First-principle calculations of thermodynamic properties of ZrC and ZrN at high pressures and high temperatures

Physica B 410 (2013) 57–62

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First-principle calculations of thermodynamic properties of ZrC and ZrN at high pressures and high temperatures Arash Abdollahi n Department of Physics, Faculty of Science, Urmia Branch, Islamic Azad University, Urmia, Iran

a r t i c l e i n f o

abstract

Article history: Received 25 June 2012 Received in revised form 27 October 2012 Accepted 31 October 2012 Available online 10 November 2012

Ab initio calculations for the thermal properties of ZrC and ZrN have been performed by using the projector augmented-wave (PAW) method within the generalized gradient approximation (GGA). Pressure–temperature-dependent thermodynamic properties including the bulk modulus, thermal expansion, thermal expansion coefficient, heat capacity at constant volume and constant pressure were calculated using three different models based on the quasi-harmonic approximation (QHA): the ¨ Debye–Slater model, Debye–Gruneisen model and full quasi-harmonic model (that requires the phonon density of states at each calculated volume). Also the empirical energy corrections are applied to the results of three models. The calculated values are in good agreement with experimental results. It is found that the full quasi-harmonic model provides more accurate estimates in comparison with the other models. & 2012 Elsevier B.V. All rights reserved.

Keywords: ZrC ZrN First-principle calculation Quasi-harmonic Thermodynamic properties High pressure

1. Introduction Zirconium nitride (ZrN) and Zirconium carbide (ZrC) ceramics are characterized by high melting point, high stiffness, wear resistance and solid-state phase stability [1–3]. They also have high hardness, high thermal and electrical conductivity and low evaporation which make them widely used for cutting tools, and also wear-resistant coatings, manufacturing of electrodes and filaments [4]. Because of the special properties of maintained ceramics, they have been extensively studied both experimentally and theoretically. Me´c- abih et al. [5] and Saha et al. [6] have studied the structure and electronic properties of ZrC and ZrN, respectively. There are a number of experimental reports on the thermal expansion of ZrC [7,8] and ZrN [9–12]. Also, several measurements have been made to determine the heat capacities of ZrC [13–16] and ZrN [17–19]. There have been a number of theoretical studies of the thermodynamic properties of ZrC and ZrN. In the framework of the quasi-harmonic approximation, the heat capacities of the cubic metal nitrides have been investigated by Iikubo et al. [20] and Wang et al. [21]. Lu et al. [22,23] evaluated the thermodynamic properties of nitrides and carbides ¨ at temperatures up to 3000 K by using the Debye–Gruneisen model [24]. Jun et al. [25] and Fu et al. [26] have investigated the thermodynamic properties of ZrC and also Hao et al. [27] have

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calculated the thermodynamic properties for ZrN, by using quasiharmonic Debye model [28]. Also there are a few theoretical studies of the structural phase transformation from the NaCl-type (B1) structure to CsCl-type (B2) structure of the ZrC and ZrN. Theoretical results for phase transition pressures are 295 GPa [27] and 303 GPa [29] for ZrC, also 205 GPa [27] for ZrN. In this paper, I report my theoretical study of the thermal properties of zirconium carbide and zirconium nitride and compare the results of Debye– ¨ Slater model [30], Debye–Gruneisen model and full quasi-harmonic model [31]. The results obtained from all three models for the bulk modulus, thermal expansion, coefficient of volume thermal expansion (CVTE), heat capacity at constant volume (CV) and constant pressure (CP) are reported and discussed. 2. Theoretical method 2.1. First-principles and phonon calculations The ab initio calculations were performed within the density functional theory (DFT), using the plane-wave pseudo-potential method as implemented in the Quantum-ESPRESSO package [32] with the ultrasoft Vanderbilt pseudo-potentials [33]. For the exchange and correlation terms in the electron–electron interaction, the generalized gradient approximation (GGA) of Perdew– Burke–Eruzerhof (PBE) [34] has been used. The plane-wave energy cutoff was 80 Ry and the k-grids used in total energy were 16  16  16 Monkhorst–Pack (MP) meshes [35]. Density

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A. Abdollahi / Physica B 410 (2013) 57–62

functional perturbation theory (DFPT) [36] as implemented in Quantum-ESPRESSO was used for phonon calculations. For the phonon calculations we have used 80 Ry energy cutoff for all atoms together with an 8  8  8 mesh of k-points. The selfconsistent threshold value for convergence was 10  10 Ry, and for

phonon calculations it was set to 10  16 Ry. The dynamical matrices are produced in a k-point grid of 12  12  12 in irreducible wedge of the Brillouin zone. For Brillouin zone integration, the first-order Methfessel–Paxton method [37] was used with a smearing width of 0.05 Ry.

Fig. 1. Equation of states of ZrC (a) and ZrN (c). The coefficient of volume thermal expansion of ZrC (b) and ZrN (d).

Fig. 2. Temperature dependence of the volume of ZrC (a) and ZrN (c). The coefficient of volume thermal expansion of ZrC (b) and ZrN (d) as a function of temperature.

A. Abdollahi / Physica B 410 (2013) 57–62

2.2. Quasi-harmonic approximation In the harmonic approximation, the free energy F(V,T) as a function of volume (V) and temperature (T) is given by: F ðV,T Þ ¼ E0 ðVÞ þkB T

X

 ½2sinh _on ðk,V Þ=2kB TÞ, k,n

where kB is the Boltzmann constant, and on(k,V) is the frequency of the phonon mode for wave vector k and volume V. phonon densities of states at several volumes (11 equally spaced volumes from 430 bohr3 to 780 bohr3 in this work) were calculated around the equilibrium volume. The calculated Helmholtz free energy versus volume was fitted to isothermal third-order finite strain equations of state (EOS). The pressure–volume curves have been described by a third-order Birch–Murnaghan EOS [38], P ¼ 3=8B0



     x2=3 1 = x10=3 Þ 3B00 x216x231=3 B00 24 Þ

where x¼(V/V0), B0 is bulk modulus at zero pressure, and B00 ¼ (dB/ dP)p ¼ 0. The entropy (S), internal energy (U) specific heat capacity (CV) at constant volume, isothermal bulk modulus (BT), thermody¨ namic Gruneisen ratio (gth), coefficient of volume thermal expansion (a), heat capacity at constant pressure (CP) and adiabatic bulk

59

modulus (BS) can be expressed as:   S ¼ @F=@T V ,

ð3Þ

U ¼ F þ TS,

ð4Þ

  C V ¼ @U=@T V , 

BT ¼ V @2 F=@V 2 

ð5Þ  T

,

ð6Þ



ð7Þ

a ¼ gth C V =ðBT V Þ,

ð8Þ

  C p ¼ C V 1 þ gth aT ,

ð9Þ

  BS ¼ BT 1 þ gth aT :

ð10Þ

gth ¼  V=ðC V T ÞÞ @ðTSÞ=@VÞT , 



In the Debye–Slater model, the phonon contribution to the free energy is determined by approximating the phonon density of states as a quadratic function of frequency up to a cutoff ¨ frequency, called the Debye frequency. In the Debye–Gruneisen model, the quasi-harmonic approximation has been used to calculate the volume dependence of the Debye frequency [31]. In order to calculate the thermal properties of ZrC, we used the

Fig. 3. Pressure dependence of the heat capacities of ZrC (a) and ZrN (c) at constant volume. The heat capacities of ZrC (b) and ZrN (d) at constant pressure as a function of pressure.

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A. Abdollahi / Physica B 410 (2013) 57–62

Fig. 4. Temperature dependence of the heat capacities of ZrC (a) and ZrN (c) at constant volume. The heat capacities of ZrC (b) and ZrN (d) at constant pressure as a function of temperature.

Table 1 Zero pressure coefficient of volume thermal expansion (a) and bulk modulus (B0) for ZrC and ZrN together with the available literature data. Present calculations

ZrC a (10  6 /K) T¼298 K T¼1000 K T¼1400 K T¼2000 K T¼2400 K B0 (GPa) ZrN a (10  6/K) T¼298 K T¼1000 K T¼2000 K T¼2500 K B0 (GPa) a

Ref. [7]. Ref. [8]. Ref. [13]. d Ref. [14]. e Ref. [42]. f Ref. [43]. g Ref. [9]. h Ref. [10]. i Ref. [12]. j Ref. [6]. k Ref. [21]. l Ref. [44]. m Ref. [45]. n Ref. [46]. b c

Debye

¨ Debye–Gruneisen

QHA

19.5 25.6 27.1 29.4 31.1 222.183

15.9 20.7 21.8 23.4 24.6 222.183

16.6 22.9 24.6 27.4 29.7 222.356

20.5 27.3 31.7 34.4 259.0

16.9 22.4 25.7 27.5 257.2

16.9 24.8 28.9 31.1 257.9

Other calculations

Experiments

218c, 225d, 238e

17.0a 22.5a, 22.3b 23.7a, 24.5b 27.8b 30.0b 223f

289j, 264k

17.4g 24.5g, 23.8h, 23.5i 30.6g 32.9g 264l, 250m, 285n

A. Abdollahi / Physica B 410 (2013) 57–62

¨ full quasi-harmonic, Debye–Gruneisen and Debye–Slater approaches as implemented in the GIBBS code [31]. The empirical energy corrections are applied to the results of three models to correct the systematic errors introduced by the functional, in the calculation of equilibrium volumes [31]. The corrected static energy (ES ) is defined as follows:

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The ZrC and ZrN crystals have a cubic close packed (NaCl-type) structure. The equilibrium lattice constant calculated for ZrC is ˚ which compares with experimental values of 4.68 A˚ a¼4.708 A, [39] and 4.698 A˚ [40]. For ZrN, the equilibrium lattice constant was ˚ which compares with experimental obtained to be a¼4.585 A, ˚ values of a¼4.577 A [17], 4.586 A˚ [10], and 4.610 A˚ [41]. The equilibrium volumes at room temperature that has been used for empirical energy corrections for ZrC and ZrN set to 4.69 A˚ and ˚ respectively, that are the mean value of experimental data. 4.59 A, The results of the full-phonon quasi-harmonic analysis for ZrC and ZrC at different temperature and pressure are plotted in Figs. 1–4.

The EOS curves at different temperatures are shown in Fig. 1(a) and (c). Fig. 1(b) and (d) shows CVTE curves as a function of pressure. It can be seen that for both compounds, CVTE drops rapidly by increasing pressure up to 50 GPa (as becomes half at 25 GPa) at very high temperature. Fig. 2(a) and (b) shows the temperature dependences of the volume up to 3000 K at pressures of 0, 50, 100 and 200 GPa. Fig. 2(b) and (d) shows the CVTE plots up to 3000 K at pressures of 0, 50, 100 and 200 GPa. It can be seen that the volume varies rapidly by increasing temperature at lower pressure. CV of ZrC and ZrN as a function of pressure are shown in Fig. 3(a) and (c). Both figures show that, at room temperature, CV highly depends on the pressure. For example, at room temperature, CV of ZrC changes from 37.6 (J/mol K) to 27.5 (J/mol K) when pressure varies from 0 to 200 GPa. Cp of ZrC and ZrN are plotted in Fig. 3(b) and (d). As can be seen from these figures, CP behaves like CV as long as the temperature is sufficiently low. But unlike CV, CP drops rapidly when pressure increases from 0 to 50 GPa at very high temperature. Temperature dependences of CV for ZrC and ZrN are shown in Fig. 4(a) and (c). Fig. 4(b) and (d) shows the variation of CP with temperature at different pressures up to 200 GPa, for ZrC and ZrN, respectively. It is apparent from the figures that at zero pressure, for both compounds, the heat capacity increases rapidly with increasing temperature for T42000 K while at elevated pressures, the heat capacity weakly depends on temperature at high temperatures. In order to compare the results of the full-phonon calculations ¨ with those obtained with the Debye–Slater and Debye–Gruneisen models, I used all three models with the same parameters for electronic calculations. The results for bulk modulus and CVTE at

Fig. 5. The coefficient of volume thermal expansion of ZrC (a) and ZrN (b) as a function of temperature. The solid, dashed, and dotted curves denote those of ¨ QHA, Debye–Gruneisen and Debye, respectively. Experimental data are also displayed for comparison.

Fig. 6. Temperature dependence of the heat capacities of ZrC (a) and ZrN (b) at constant pressure. The solid, dashed, and dotted curves denote those of QHA, ¨ Debye–Gruneisen and Debye, respectively. Experimental data are also displayed for comparison.

E  sðVÞ ¼ ES ðVÞ þ DPV

ð11Þ

where ES is static energy and DP is the pressure. The correction used in this work is done by choosing the proper pressure to provide the equilibrium volume at room temperature equal to the experimental results. The calculation procedure is described in more detail in [31].

3. Results and discussion

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zero pressure together with the available literature data are presented in Table 1. The results show that all of three models lead to approximately the same values of bulk modulus that are in good agreement with the experimental values. The values of the CVTE obtained from three models are different from each other especially when temperature is higher than room temperature. The results of the quasi-harmonic Debye analysis is higher while ¨ the Debye–Gruneisen model gives lower values than the experimental values. It has been found that the full-phonon calculations give the results in reasonably good agreement with experiments as compared to other two models. Differences between calculated and experimental values [7] of the volume thermal expansion coefficient of ZrC at 1000 K are about 1.8%, 8.0% and 13.8% for ¨ quasi-harmonic, Debye–Slater and Debye–Gruneisen models, respectively. The same quantities for ZrN are about 1.1%, 11.3% and 8.7% in comparison with the experimental data [9]. Temperature dependence of the coefficient of volume thermal expansion of ZrC and ZrN obtained from full-phonon calculations, ¨ Debye and Debye–Gruneisen models together with the experimental data are shown in Fig. 5(a) and (b), respectively. In Fig. 6, the results of three models for heat capacities at constant pressure for ZrC (a) and ZrN (b) are compared with experimental data. As can be seen from the figure, the full-phonon calculations give better results than those obtained from the Debye and ¨ Debye–Gruneisen models.

4. Conclusions The first-principle calculations have been performed to obtain thermodynamic properties of zirconium carbide and zirconium nitride ceramics at high pressure (up to 200 GPa) and high temperature (up to 3000 K). Some basic thermodynamic quantities such as bulk modulus, heat capacity at constant volume and constant pressure, thermal expansion and coefficient of volume thermal expansion have been calculated as a function of pressure and temperature based on the full-phonon quasi-harmonic, ¨ Debye and Debye–Gruneisen approximations. The empirical energy corrections are applied to the results of three models to correct the systematic errors introduced by the functional. The results obtained from full-phonon quasi-harmonic analysis show that this method can be used to determine thermodynamic properties of both compounds with a reasonable accuracy. Comparing the results of full-phonon calculations for heat capacity and coefficient of volume thermal expansion with those obtained ¨ by Debye and Debye–Gruneisen approximations indicates that, full-phonon calculations produce more accurate estimates than other two models. The results obtained from full-phonon calculation for coefficient of volume thermal expansion show that estimated error for ZrC and ZrN at temperatures between 298 K and 1500 K is not more than about 4% and 3%, respectively, in comparison with the experimental data [7,9].

Acknowledgments The author wishes to acknowledgement Dr. M. Oloumi for his useful advice and suggestions. The author also would like to thank Dr. M.A. Blanco and his co-workers for their GIBBS code.

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