Temperature- and pressure-dependent elastic properties, thermal expansion ratios, and minimum thermal conductivities of ZrC, ZrN, and Zr(C0.5N0.5)

Author’s Accepted Manuscript Temperature- and pressure-dependent elastic properties, thermal expansion ratios, and minimum thermal conductivities of ZrC, ZrN, and Zr(C0.5N0.5) Jiwoong Kim, Yong Jae Suh www.elsevier.com/locate/ceri

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S0272-8842(17)31408-6 http://dx.doi.org/10.1016/j.ceramint.2017.06.195 CERI15717

To appear in: Ceramics International Received date: 12 May 2017 Revised date: 22 June 2017 Accepted date: 30 June 2017 Cite this article as: Jiwoong Kim and Yong Jae Suh, Temperature- and pressuredependent elastic properties, thermal expansion ratios, and minimum thermal conductivities of ZrC, ZrN, and Zr(C0.5N0.5) , Ceramics International, http://dx.doi.org/10.1016/j.ceramint.2017.06.195 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Temperature- and pressure-dependent elastic properties, thermal expansion ratios, and minimum thermal conductivities of ZrC, ZrN, and Zr(C0.5N0.5)

Jiwoong Kima,b, Yong Jae Suha,b,* a

Mineral Resources Research Division, Korea Institute of Geoscience and Mineral Resources, 124 Gwahang-no, Yuseong-gu, Daejeon 34132, Republic of Korea

b

NT-IT, Korea University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea

*

Corresponding Author. Tel.: +82-42-868-3570; Fax: +82-42-868-3415; E-mail address: [email protected].

Abstract We investigated the temperature- and pressure-dependent properties of ZrC, ZrN, and Zr(C0.5N0.5) solid solution using first-principles calculations with Debye–Gruneisen theory. The properties investigated included elastic moduli, thermal expansion coefficients, and thermal conductivities. Equilibrium volumes were determined by obtaining the energy–volume (E–V) curves of ZrC, ZrN, and Zr(C0.5N0.5) at different temperatures and external pressures. We adopted quasi-harmonic and quasi-static approximations to examine the influence of temperature and pressure on the elastic properties. Throughout the temperature and pressure ranges studied, the bulk modulus of the ZrN phase was higher than that of the other two phases, whereas its shear and Young’s moduli were lower. An increase in nitrogen content resulted in an increase in the thermal expansion coefficient. The hardness, anisotropic properties, and minimum thermal conductivities of Zr(C0.5N0.5) and ZrC were similar, whereas those of the Zr(C0.5N0.5) phase were somewhat larger. Given its elastic properties, anisotropy, hardness, and minimum thermal conductivity, Zr(C0.5N0.5) is the best of the three phases for extreme environmental applications. Our results provide fundamental and useful information on the ZrC, ZrN, and Zr(C0.5N0.5) phases.

Keywords: C: Mechanical properties; C: Thermal properties; E: Refractories; Carbonitride

1

Introduction Transition metal carbides and nitrides are of great interest because of their excellent properties, such as superior hardness, good chemical stability, and high melting point [1-7]. Hence transition metal carbides, nitrides, and their solid solutions have recently been widely employed for applications in extreme environments [8-12]. For example, cutting tools used in extreme tribological environments operate at temperatures ranging from 1000 to 1200 K and therefore require both excellent mechanical properties and high-temperature durability. Another example is a fourth-generation gas-cooled fast reactor that operates at temperatures between 1300 and 1800 K [13, 14]. Only Zr-based transition metal carbides have the high-temperature durability and chemical stability necessary for this application [13, 15, 16]. Accordingly, knowledge of the mechanical and thermophysical properties of transition metal carbides and nitrides under extreme conditions is required. It is difficult and costly to obtain these properties experimentally using specialized equipment, making alternative routes such as theoretical calculations attractive. Recent advances in computing power and computational theory, particularly in first-principles calculations, have allowed the prediction of various extreme properties of transition metal carbides and nitrides [17-20]. Although several studies have investigated the mechanical and elastic properties of ZrC, ZrN, and their solid solutions using first-principles calculations, few have focused on their temperature-dependent properties, much less on their pressure-dependent properties [21-25]. For example, Hao et al. [21] calculated the electronic, elastic, and thermophysical properties of ZrC and ZrN at temperatures and pressures up to 1500 K and 100 GPa, respectively, and Abdollahi [26] calculated the temperature- and pressuredependent themophysical properties of these compounds. The temperature- and pressure-dependent elastic and thermophysical properties of Zr(C1-xNx) solid solutions remain poorly studied, and there is essentially no information regarding other elastic and thermal properties, such as isotropy and thermal conductivity. In this study, we investigated the temperature- and pressure-dependent elastic and thermophysical properties, as well as the thermal conductivity, of ZrC, ZrN, and their solid solution phase using first-principles calculations based on Debye–Gruneisen theory [25, 27]. Our results provide fundamental information on Zr-based transition metal carbides, nitrides, and their solid solutions, which will be useful for extreme environment applications.

Calculation The Vienna ab initio simulation package was used to obtain the temperature- and pressure-dependent properties of ZrC, ZrN, and Zr(C0.5N0.5) solid solutions [28-30]. We obtained the random disordered solid solution for Zr(C0.5N0.5) by adopting the special 2

quasi-random structure (SQS) model proposed by Zunger et al. [31]. Exchangecorrelations were implemented using generalized gradient approximation (GGA). To integrate the Brillouin zone, we used the Monkhorst pack 13 ´ 13 ´ 13 and 7 ´ 11 ´ 3 kpoints for the B1 (cubic) structure of ZrC and ZrN, and the SQS model of Zr(C0.5N0.5), and obtained these k-points from the k-points convergence test shown in Fig. 1. Accurate results were obtained by using a high-energy cutoff of 500 eV with a precise energy convergence of 0.01 eV/Å. Integration was conducted using a tetrahedron method with Bloch corrections. The influence of temperature was examined using Debye–Gruneisen theory and quasi-harmonic approximation (QHA) [25, 27, 32-34]. Equilibrium volumes at different temperatures and pressures were obtained by adopting the Birch–Murnaghan equation of state with five parameters [18]:

E(V) =  + V / + V !/ + "V  + #V $/

(1)

where E(V) represents the free energy at a certain volume V, and a to e are the fitting parameters. The non-equilibrium Gibbs free energy G* can be obtained as follows [27]:

G∗ (x, P, T) = E&'& (x) + PV(x) + A&'& (x, T)

(2)

where Etot and x are the total electronic energy and configuration vector, respectively, and PV(x) is the pressure and volume effect on the total Helmholtz free energy, Atot(x,T). The Helmholtz free energy is described as follows:

A&'& (x, T) = E* (x, T) − TS* (x, T) + E.0 (x, T) − TS.0 (x, T)

(3)

where ED and SD are the internal energy and entropy, respectively, obtained from the Debye vibrational lattice, and Eel and Sel are the electronic contributions to the internal energy and entropy, respectively. Eel and Sel were excluded in the current study [17]. The elastic and thermal expansion properties were obtained from strain–stress relations and quasi-static approximation [35-38]. Hardness was obtained by the following empirical formulae [39, 40]:

3

Hv1 = 0.92k1.15 6 7.57$

(4)

Hv = 2(8  6)7.:$: − 3

(5)

where k is Pugh’s constant (G/B), and G and B are the shear and bulk modulus, respectively. We also investigated the anisotropic properties of ZrC, ZrN, and Zr(C0.5N0.5) using the pertinent equations employed in previous studies [41-43]. Finally, we examined the minimum thermal conductivity at high temperature in the same way as proposed by Clarke [44].

Results and discussion B1 structured transition metal carbides and nitrides have two sub-lattices that are occupied by transition metals and non-metals. When solid solution carbo-nitrides decompose, carbon and nitrogen atoms situated on identical sub-lattices mix randomly with each other [34, 45]. We adopted SQS models to depict the random mixing of carbon and nitrogen atoms in the B1 sub-lattice, and Fig. 2 shows the computational models employed to obtain the temperature- and pressure-dependent properties of ZrC, ZrN, and Zr(C0.5N0.5). The ZrC and ZrN models have identical crystal structures (B1), whereas the model for Zr(C0.5N0.5) shows a monoclinic lattice system (Pm, space group no. 6 symmetry). The applicability of SQS models in describing elastic and thermophysical properties has been demonstrated [46-50], despite the different crystal structures and low symmetries of these models. The reproducibility of adopted models was verified by inspecting detailed structural information of the SQS model for Zr(C0.5N0.5) (Table 1). To ensure the reliability of our calculations, we first compared our calculated results at a temperature of absolute zero and a pressure of 10–4 GPa (Table 2) with those previously obtained experimentally and theoretically [21, 23, 24, 34, 51-56]. The equilibrium structures were determined from energy–volume curves. The calculated results represent slightly larger lattice parameters and volumes than previous experimental and theoretical values, and thus our elastic modulus values are slightly lower than those reported previously. This is a characteristic feature of the generalized gradient approximation approach that employs exchange correlations in density functional theory calculations [17, 53, 57]. Regardless, our results are in overall agreement with those of previous studies.

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Our calculated value for the elastic modulus of Zr(C0.5N0.5) is higher than other values previously calculated. This is strongly related to the valence electron concentration (VEC) per cell. In a previous electronic band structure analysis [58], the transition metal carbonitride system exhibited maximum elastic properties at a VEC of ~8.4 per cell. Therefore, an approximate atomic ratio of C:N of 1 may provide the maximum elastic modulus value for this system. This trend of Zr(C1-xNx) solid solutions was also observed in a previous theoretical investigation [23]. All results from static calculations (T = 0) at 10–4 GPa indicate high reliability of the computational methods and models employed in the present study. Based on the static calculation results, we calculated the Debye temperature, Pugh’s ratio, isotropic index, hardness, and high-temperature minimum thermal conductivity values of ZrC, ZrN, and Zr(C0.5N0.5) (Table 3) [23, 24, 34, 59, 60]. Generally, the Debye temperature indicates material rigidness: i.e., the higher the Debye temperature, the more rigid the material. Using this criterion, Zr(C0.5N0.5) is the most rigid of the three materials and ZrN is the softest. This trend is also observed in the Pugh’s constant and hardness values. A Pugh’s constant larger than 0.571 indicates brittleness, and thus the ZrN phase is more ductile than the other phases, whereas the hardness values indicate that Zr(C0.5N0.5) is the hardest of the three phases. The thermal conductivity of solid materials varies with temperature and pressure because the phonon mean free paths and vibration properties of material depend on temperature and pressure. At high temperature, solid materials converge to their minimum value of thermal conductivity, as proposed by Clarke [61]. Minimum thermal conductivity is proportional to the mean acoustic velocity, and this velocity is influenced by the rigidness of material. Therefore, the results of the present study rate the minimum thermal conductivities in the order Zr(C0.5N0.5) > ZrC > ZrN, consistent with the elastic properties mentioned above. Anisotropy is an important factor that affects the mechanical properties of structural materials. In particular, anisotropy directly influences crack initiation, propagation, and lattice distortion, and thus information regarding the anisotropic properties of structural materials is essential. Universal and shear anisotropic indexes (Table 3) indicate that ZrC is the most isotropic of the three phases, whereas ZrN shows high anisotropy. The anisotropic values of the ZrC and ZrC0.5N0.5 phases are similar, but there are significant differences between the values of these two phases and that of ZrN. As a result, anisotropy increased in the order ZrC ~ ZrC7.: N7.: ≪ ZrN. The thermal expansion properties of ZrC, ZrN, and Zr(C0.5N0.5) are shown in Fig. 3. For convenience, we divided the temperatures into three ranges. In the lowertemperature ranges (I and II), Zr(C0.5N0.5) exhibited the lowest thermal expansion of the three phases. As the temperature increased above 1000 K, the thermal expansion of the ZrC phase became the lowest, suggesting that ZrC is the best material for hightemperature applications in terms of its thermal expansion characteristics.

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The effects of pressure on the thermal expansion of ZrC, ZrN, and Zr(C0.5N0.5) are shown in Fig. 4. An increase in external pressure depressed the thermal expansion of all three phases, showing similar volume compression behavior in each case. Pressure effects became significant with increasing temperature. Dramatic compression was detected for each phase up to 40 GPa external pressure, whereas above this pressure the compression behavior became insignificant. Linear fittings of the 40 GPa data in Fig. 4 reveal that despite high external pressure, the thermal expansion behavior of all three phases remained unchanged. The effects of temperature and external pressure on the elastic moduli of ZrC, ZrN, and Zr(C0.5N0.5) (Fig. 5) exhibit that all the moduli decreased as the temperature increased. On the other hand, an increase in external pressure resulted in the elastic moduli of the three phases decreasing. In general, the bulk modulus is proportional to the number of valence electrons or electron density [18, 34]. The calculated bulk moduli agreed well with this general trend because the number of valence electrons increased with increasing nitrogen content. In addition, the bulk moduli appear to be linearly proportional to the external pressure, regardless of temperature. Finally, the shear and Young’s moduli showed trends similar to that of the bulk moduli although they deviated somewhat from proportionality to changes in pressure. At higher pressures, degradation of the shear and Young’s moduli decreased with increasing temperature. It should be noted that Zr(C0.5N0.5) showed the highest elastic moduli of the three phases over all temperatures and pressures studied, but not the highest bulk modulus.

Conclusions We investigated the temperature- and pressure-dependent elastic properties, thermal expansion, and minimum thermal conductivities of the ZrC, ZrN, and Zr(C0.5N0.5) phases using first principles calculations. Zr(C0.5N0.5) showed the highest elastic moduli, except for the bulk modulus. In particular, its Debye temperature, Pugh’s constant, hardness, and temperature- and pressure-dependent shear and Young’s moduli were excellent, indicating that Zr(C0.5N0.5) is a promising material for applications in extreme environments. Further control of the ratio between carbon and nitrogen might lead to a better-quality Zr(C1-xNx) phase. We believe that these results provide a rigorous foundation for understanding the ZrC, ZrN, and Zr(C0.5N0.5) phases, and a useful guideline on how to develop new ZrC- and ZrN-related materials for use in extreme environments.

Acknowledgements

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This research was supported by the Basic Research Project of the Korea Institute of Geoscience and Mineral Resources (KIGAM) funded by the Ministry of Science, ICT and Future Planning of Korea.

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Fig. 1. K-points convergence test of the Fm-3m (Cubic, B1) and Pm (Monoclinic, SQS) models: a) and b) Variations of total energy and lattice parameter for the B1 structure model, respectively. c) and d) Variations of total energy and volume for the SQS model, respectively.

Fig. 2. Schematic model structures of the B1 and SQS models: a) B1 (cubic) model and b) SQS (monoclinic) model.

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Fig. 3. Temperature-dependent volume variations of the ZrC, ZrN, and Zr(C0.5N0.5) phases: a) entire temperature range and b–d) temperatures of ranges I, II, and III, respectively.

Fig. 4. Temperature- and pressure-dependent volume variations of the ZrC, ZrN, and Zr(C0.5N0.5) phases: a) ZrC, b) ZrN, and c) Zr(C0.5N0.5). d) Linear fittings of the volume variations at 40 GPa.

Fig. 5. Temperature- and pressure-dependent elastic bulk, shear, and Young’s moduli of the ZrC, ZrN, and Zr(C0.5N0.5) phases: a) bulk, b) shear, and c) Young’s moduli.

Table 1. Detailed structural information of the SQS model for randomly distributed Zr(C0.5N0.5) solid solutions Chemical formula

Zr(C0.5N0.5)

Space group

Pm (No. 6)

5.6964 0.0000 0.0000 Lattice vectors

0.0263 3.2901 0.0000 -1.9012 0.0000 10.7337

Unit cell volume (Å3) / Density (g/cm3)

201.17 / 6.883

0.8747 0.0000 0.6278 C

0.9998 0.5000 0.4998 Ti

0.3754 0.0000 0.6279 C

0.2478 0.0000 0.7488 Ti

0.8734 0.5000 0.1235 C

0.7509 0.5000 0.7486 Ti

0.1267 0.0000 0.3765 C

0.2500 0.5000 0.2504 Ti

0.3766 0.5000 0.6265 N

0.4999 0.0000 0.5003 Ti

0.6233 0.0000 0.8735 N

0.0004 0.0000 0.0011 Ti

Fractional atomic positions

12

0.1250 0.5000 0.8722 N

0.5007 0.5000 0.0015 Ti

0.6249 0.5000 0.3779 N

0.7507 0.0000 0.2496 Ti

Table 2. Calculated structural information and elastic moduli of ZrC, ZrN, and Zr(C0.5N0.5) Model

Lattice [Å]

ZrC

ZrN

Zr(C0.5N0.5)

P

Volume [ŝ ]

Bulk M.

Shear M.

Young’s M.

Poisson’s ratio

4.718

105.04

217.9

154.8

375.6

0.213

4.705a

104.16a

219c

~160c

~390c

0.20a

4.698b

103.69b

239f

162h

386h

~0.202c

4.689c

103.10c

223g

4.688d

103.03d

207h

4.604

97.60

243.98

135.4

342.6

0.266

4.591a

96.77a

240h

~147c

~360c

0.27a

4.564c

95.07c

246i

130i

332.7i

~0.25c

4.57e

95.44e

4.664

101.47

229.8

165.0

399.3

0.210

~4.65c

~100.55c

~233c

~170c

~405c

~0.21c

376j

The lattice parameters of Zr(C0.5N0.5) were calculated from the volume obtained from the SQS model. a

Ref. 21

b

Ref. 51

c

Ref. 23

d

Ref. 24

e

Ref. 52

f

Ref. 53

13

g

Ref. 54

h

Ref. 55

i

Ref. 34

j

Ref. 56

Table 3. Debye temperature, Pugh’s constant, anisotropic index, empirical hardness, and high-temperature minimum thermal conductivity values of ZrC, ZrN, and Zr(C0.5N0.5) at a temperature of absolute zero and a pressure of 10–4 GPa Model

ZrC

Debye temperature [K]

Pugh’s constant (k = G/B)

Anisotropic index

Empirical hardness

(Au, A1, A2, A3)

(Hv1, [GPa]

Hv2)

678.9

0.7104

Au: 0.0235

Hv1: 22.19

680a*

0.7306d

A1: 0.8697

Hv2: 22.64

Minimum thermal conductivity (kmin) [W/mK]

1.637

A2: 0.8699

627a**

A3: 0.8701

ZrN

624.8

0.5544

Au: 0.4348

Hv1: 15.64

667b

0.5285e

A1: 0.5537

Hv2: 15.14

582c

1.568

A2: 0.5543 A3: 0.5531

Zr(C0.5N0.5)

693.1

0.7188

Au: 0.0435

Hv1: 23.54

A1: 1.1135

Hv2: 24.00

1.690

A2: 1.1046 A3: 0.9516 Au: Universal anisotropic index; Au = 0 and Au = 1 represent the isotropy and anisotropy of the crystal, respectively.

14

A1~3: Shear anisotropic indexes; A1–3 = 1 represents perfect isotropy of the crystal. Any deviation from unity indicates anisotropy of the crystal. a

Ref. 24, a* from the average of the spring constants for Zr and C, a** from the thermal expansion value

b

Ref. 59

c

Ref. 60

d

Ref. 23

e

Ref. 34

15

Figure 1

a)

-19.6

-19.4

-19.2

-19.0

-18.8

-18.6

-18.4

-18.2

-159.90

-159.85

-159.80

-159.75

-159.70

-159.65

-159.60

c)

-159.55

Total electronic energy (eV)

Total electronic energy (eV)

1

1

2

3

2

4

3

6

Kpoints

Kpoints

5

4

8

5

Selected

7

Selected

9

6

10

b)

d)

4.71

4.72

4.73

4.74

4.75

4.76

4.77

201.0

201.1

201.2

201.3

201.4

201.5

Lattice parameter (A) Volume (A3)

1

1

2 3

2

4

3

6

Kpoints

Kpoints

5

4

9

5

6

Kpoints 1: 3x3x1 2: 3x5x3 3: 5x7x3 4: 5x9x3 5: 7x11x5 6: 9x13x5

8

Selected

7

Selected

Kpoints 2: 2x2x2 3: 3x3x3 4: 5x5x5 5: 7x7x7 6: 9x9x9 7: 11x11x11 8: 13x13x13 9: 15x15x15

10

Figure 2

Fig. 2

a)

C

b) C

N

a)

1.000

1.002

1.004

1.006

1.008

1.00

1.01

1.02

1.03

1.04

Figure 3

Volume expansion (V/V0)

Volume expansion (V/V0)

0

b)

50

500

100

1000

200

250

300

Temperature [K]

150

350

1500

400

Range III

Temperature [K]

ZrC ZrN ZrC0.5N0.5

Range I

0

Range I

Range II

ZrC ZrN ZrC0.5N0.5

c)

450

500

1.004 500

1.006

1.008

1.010

1.012

2000 1.014

600

800

Temperature [K]

700

ZrC ZrN ZrC0.5N0.5

Range II

900

1000

1200

1600

Temperature [K]

1400

ZrC ZrN ZrC0.5N0.5

Range III

d)

1000

1.016

1.020

1.024

1.028

1.032

1.036

1.040

1800

2000

Figure 4

a)

c)

Volume expansion (V/V0)

Volume expansion (V/V0)

0.995

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.040

1.045

0.995

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.040

1.045

0

0

500

1000

Temperature [K]

1000

Temperature [K]

10-4 GPa 20 GPa 40 GPa 60 GPa 80 GPa 100 GPa

500

10-4 GPa 20 GPa 40 GPa 60 GPa 80 GPa 100 GPa

1500

1500

2000

2000

b)

d)

Volume expansion (V/V0) Volume expansion (V/V0) 1.000

1.005

1.010

1.015

1.020

1.025

0.995

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.035

1.040

1.045

0

0

Slope (b) ZrC: 0.997 ZrN: 0.998 ZrC0.5N0.5: 0.997

500

1000

Temperature [K]

1000

Temperature [K]

Y=a+bX (40 GPa)

500

10-4 GPa 20 GPa 40 GPa 60 GPa 80 GPa 100 GPa

1500

ZrC ZrN ZrC0.5N0.5

1500

2000

2000

Figure 5

a)

b)

Bulk modulus (GPa)

c)

Shear modulus (GPa)

Young's modulus (GPa)

300

400

500

600

700

125

150

175

200

225

250

275

200

300

400

500

600

0

0

0

20

0K 300 K 1000 K 2000 K

20

0K 300 K 1000 K 2000 K

20

0K 300 K 1000 K 2000 K

40

40

40

60

60

60

80

ZrC

80

ZrC

80

ZrC

100

100

100

0

0

0

20

40

40

40

60

60

60

80

ZrN

80

ZrN

80

ZrN

External pressure (GPa)

0K 300 K 1000 K 2000 K

20

0K 300 K 1000 K 2000 K

20

0K 300 K 1000 K 2000 K

100

100

100

0

0

0

20

0K 300 K 1000 K 2000 K

20

0K 300 K 1000 K 2000 K

20

0K 300 K 1000 K 2000 K

40

40

40

80

100

80

100

60

80

100

ZrC0.5N0.5

60

ZrC0.5N0.5

60

ZrC0.5N0.5