 1000 K, which is near the Debye temperature of the VN, it experiences a 40% increase.
Acknowledgment The authors gratefully acknowledge the support of the Nano Research and Training Center of Iran (NRTC.ir) under Grant No. 15.97.06.25. Computational resources were provided by the Davami Computing Cluster belonging to Tafresh University which is supported by the Nano Research and Training Center of Iran (NRTC.ir). Appendix A. Supplementary data
4. Conclusions
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.matchemphys.2019.02.082.
In this paper using the PBEsol functional alongside the PAW pseudopotentials, we have computed the elastic constants of rock-salt VN. Mechanical parameters such as bulk, Young and shear moduli and Poisson's ratio have been calculated based on the continuum theory and using the Voigt, Reuss and Hill methods. The elastic constants C11 = 580.97, C12 = 205.02 and C44 = 108.92 GPa have been obtained. It has been determined that the results calculated using the three methods are very close to each other. A ground state bulk modulus of 330.33 GPa and Young's modulus equal to 370.26 GPa classifies the cubic VN as a hard material. Additionally, the shear modulus was calculated to be between 130.95 and 140.54 GPa and the ductility ratio is about 2.35–2.52 which predicts good malleability for VN. On the other hand, Young's modulus of the cubic VN along its V-N bonds are much greater than its Young modulus along other directions indicating that VN has higher resistance under tensions along the < 100 > , < 010 > and < 001 > directions. Therefore, for hard coating purposes using VN it is better to perform coatings and sputtering along the mentioned directions. In addition to its toughness and good
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Materials Chemistry and Physics journal homepage: www.elsevier.com/locate/matchemphys
Ab initio investigation of mechanical and thermodynamic properties of vanadium-nitride
T
Bahram Abedi Ravana, Mahdi Faghihnasirib,∗, Homayoun Jafaric a
Department of Basic Sciences, Shahid Sattari Aeronautical University of Science and Technology, Tehran, Iran Computational Materials Science Laboratory, Nano Research and Training Center, NRTC, Iran c Department of Physics, Iran University of Science and Technology, Tehran, Iran b
H I GH L IG H T S
and thermodynamic properties of vanadium nitride have been studied. • Mechanical ratio between 2.35 and 2.52 is predicted. • Ductility • Expansion coefficient reaches to 2.63 × 10 1/K at room temperature. −5
A R T I C LE I N FO
A B S T R A C T
Keywords: Ab-initio Mechanical properties Bulk modulus by temperature Vanadium nitride
Density functional theory-based calculations in combination with the continuum approach are used to study the mechanical properties of vanadium nitride (VN) and the method of Debye-Grüneisen is used to investigate its thermodynamic characteristics. Elastic constants of the structure C11 = 580.97, C12 = 205.02 and C44 = 108.92 GPa are obtained. Ductility ratio between 2.35 and 2.52 is calculated which predicts the VN compound to be malleable and structurally stable and it is categorized among tough coating materials. Also, thermodynamic behavior, bulk modulus, specific heat capacity in constant pressure and temperature expansion coefficient of VN are rigorously investigated. Results indicate that as the temperature increases, the bulk modulus decreases. At room temperature the bulk modulus reaches 294.53 GPa, at 400 K it is 284.24 GPa and finally reduces to 221.25 GPa at 1000 K. Hence, compared to the ground state calculations the bulk modulus obtained here is fairly consistent with the experimental data.
1. Introduction Adding nitrogen to a crystalline lattice made of transition metals, creates special nitride compounds which have received considerable research interests in the past decades [1]. The nitride compounds such as ZrN [2], HfN [3], TiN [4], ScN [5], VN [6] and CrN [7] are usually crystallized in the rock-salt (NaCl-type) structure. Among these nitride compounds the vanadium nitride (VN) has received great attention because of its unique electronic properties and great thermal and electrical conductivity and also its promising catalytic features [8]. VN is a potential material of choice for fabricating the electrodes of supercapacitors [9]. The “metal-metal” and “metal-nitrogen” bonds in this category of nitrogen compounds causes the materials to have better chemical stability and interesting ceramic properties [10,11]. These compounds possess special physical and chemical features which make them
∗
desirable for many practical applications. In recent years, scientists have made great efforts towards discovery of tough materials in order to use them as thin protection layers over different surfaces. Coating of hard materials on surfaces which are prone to damage or on cutting tools protects them against corrosion or abrasion and consequently increases their lifetime [12,13]. Hard materials are susceptible to tensions beyond their harmonic region. This low tensile strength leads to their cracking under heavy tensile loads. For hard coating applications the coating material is preferred to be malleable, a property that is directly proportional to the material's bulk modulus and inversely proportional to its shear modulus. Being hard and structurally strong alongside formability puts a material into the high-toughness category of materials [12]. Many studies have been carried out investigating mechanical properties and toughness of the VN compound but still, compared to other nitride compounds, in some cases controversies can be seen in the
Corresponding author. E-mail address: [email protected] (M. Faghihnasiri).
https://doi.org/10.1016/j.matchemphys.2019.02.082 Received 22 November 2018; Received in revised form 17 February 2019; Accepted 19 February 2019 Available online 21 February 2019 0254-0584/ © 2019 Elsevier B.V. All rights reserved.
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reported results. For instance, the elastic constant C11 of VN is experimentally measured to be 533 GPa [14], while theoretical calculations have predicted this parameter to be between 591 and 689 GPa [15,16]. Additionally, very scarce data is available regarding thermodynamic properties of this material. Hence, in this study using the most accurate computational approaches we have calculated the mechanical properties and thermodynamic characteristics of VN. The electronic and mechanical calculations are conducted based on the density functional theory (DFT) and the thermodynamic properties are investigated using the continuum theory and the Debye-Grüneisen approach.
Table 1 Lattice constant (a) of the cubic VN obtained in this work alongside previous theoretical and experimental results. Method
a (Å)
Ref.
PBEsol Exp. acoustic Exp. HP-XRD LDA LDA GGA GGA
4.08 4.13 4.13 4.05 4.06 4.10 4.12
This work [14] [20] [21] [22] [23] [21]
2. Computational details First-principle study by DFT calculations were carried out with the QUANTUM ESPRESSO package [17]. The exchange-correlation term is described by adopting generalized gradient approximation (GGA) using Perdew-Burke-Ernzerhof revised for solids (PBEsol) functional [18]. This approximation is appreciably accurate for investigating solid lattice parameters and crystalline structures. To reduce the required wave functions for describing the electrons near the atomic cores, projectoraugmented wave (PAW) pseudopotentials are applied to replace the core electron wave functions. Electronic configurations which have been used for atoms V and N were [Ar] 3 d3 4s2 and [He] 2s2 2p3, respectively. The kinetic energy cutoff for the plane wave basis set was chosen to be 700 eV which is computationally enough for this structure and a Brillouin zone sampling using Monkhorst-Pack [19] k-mesh of 6 × 6 × 6 is employed. Atomic positions were relaxed until the energy differences were converged within 10−6 eV and the maximum Hellmann–Feynman force on any atom was below 10−5 eV/Å.
Fig. 2. Calculated phonon-dispersion curves for VN.
3. Results and discussion Unit cell of the cubic VN containing 8 atoms, 4 vanadium and 4 nitrogen atoms, can be seen in Fig. 1. At first, lattice vectors and atomic positions of VN are optimized. Lattice vectors obtained using the PBEsol approximation are reported in Table 1 and the calculated results are compared with the similar structures. Furthermore, the stability of VN has been investigated at its equilibrium state and can be seen in phonon dispersion graphs shown in Fig. 2. Since, any negative modes have not been observed in this graph, they elucidate the dynamically stability properties of our structure. Also, we have investigated the electronic band structure and density of states (DOS) of the cubic VN, which, as shown in Fig. 3, illustrates that the structure at its ground state has no band gap and hence no interesting electronic properties. Inspecting the electronic band structure and DOS of the structurally optimized cubic VN reveals that not only the material is metal but also there exist lots of states both on and around the Fermi level. In the next step, to calculate the elastic constants of the cubic VN we used the method explained in our previous works [24–27], as brought
Fig. 3. Electronic band structure (left) along high-symmetry points in reciprocal space and DOS (right) of the cubic VN.
in Supporting Information (SI). According to this method the unit cell is subjected to a set of deformations based on the crystal structure so that elastic constants can be calculated. Due to symmetry of the cubic structures there are only three independent elastic constants C11, C12 and C44 which are calculated using the following cubic, orthorhombic, and monoclinic deformation tensors proposed by Jamal et al. [28].
Dcubic
⎡η 0 ⎡η 0 0 ⎤ 0 −η = ⎢ 0 η 0 ⎥, Dortho = ⎢ ⎢ ⎥ ⎢ ⎢0 0 ⎣0 0 η ⎦ ⎣
0 ⎤ ⎡0 η 0 ⎥ η 0 , Dmono = ⎢ ⎢ η2 ⎥ ⎢0 0 ⎥ 1 − η2 ⎦ ⎣
0 ⎤ 0 ⎥ η2 ⎥ ⎥ 1 − η2 ⎦
Each deformation mode contains −10% to 20% strains which are applied on the structure. The three nonzero second order elastic constants of this structure are C11, C12 and C44. Based on these elastic constants, the bulk modulus and shear modulus can be directly calculated, Eqs. S3 and S4. Consequently, Young's modulus (E) and Poisson's ratio (ν) can be calculated using the bulk and shear moduli, Eqs. S5 and S6. Utilizing elastic constants, many mechanical and thermodynamic parameters can be computed. Elastic constants can also help us obtain Young's, bulk and shear moduli and the Poisson ratio. Moreover, speed of the sound and Debye's temperature inside the crystal can also be calculated.
Fig. 1. Schematic view of cubic VN atomic configuration and its unit cell (black solid line). 238
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Fig. 4. Energy-strain plots for three different types of deformations. (a) Cubic, (b) Ortho and (c) Mono.
elastic moduli and Poisson's ratio obtained from three approaches of Voigt, Reuss and Hill are very close to each other. Bulk modulus of the cubic VN is the same in the three approaches (330.33 GPa), Young's modulus faces 22 GPa differences, shear modulus has around 9.59 GPa variations and alterations the Poisson ratio is of order 0.01. Also in Table 2 we can observe a clear discrepancy between the computed mechanical parameters and the experimental data reported in the literature. However, most reported values for the bulk modulus are within the range of 299–333 GPa and we see that, in comparison to the LDA, the GGA approximation is more accurate and gives 303 GPa which is closer to the experimental data. The bulk modulus obtained in this work using PBEsol exchange-correlation term and PAW approximation is 330.33 GPa. We believe the inconsistency with the experimental data comes from the fact that the PBEsol approximation has calculated the ground state lattice constant of the cubic VN equal to 4.08 Å while the
As previously mentioned, in order to calculate second order elastic constants, three different deformations will be applied on the structure. In Fig. 4 it can be seen that no phase transition and no phase shifting occurs under the applied strains. Fig. 4 alongside Eqs. S1 and S2 can help us to calculate elastic constants of the structure, as presented in Table 2 in which for the cubic VN they are C11 = 580.97, C12 = 205.02 and C44 = 108.92 GPa. For mechanical stability of the structure the elastic stiffness constants matrix should satisfy the Born stability criteria. The Born stability criteria for cubic structures are as follows: C11 > 0, C11-C12 > 0, C11+2C12 > 0, C44 > 0.
(1)
It can be seen that the elastic constants of Table 2 satisfy the Born conditions. In the next step, using Eqs. S3-S6, mechanical moduli and Poisson's ratio are calculated (Table 2). Results show the fact that all the
Table 2 Second order elastic constants (Cij in GPa), bulk (B), Young's (E) and shear (G) moduli (in GPa). Shear, compressional and average sound velocities (m/s), Debye temperature (θD in K), ductility (B/G) and Poisson's ratio (ν) of the cubic VN. This work Voigt C11 C12 C44 B E G Vt Vl Vm θD B/G ν
580.97 205.02 108.92 330.33 370.26 140.54 4704 9030 6244 1043 2.35 0.31
Exp. Reuss
Hill
[14]
GGA [20]
[21]
[29]
[16]
[30]
[15]
333.16
621 172 126 321
591 159 137 303
628.70 144.63 147.41 305.98
165
164
185.26
689.69 138.69 125.57 320.08 437.80 172 5287.05 94447.30 5875.86 848.44 1.86
533 135 133 330.33 347 130.95 4541 8917 1012 1012 2.52 0.32
330.33 358.17 135.75 4623 8974 6152 1028 2.43 0.31
299.74
239
LDA [22]
382.51
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Fig. 5. 3D curves of (a) Young and (b) Shear moduli, (c) Poisson ratio and (d) linear compressibility of the cubic VN.
experimental value is reported to be 4.13 Å which led to a 0.05 Å underestimation in the lattice constant calculation. Young's modulus obtained in this work (∼370.26 GPa) for the cubic VN is greater than its bulk modulus. This result indicates that the cubic VN under uniaxial strains is harder and shows higher stiffness or in specific directions it shows greater mechanical toughness. To better analyze this phenomenon, the three dimensional (3D) Young's modulus, shear modulus, Poisson's ratio and linear compressibility graphs are plotted in Fig. 5 (using the ELATE software [29]). We can observe that Young's modulus is anisotropic and has greater values along directions parallel to the V-N bonds. Maximum and minimum values of Young's modulus in the graph are 474.01 and 294.4 GPa, respectively (Table 3) which explain the crystal's anisotropic behavior. Maximum Young's
modulus of 437.80 GPa, calculated using the GGA approximation, is reported by Chauhan and coworkers [14]. In Fig. 5 we observe that the shear modulus and Poisson's ratio of the cubic VN are also, as stated in Table 3, anisotropic where we read a maximum and minimum shear modulus of 187.9 and 108.92 GPa, respectively. The maximum and minimum values for Poisson's ratio are respectively 0.49 and 0.17 whereas it can be seen that the linear compressibility (defined by 1/B) is completely isotropic. Furthermore, parameters such as speed of sound and Debye's temperature are presented in Table 2 which correspond to the chemical bonds of the structure and are calculated using the second order elastic constants. Debye's temperature is directly proportional to the elastic
Table 3 Minimum and maximum values of Young's modulus, linear compressibility, shear modulus and Poisson's ratio. Young's modulus (GPa)
Linear compressibility (TPa−1)
Shear modulus (GPa)
Poisson's ratio
Emin
Emax
βmin
βmax
Gmin
Gmax
νmin
νmax
294.4
474.01
1.0091
1.0091
108.92
187.97
0.17896
0.49288
240
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strains, PK2 strain curves are calculated using Eq. S(11). True stress is computed using the QUANTUM ESPRESSO package under DFT framework. Employing DFT alongside the PBEsol approximation, the PK2 curves are calculated and the results are depicted in Fig. 6. These curves are plotted to display the symmetric deformation matrix Dcubic applied on VN. In these diagrams, the maxima of the curves reveal the ultimate stress which a material can withstand under the associated strain, and the corresponding strain is the ultimate strain. The ultimate strain of a pristine material is always larger than the most durable strain of the same material in realistic conditions where there exist thermal vibrations or crystal imperfections. Hence, a material is meta-stable beyond its ultimate strain and will be destroyed under crystalline defects, vacancies or thermal vibration. Below the ultimate strain is called “elastic region” and above that is “plastic region”. Within the elastic region, the structure can return to its initial state after eliminating the external stress but in the plastic region the structure cannot reverse. The ultimate strain and the corresponding stress for the cubic VN are 6.5% and 0.27 eV/Å3, respectively. Compressive strain increases the stress more than the tensile strain, as is obvious in the PK2 curves in Fig. 6 where, for instance, we can see the amount of stress for a −5% strain is twice as much as the stress for a 5% strain. This behavior can be explained based on the Leonard-Jones potential which describes the interaction energy between molecules and atoms. According to formula S12, this potential is formed based on repulsion component of order 12 and attraction component of order 6. According to this equation, variation of Leonard-Jones potential is of order 12 in short distances causing more intense interactions between atoms in close vicinity to each other. It should be mentioned that compressive strain more than 10% is not accessible due to the sharp stress variation and, furthermore, the structure would not be stable under these stresses. For this reason, we have not investigated the strains more than −10%. In addition to what is already mentioned, as Pugh and coworkers [31] have reported the ratio of bulk to shear modulus (B/G) illustrates the ductility or brittleness of materials. If B/G > 1.75 then the material is said to be ductile but if this ratio is less than 1.75 then the material is categorized as brittle or fragile. For the cubic VN the B/G ratio calculated using the PBEsol approximation is 2.35 (Table 2) which proves the ductility of this structure. Thermodynamic properties. By computing the Equation of State (EoS) of a material, one can investigate its behavior at high pressures. Having pressure versus volume variations and applying the BirchMurnaghan EoS we can calculate the bulk modulus and its pressure derivative B’. To calculate the EoS and other thermodynamic parameters of cubic VN, we employed the Gibbs2 code [32,33]. This code utilizes the fourth order Birch-Murnaghan equation and Debye–Grüneisen model to calculate the thermodynamic properties up to 1000 K, which is lower than the calculated Debye temperature (1043 K). Thus, using static energy or the ground state energy, and nonequilibrium Gibbs function defined in Eq. S(13) we can, with the help of the Debye–Grüneisen model, compute free energy at any temperature. At the first step, using the thermodynamic calculations we investigate variations of the bulk modulus as a function of temperature. Results are displayed in Fig. 7 where we see that as the temperature increases the bulk modulus decreases. Here, the bulk modulus is based on the third-order Birch-Murnaghan isothermal equation of state [34] which is slightly different from the one computed in the previous section due to the different approaches employed. However, the bulk modulus calculated using this method at zero temperature is 301.68 GPa which increases gradually with temperature. At room temperature the bulk modulus is about 294.53 K. It reduces as the temperature increases until at 400 K it becomes 284.24 GPa and finally reduces to 221.25 GPa at 1000 K. As it is obvious from the slope of the curve in Fig. 7, variations near
Fig. 6. Stress-strain curves for the cubic VN along two axial (Σa and Σb) and one biaxial (Σc) deformations. The elastic-plastic transition point is indicated by red line separator between white-yellow regions. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
Fig. 7. Temperature dependence of the bulk modulus of the cubic VN.
Fig. 8. Specific heat capacity obtained in this work for constant volume (Cv) and constant pressure (Cp) of cubic VN against temperature.
constants, heat capacity and melting point. It can be calculated using low temperature elastic constants. Debye's temperature, Vt and Vl , which are shear and compressional velocities and average sound velocity (Vm ) can be obtained via Eqs. S7-S10. The calculated results are given in Table 2. Debye's temperature of the cubic VN structure is obtained about 1012–1043 K, which is comparable with theoretical values 848.88 [14] and 838 K [30] and the experimental amount of 772 K [31]. Since for temperatures lower than Debye's temperature it is possible to calculate thermodynamic properties of crystalline structures, using the Debye–Grüneisen approach, all the thermodynamic properties of cubic VN are presented further. Within the continuum approach, the second Piola-Kirchhoff (PK2) can be used to investigate the effect of large strains on structure's behavior. In order to investigate the structure's behavior under large 241
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Fig. 9. Thermal expansion (α ) and volume (strain) variation curves against temperatures up to the Debye temperature for cubic VN.
ductility, studying the PK2 plots revealed that VN has considerable elasticity of about 8%. Then, using the elastic constants and mechanical parameters, we calculated the Debye temperature of VN. Since thermodynamic behavior of materials at temperatures lower than Debye's temperature can be investigated using the Debye–Grüneisen approach, we used this method to investigate temperature dependence of the bulk modulus, specific heat capacity and thermal expansion coefficient of the cubic VN. The problem was that the experimentally measured value for the bulk modulus of VN is 299.74 GPa, but the theoretical calculations have in cases by far overestimated it. Our calculations indicated that the bulk modulus decreases with temperature. It was found out that at room temperature the bulk modulus was about 294.53 GPa which is fairly consistent with the experimentally measured value of 299.74 GPa. Furthermore, according to our calculations as the temperature was raised to 400 K the bulk modulus plummeted to 284.24 GPa and finally it dropped to 221.25 GPa at 1000 K. At last, we observed that as the temperature increases the material undergoes a thermal expansion which is equivalent to the applied strain such that at room temperature the material experiences a 19% strain.
the ground state temperature are very moderate and at about 150 K the slope begins to decline more steeply. For temperatures higher than approximately 400 K we observe a linear reduction behavior in the bulk modulus. Specific heat coefficients at constant volume (CV ) and constant pressure (Cp ) are also plotted in Fig. 8. Specific heat illustrates vibrational properties of the structure. CV and Cp of VN at atmospheric pressure and room temperature are 133.43 and 135.95 J mol−1 K−1, respectively. The relationship between CV and Cp is defined as S14. In Fig. 8 we can clearly see that at low temperatures CV and Cp are very close to each other and the two curves are fairly overlapping; however, as the temperature increases the difference between these two coefficients gradually rises or they split up by increasing the expansion coefficient. This is the main reason that at temperatures higher than 300 K these two curves separate from each other such that Cp becomes greater than CV . Thermal expansion curve in terms of temperature is depicted in Fig. 9. It is easy to see that as the temperature increases the expansion coefficient rises up such that at room temperature it becomes 2.63 × 10−5 1/K. It continues going up until at 400 K it reaches to 3.19 × 10−5 1/K and finally hits 5.13 × 10−5 1/K at 1000 K. Using the obtained volume expansion coefficient defined as Eq. S (15), length variation and the strain, Eq. S(16), due to the temperature increase can be calculated for the VN structure. In Fig. 9 we observe that by raising the temperature to room temperature VN undergoes a 19.9% strain with respect to its equilibrium state. This strain increases to 23.3% at 400 K and eventually at 1000 K, which is near the Debye temperature of the VN, it experiences a 40% increase.
Acknowledgment The authors gratefully acknowledge the support of the Nano Research and Training Center of Iran (NRTC.ir) under Grant No. 15.97.06.25. Computational resources were provided by the Davami Computing Cluster belonging to Tafresh University which is supported by the Nano Research and Training Center of Iran (NRTC.ir). Appendix A. Supplementary data
4. Conclusions
Supplementary data to this article can be found online at https:// doi.org/10.1016/j.matchemphys.2019.02.082.
In this paper using the PBEsol functional alongside the PAW pseudopotentials, we have computed the elastic constants of rock-salt VN. Mechanical parameters such as bulk, Young and shear moduli and Poisson's ratio have been calculated based on the continuum theory and using the Voigt, Reuss and Hill methods. The elastic constants C11 = 580.97, C12 = 205.02 and C44 = 108.92 GPa have been obtained. It has been determined that the results calculated using the three methods are very close to each other. A ground state bulk modulus of 330.33 GPa and Young's modulus equal to 370.26 GPa classifies the cubic VN as a hard material. Additionally, the shear modulus was calculated to be between 130.95 and 140.54 GPa and the ductility ratio is about 2.35–2.52 which predicts good malleability for VN. On the other hand, Young's modulus of the cubic VN along its V-N bonds are much greater than its Young modulus along other directions indicating that VN has higher resistance under tensions along the < 100 > , < 010 > and < 001 > directions. Therefore, for hard coating purposes using VN it is better to perform coatings and sputtering along the mentioned directions. In addition to its toughness and good
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