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Structural, electronic, thermophysical properties and bond stiffness of ternary ceramic ScAl3C3 and UAl3C3: Ab initio calculations
Materials Letters 255 (2019) 126610
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Materials Letters journal homepage: www.elsevier.com/locate/mlblue
Structural, electronic, thermophysical properties and bond stiffness of ternary ceramic ScAl3C3 and UAl3C3: Ab initio calculations Feng Shiquan a,⇑,1, Zhao Jianling b, Su Yuling a,⇑,1, Zhang Weibin c, Cheng Xinlu d a
School of Physics and Electronic Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China Division of Radiation Physics, State Key Laboratory of Biotherapy and Cancer Center, West China Hospital, Sichuan University, Chengdu 610041, China c School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China d Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China b
a r t i c l e
i n f o
Article history: Received 22 May 2019 Received in revised form 24 July 2019 Accepted 31 August 2019 Available online 31 August 2019 Keywords: Ternary ceramic First-principles calculations Elastic property Thermal property
a b s t r a c t In this work, we perform the investigation of the structural, electronic, elastic and thermal properties as well as bonding stiffness of ScAl3C3 and UAl3C3 by means of first-principles calculations. Mechanical stability of these two ceramics is discussed by elastic constant calculation. The G/B ratio and Poisson ratio are calculated to discuss their damage tolerance, fracture toughness and the bond composition. By computing the density of states and the bond stiffness of different bonds in ScAl3C3 and UAl3C3, we discuss their bonding properties. The ratio of the weakest bond stiffness to the strongest bond stiffness is presented to discuss their mechanical properties. What’s more, by calculating longitudinal, transverse sound, mean sound velocities, and the Debye temperatures, we discussed the thermal properties of ScAl3C3 and UAl3C3. Ó 2019 Elsevier B.V. All rights reserved.
1. Introduction Due to their high melting point, high hardness, good wear resistance and chemical inertness, binary TiC, NbC, ZrC and HfC ceramics are considered as candidates for thermal protection materials in hypersonic vehicles [1–3]. But their intrinsic brittleness and poor oxidation resistance greatly inhibit the applications of these compounds in industrial fields [3,4] (Fig. 1). As a new type of ceramic materials stacked by M-X octahedron slabs and weakly bonded A slabs, Mn+1AXn (MAX) compounds have attracted tremendous attention for their excellent properties recently. MAX possess layered-hexagonal structure alternately, where M is an early transition metal, A is a group 13–16 element, and X is mainly C or N. Some of them have a better performance than binary ceramics as TiC, NbC, ZrC and HfC in oxidation resistance and damage tolerance [4]. In the past new decades, nearly one hundred MAX phases [5–9] have been successful discovered by different methods since the discovery of Ti3SiC2 by Nowotny and his colleagues [10]. Recent years, by adding the element Al to binary ceramics, a new family of layered ternary carbides (MC)mAl3C2 (where M is metal atom, m = 1, 2, 3. . .) have been synthesized successively ⇑ Corresponding authors. 1
E-mail addresses: [email protected] (S. Feng), [email protected] (Y. Su). Feng Shiquan and Zhao Jianling contributed equally to this work.
https://doi.org/10.1016/j.matlet.2019.126610 0167-577X/Ó 2019 Elsevier B.V. All rights reserved.
[11–13]. Among them, the most typical representatives are Zr-AlC and Hf-Al-C carbides. Studies [14–16] show that these layered ternary carbides have superior oxidation resistance and fracture toughness compared to ZrC and HfC. And it makes them potential refractories for the high stiffness and strength of Zr-Al-C and Hf-AlC carbides. ScAl3C3 and UAl3C3 have similar crystal structures with (MC)mAl3C2 ceramics (M = Hf, Zr) [11]. (MC)mAl3C2 (M = Hf, Zr) possess the advantages of high melting point, high oxidation resistance and fracture toughness. Similarly, whether ScAl3C3 and UAl3C3 own these merits? There are few studies [17] on their related properties, especially for ScAl3C3. In this work, we focused on the mechanical, electronic, elastic, thermal properties and bond characters of ScAl3C3 and UAl3C3, and compared to (HfC)mAl3C2 and (ZrC)mAl3C2 phases.
2. Computational methods and details In present work, we performed first-principles calculations to study the electronic, elastic and thermophysical properties of ternary ceramic ScAl3C3 and UAl3C3 [18] in the ab initio plane-wave code ABINIT. The crystal structures of these two ceramic were optimized by the conjugate gradient (CG) algorithm at the hydrostatic pressure range from 0 to 80 GPa. And the generalized gradient approximation designed by Perdew, Burke, and Ernzerhof (PBE)
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Fig. 1. Crystal structures of (a) ScAl3C3 and (b) UAl3C3.
[19,20] is considered as the exchange-correlation energy in these calculations. The cut-off energy was set to 550 eV and the k points grid was chose as 7 7 2. The Brillouin zone sampling was performed using Monkhorst-pack grid [21]. In the structural optimization, the systems are relaxed until the total energy and ionic Hellmann-Feynman force less than 5 107 eV/atom and 0.005 eV/Å, respectively. The maximum ionic displacement and maximum stress are set to less than 5 104 Å and 0.02 GPa, respectively. 3. Results and discussion 3.1. Equilibrium structure In this work, we set up theoretical crystal model of ScAl3C3 and UAl3C3 based on experimental data from Ref. [11]. Both these two structures are composed of layered stacking composed of MC and Al3C2. They crystallize with hexagonal unit cells, space group P63mc. The experimental lattice parameters are a = 3.352(1) Å and c = 16.778(2) Å for ScAl3C3, while a = 3.387(1) Å and c = 17.393(2) Å for UAl3C3. We optimized these two crystal structures by DFT calculations. For the ScAl3C3 phase, the calculated lattice parameters a and c are 3.357 Å and 16.784 Å, respectively. For the UAl3C3 phase, theoretical a = 3.376 Å and c = 17.441 Å, respectively. Compared with the experimental values, the computed lattice constants for both ScAl3C3 and UAl3C3 are consistent with experimental data within 0.5% deviation. Therefore, the present calculations can reliably reproduce the crystal structure of ScAl3C3 and UAl3C3. As hexagonal crystals, both ScAl3C3 and UAl3C3 satisfy the mechanical stability criterion [22] as following,
C 12 > 0; C 11 C 12 > 0; C 33 > 0; C 44 > 0; ðC 11 þ C 12 ÞC 33 2C 213 > 0 ð1Þ For ScAl3C3, C12 = 94 > 0, C11-C12 = 192 > 0, C33 = 282 > 0, C44 = 147 > 0, (C11 + C12) C33-2C213 = 126118 > 0; For UAl3C3, C12 = 302 > 0, C11-C12 = 274 > 0, C33 = 293 > 0, C44 = 147 > 0, (C11 + C12) C33-2C213 = 96184 > 0. Both of them satisfy the mechanical stability criterion, which suggests that the hexagonal phase ScAl3C3 and UAl3C3 are mechanical stable. More details about the calculated elastic constants of ScAl3C3 and UAl3C3 would be discussed in Sec 3.4. 3.2. Electronic structure and chemical bonding To explore the nature of the interatomic bonding, the electronic properties of ScAl3C3 and UAl3C3 are calculated. Fig. 2 presents the
band structure, total and partial electronic density of states (PDOS) of ScAl3C3 and UAl3C3. For ScAl3C3, there is a very small band gap (0.063 eV); while for UAl3C3, there are several energy bands cross the Fermi level. Due to broadening effect, it can be seen in Fig. 2 that there exist electronic density of states at the Fermi levels for both ScAl3C3 and UAl3C3. So it can be said that both ScAl3C3 and UAl3C3 possess metallic-like nature. It is seen that Sc d states, Al p states and C p states mainly contribute to this behavior for ScAl3C3. For UAl3C3, the f orbitals of U atoms are mainly contributions for the electronic conduction behavior. In addition, there are four peaks in valence bands and one peak in conduction bands for both ScAl3C3 and UAl3C3. The lowest valence bands are mainly contributed by C-s electron with Al-p electron. The similar shape of C-s state and Al-p states indicates a hybridization of C-s and Al-p states to form Al-C bonds. Similarly, it can be seen that the other three peaks in valence bands are the hybridization of electron states between C and Al atoms. For the peak in conduction bands, it can be considered as a result of the hybridization of Sc-d and C-p states to form Sc-C bonds for ScAl3C3, and the hybridization of U-f and C-p states to form U-C bonds for UAl3C3. The peaks in the density of states for ScAl3C3 are sharper than that for UAl3C3, which suggests that the chemical bonds in ScAl3C3 are stronger than in UAl3C3. 3.3. Compressibility and bond stiffness By fitting the pressure dependence of normalized cell volume V/V0 to the Birch-Murnaghan equation of state [23], the bulk moduli B of ScAl3C3 and UAl3C3 are estimated as 180 and 187 GPa, respectively (Fig. 3). As bulk modulus B is a physical quantity to measure the resistance of a material against the volume change under a hydrostatic pressure, this result indicates UAl3C3 has a better performance in resisting external pressure than ScAl3C3. According to the theoretical model proposed by He et al. [24,25], the bond stiffness k can be obtained by fitting the relationship between the relative bond lengths d/d0 (d0 is the bond length at 0 GPa) and the hydrostatic pressure P to a form of polynomial function (d/d0 = C0 + C1P + C2P2, where Ci (i = 0, 1, 2) are the quadratic fitting coefficients) (Fig. 4). And then the bond stiffness k can be expressed as,
dðd=d0 Þ 1 ¼ jC 1 þ 2C 2 Pj1 k ¼ dP
ð2Þ
Using this model, we calculated the quadratic fitting coefficients and bond stiffness of different bonds in ScAl3C3 and UAl3C3 at 0 GPa, and listed them in Table 1. It can be seen that the U-C bonds overall have higher stiffness than Al-C bonds for UAl3C3. But for ScAl3C3, an interesting phenomenon is that the Al2-C2 bonds (667GPa) possess the highest bond stiffness. Another obvious phenomenon is the Al1-C1, Al2-C2 and Al3C3 bonds is much stronger than Al1-C2 and Al3-C2 bonds in Al-C slab for both ScAl3C3 and UAl3C3. Similar results are obtained for (MC)nAl3C2 (M = Hf and Zr) [26,27]. In addition, the ratios of the bond stiffness of the weakest bond to that of the strongest bond for both ScAl3C3 and UAl3C3 are larger than 1/2. According to previous studies [26,28], it means these two compounds are ceramics with low damage tolerance and fracture toughness. 3.4. Elastic stiffness and hardness In above sections, we have revealed the chemical bonding nature of ternary ceramic ScAl3C3 and UAl3C3. It is known that the bonding properties have a close connection with elastic modulus. To disclose the relationship between chemical bonding and elastic stiffness, we calculated and listed the second-order elastic
S. Feng et al. / Materials Letters 255 (2019) 126610
Fig. 2. Band structures of ScAl3C3 (a) and UAl3C3 (b); total and partial electronic density of states of ScAl3C3 (c) and UAl3C3 (d).
Fig. 3. Normalized lattice constants of (a) ScAl3C3 (b) UAl3C3.
3
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Fig. 4. Normalized bond length as function of pressure for (a) ScAl3C3 (b) UAl3C3.
Table 1 The quadratic fitting coefficients and bond stiffness (GPa) of different bonds in (a) ScAl3C3 (b) UAl3C3. ScAl3C3
C1 (1 0 3)
C2 (1 0 6)
k
UAl3C3
C1 (1 0 3)
C2 (1 0 6)
k
Sc-C1 Sc-C3 Al1-C1 Al1-C2 Al2-C2 Al3-C2 Al3-C3
1.73 1.68 1.53 2.02 1.50 2.02 1.58
7.43 6.98 6.18 9.43 6.33 9.44 6.70
578 595 654 495 667 495 633
U-C1 U-C3 Al1-C1 Al1-C2 Al2-C2 Al3-C2 Al3-C3
1.32 1.33 1.60 1.99 1.59 1.91 1.74
5.70 5.68 5.66 9.43 6.17 8.44 7.52
758 752 625 503 629 524 575
constants, Pugh’s ratio, Poisson’s ratio m and elastic stiffness of ScAl3C3 and UAl3C3 in Table 2. Table 2 presents the calculated elastic constants of ScAl3C3 and UAl3C3 as well as some Hf- and Zr-containing (MC)nAl3C2. The estimated B of ScAl3C3 and UAl3C3 are slightly smaller than those we obtained by fitting Eq(1) of the EOS, but the error is within the allowable range. From Table 2, it can be seen that C11 of ScAl3C3 is much higher than that of C33, which is consistent with Hf- and Zr-containing (MC)nAl3C2 as well as other MAX phases [24]. While it is notice that the reverse situation has occurred on UAl3C3. Its C11 is lower than C33, which indicates that the resistance to tensile deformation along the axis is lower than that of the c axis. In addition, the bulk and shear modulus of ScAl3C3 and UAl3C3 are much lower that their counterpart of Hf- and Zr-containing (MC)nAl3C2, which may be due to the different crystal structure and chemical bonding of M-C (M = Hf, Zr, U, Sc) bonds. The relative bond strength order of these four bonds is Zr-C > Hf-C > Sc-C > U-C. Compared to Hf- and Zr-containing (MC)nAl3C2, the lower bond strength of Sc-C and U-C bonds leads to the lower bulk and shear moduli for ScAl3C3 and UAl3C3. Pugh’s ratio, shear to bulk modulus, G/B is commonly used as a criterion to distinguish the ductility or brittleness of materials. From Table 2, the Pugh’s ratios of ScAl3C3, UAl3C3 and other Hfand Zr-containing (MC)nAl3C2 are higher than 0.571 implying their brittle nature. But UAl3C3 possess a much lower G/B ratio than other compounds, suggesting the toughness of UAl3C3 is better than them. In addition, the higher G of ScAl3C3, Hf- and Zrcontaining (MC)nAl3C2 reflect they have a better shear deformation resistance than UAl3C3.
Poisson ratio value m is an important parameter to evaluate the degree of the covalent bonding. For covalent crystals, Poisson ratio is small (m = 0.1), whereas it is about 0.25 for ionic compounds [32]. From Table 2, we can see the Poisson ratios of ScAl3C3, UAl3C3 and other Hf- and Zr-containing (MC)nAl3C2 are between 0.1 and 0.25, which suggests that these two ceramics possess partial ionic and covalent component. Hardness is considered as an important parameter to quantify the crystal structures’ resistance ability to various kinds of deformation. And it is related to the chemical bond types in crystals. Chen et al. proposed an empirical theoretical model to predict the hardness of metals, intermetallics and ceramics [33] by shear and bulk modulus. In this model, the Vickers hardness can be calculated by following formula,
0:585 2 Hv ¼ 2 k G 3
ð3Þ
where k = G/B is the Pugh’s modulus ratio. Using this model, we obtained the calculated hardness of ScAl3C3 and UAl3C3 are 26.7 and 17.3 GPa, respectively. High intrinsic hardness indicates that these two ceramics are relatively ‘hard’ materials. 3.5. Thermophysical properties Debye temperature is an important thermal quantity. It can be used to predict the melting temperature of the material. The Debye temperature, HD, can be calculated from the average sound velocity of a polycrystalline material as follow,
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Table 2 Calculated elastic constants cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus modulus E (GPa), G/B, Poisson’s ratio m and hardness of ScAl3C3 and UAl3C3. Compound
C11
C12
HfC Hf2Al3C4 Hf3Al3C5 Zr2Al3C4 Zr3Al3C5 YAl3C3 UC UAl3C3
525 413 425 420 429 359 169 304 302 368
108 116 120 109 110 80 146 115 110 94
ScAl3C3
C13
C33
95 99 88 93 63
347 362 366 378 291
97 100 68
326 282 293
C44 160 171 175 171 179 143 47 154 147 147
C66 148 152 156 160 139 95 96 137
B
G
E
G/B
m
247 198 205 196 202 168 154 172 167 164
180 156 160 161 166 138 61
489 371 381 379 391 325 162
0.73 0.79 0.78 0.82 0.82 0.82
0.17 0.19 0.19 0.18 0.18 0.18 0.34
114 139
279 325
0.68 0.85
0.22 0.17
Hv
12.6
17.3 26.7
Ref [26] [26] [26] [29] [29,30] [14] [31] [17] This work This work
Table 3 Our calculated density; longitudinal, transverse sound, mean sound velocities, and the Debye temperatures obtained from the mean sound velocities of ScAl3C3 and UAl3C3; and corresponding data for YAl3C3, Zr2Al3C4 and Zr3Al3C5 obtained from previous studies. Compounds
B (GPa)
G (GPa)
q (g/cm3)
vl (m/s)
vt (m/s)
va (m/s)
HD (K)
Ref
YAl3C3 Zr2Al3C4 Zr3Al3C5 ScAl3C3 UAl3C3
158
139
164 167
139 114
3.90 4.80 5.28 3.29 6.62
9382 9249 8954 10,311 6943
5970 5792 5607 6505 4151
6563 6379 6175 7158 4593
837 830 806 938.8 593
[13] [32] [32] This work This work
HD ¼
1 h 3n NA q 3 va kB 4p M
ð4Þ
where h is the Plank’s constant, kB is the Boltzmann’s constant, NA is Avogadro’s number, q is the mass density, M is the molecular weight, and n is the number of atoms in the molecule, va is the average sound velocity of a material. While the average sound velocity is defined as,
va ¼
13 1 2 1 þ 3 v 3t v 3l
ð5Þ
where vt and vl are the transverse and longitudinal sound velocities, respectively. The longitudinal and transverse sound velocities can be calculated from the shear and bulk moduli, and the mass density (q) by following equations,
vl ¼ vt ¼
1 3B þ 4G 2 3q
12 G
q
ð6Þ
ð7Þ
According to above Debye model, the longitudinal sound vl, transverse sound vt, mean sound velocities va and the Debye temperatures HD of ScAl3C3 and UAl3C3 are calculated and presented in Table 3. The Debye temperatures for YAl3C3, Zr2Al3C4 and Zr3Al3C5 obtained from previous studies are also listed for comparison. It is obvious that the Debye temperature of ScAl3C3 is even higher than YAl3C3, Zr2Al3C4 and Zr3Al3C5. According to the Debye-Lindemann theory, the relationship between the melting temperature Tm and the Debye temperature HD can be expressed as Tm = AH2D (A depends on the density and the atomic mass) [34], which means ScAl3C3 may even possess a higher melting point than YAl3C3, Zr2Al3C4 and Zr3Al3C5. The Debye temperatures of UAl3C3 is lower than YAl3C3, Zr2Al3C4 and Zr3Al3C5, but it’s still a higher value, so the melting temperature Tm of UAl3C3 is high. This means UAl3C3 also exhibits a relatively good high temperature resistance.
4. Conclusion In this paper, we investigated the related physical properties of ScAl3C3 and UAl3C3 to discuss their potential applications. Some meaningful results are summarized as follows, 1) Calculated structural parameters are in agreement with the experimental results. Both of these two compounds satisfy the mechanical stability criterion. 2) Both ScAl3C3 and UAl3C3 presents metallic-like nature. For ScAl3C3, Sc d states, Al p states and C p states mainly contribute to its metallic behavior. For UAl3C3, the f and d orbitals of U atoms are mainly contributions for the electronic conduction behavior. In addition, the chemical bonds in ScAl3C3 are stronger than in UAl3C3. 3) The study of bond stiffness, elastic modulus, G/B ratio and hardness show that both ScAl3C3 and UAl3C3 are ceramics with low damage tolerance, low fracture toughness and high hardness. 4) The calculations of Debye temperature show that these two ceramics have high melting point, they exhibit relatively good high temperature resistance. To sum up, the results of this work offer a theoretical guidance for the potential industrial applications of ternary ceramic ScAl3C3 and UAl3C3. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments Supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 18A140036) and the National Natural Science Foundation of China (Grant No. 61571403).
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Materials Letters journal homepage: www.elsevier.com/locate/mlblue
Structural, electronic, thermophysical properties and bond stiffness of ternary ceramic ScAl3C3 and UAl3C3: Ab initio calculations Feng Shiquan a,⇑,1, Zhao Jianling b, Su Yuling a,⇑,1, Zhang Weibin c, Cheng Xinlu d a
School of Physics and Electronic Engineering, Zhengzhou University of Light Industry, Zhengzhou 450002, China Division of Radiation Physics, State Key Laboratory of Biotherapy and Cancer Center, West China Hospital, Sichuan University, Chengdu 610041, China c School of Physics and Optoelectronic Engineering, Yangtze University, Jingzhou 434023, China d Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China b
a r t i c l e
i n f o
Article history: Received 22 May 2019 Received in revised form 24 July 2019 Accepted 31 August 2019 Available online 31 August 2019 Keywords: Ternary ceramic First-principles calculations Elastic property Thermal property
a b s t r a c t In this work, we perform the investigation of the structural, electronic, elastic and thermal properties as well as bonding stiffness of ScAl3C3 and UAl3C3 by means of first-principles calculations. Mechanical stability of these two ceramics is discussed by elastic constant calculation. The G/B ratio and Poisson ratio are calculated to discuss their damage tolerance, fracture toughness and the bond composition. By computing the density of states and the bond stiffness of different bonds in ScAl3C3 and UAl3C3, we discuss their bonding properties. The ratio of the weakest bond stiffness to the strongest bond stiffness is presented to discuss their mechanical properties. What’s more, by calculating longitudinal, transverse sound, mean sound velocities, and the Debye temperatures, we discussed the thermal properties of ScAl3C3 and UAl3C3. Ó 2019 Elsevier B.V. All rights reserved.
1. Introduction Due to their high melting point, high hardness, good wear resistance and chemical inertness, binary TiC, NbC, ZrC and HfC ceramics are considered as candidates for thermal protection materials in hypersonic vehicles [1–3]. But their intrinsic brittleness and poor oxidation resistance greatly inhibit the applications of these compounds in industrial fields [3,4] (Fig. 1). As a new type of ceramic materials stacked by M-X octahedron slabs and weakly bonded A slabs, Mn+1AXn (MAX) compounds have attracted tremendous attention for their excellent properties recently. MAX possess layered-hexagonal structure alternately, where M is an early transition metal, A is a group 13–16 element, and X is mainly C or N. Some of them have a better performance than binary ceramics as TiC, NbC, ZrC and HfC in oxidation resistance and damage tolerance [4]. In the past new decades, nearly one hundred MAX phases [5–9] have been successful discovered by different methods since the discovery of Ti3SiC2 by Nowotny and his colleagues [10]. Recent years, by adding the element Al to binary ceramics, a new family of layered ternary carbides (MC)mAl3C2 (where M is metal atom, m = 1, 2, 3. . .) have been synthesized successively ⇑ Corresponding authors. 1
E-mail addresses: [email protected] (S. Feng), [email protected] (Y. Su). Feng Shiquan and Zhao Jianling contributed equally to this work.
https://doi.org/10.1016/j.matlet.2019.126610 0167-577X/Ó 2019 Elsevier B.V. All rights reserved.
[11–13]. Among them, the most typical representatives are Zr-AlC and Hf-Al-C carbides. Studies [14–16] show that these layered ternary carbides have superior oxidation resistance and fracture toughness compared to ZrC and HfC. And it makes them potential refractories for the high stiffness and strength of Zr-Al-C and Hf-AlC carbides. ScAl3C3 and UAl3C3 have similar crystal structures with (MC)mAl3C2 ceramics (M = Hf, Zr) [11]. (MC)mAl3C2 (M = Hf, Zr) possess the advantages of high melting point, high oxidation resistance and fracture toughness. Similarly, whether ScAl3C3 and UAl3C3 own these merits? There are few studies [17] on their related properties, especially for ScAl3C3. In this work, we focused on the mechanical, electronic, elastic, thermal properties and bond characters of ScAl3C3 and UAl3C3, and compared to (HfC)mAl3C2 and (ZrC)mAl3C2 phases.
2. Computational methods and details In present work, we performed first-principles calculations to study the electronic, elastic and thermophysical properties of ternary ceramic ScAl3C3 and UAl3C3 [18] in the ab initio plane-wave code ABINIT. The crystal structures of these two ceramic were optimized by the conjugate gradient (CG) algorithm at the hydrostatic pressure range from 0 to 80 GPa. And the generalized gradient approximation designed by Perdew, Burke, and Ernzerhof (PBE)
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Fig. 1. Crystal structures of (a) ScAl3C3 and (b) UAl3C3.
[19,20] is considered as the exchange-correlation energy in these calculations. The cut-off energy was set to 550 eV and the k points grid was chose as 7 7 2. The Brillouin zone sampling was performed using Monkhorst-pack grid [21]. In the structural optimization, the systems are relaxed until the total energy and ionic Hellmann-Feynman force less than 5 107 eV/atom and 0.005 eV/Å, respectively. The maximum ionic displacement and maximum stress are set to less than 5 104 Å and 0.02 GPa, respectively. 3. Results and discussion 3.1. Equilibrium structure In this work, we set up theoretical crystal model of ScAl3C3 and UAl3C3 based on experimental data from Ref. [11]. Both these two structures are composed of layered stacking composed of MC and Al3C2. They crystallize with hexagonal unit cells, space group P63mc. The experimental lattice parameters are a = 3.352(1) Å and c = 16.778(2) Å for ScAl3C3, while a = 3.387(1) Å and c = 17.393(2) Å for UAl3C3. We optimized these two crystal structures by DFT calculations. For the ScAl3C3 phase, the calculated lattice parameters a and c are 3.357 Å and 16.784 Å, respectively. For the UAl3C3 phase, theoretical a = 3.376 Å and c = 17.441 Å, respectively. Compared with the experimental values, the computed lattice constants for both ScAl3C3 and UAl3C3 are consistent with experimental data within 0.5% deviation. Therefore, the present calculations can reliably reproduce the crystal structure of ScAl3C3 and UAl3C3. As hexagonal crystals, both ScAl3C3 and UAl3C3 satisfy the mechanical stability criterion [22] as following,
C 12 > 0; C 11 C 12 > 0; C 33 > 0; C 44 > 0; ðC 11 þ C 12 ÞC 33 2C 213 > 0 ð1Þ For ScAl3C3, C12 = 94 > 0, C11-C12 = 192 > 0, C33 = 282 > 0, C44 = 147 > 0, (C11 + C12) C33-2C213 = 126118 > 0; For UAl3C3, C12 = 302 > 0, C11-C12 = 274 > 0, C33 = 293 > 0, C44 = 147 > 0, (C11 + C12) C33-2C213 = 96184 > 0. Both of them satisfy the mechanical stability criterion, which suggests that the hexagonal phase ScAl3C3 and UAl3C3 are mechanical stable. More details about the calculated elastic constants of ScAl3C3 and UAl3C3 would be discussed in Sec 3.4. 3.2. Electronic structure and chemical bonding To explore the nature of the interatomic bonding, the electronic properties of ScAl3C3 and UAl3C3 are calculated. Fig. 2 presents the
band structure, total and partial electronic density of states (PDOS) of ScAl3C3 and UAl3C3. For ScAl3C3, there is a very small band gap (0.063 eV); while for UAl3C3, there are several energy bands cross the Fermi level. Due to broadening effect, it can be seen in Fig. 2 that there exist electronic density of states at the Fermi levels for both ScAl3C3 and UAl3C3. So it can be said that both ScAl3C3 and UAl3C3 possess metallic-like nature. It is seen that Sc d states, Al p states and C p states mainly contribute to this behavior for ScAl3C3. For UAl3C3, the f orbitals of U atoms are mainly contributions for the electronic conduction behavior. In addition, there are four peaks in valence bands and one peak in conduction bands for both ScAl3C3 and UAl3C3. The lowest valence bands are mainly contributed by C-s electron with Al-p electron. The similar shape of C-s state and Al-p states indicates a hybridization of C-s and Al-p states to form Al-C bonds. Similarly, it can be seen that the other three peaks in valence bands are the hybridization of electron states between C and Al atoms. For the peak in conduction bands, it can be considered as a result of the hybridization of Sc-d and C-p states to form Sc-C bonds for ScAl3C3, and the hybridization of U-f and C-p states to form U-C bonds for UAl3C3. The peaks in the density of states for ScAl3C3 are sharper than that for UAl3C3, which suggests that the chemical bonds in ScAl3C3 are stronger than in UAl3C3. 3.3. Compressibility and bond stiffness By fitting the pressure dependence of normalized cell volume V/V0 to the Birch-Murnaghan equation of state [23], the bulk moduli B of ScAl3C3 and UAl3C3 are estimated as 180 and 187 GPa, respectively (Fig. 3). As bulk modulus B is a physical quantity to measure the resistance of a material against the volume change under a hydrostatic pressure, this result indicates UAl3C3 has a better performance in resisting external pressure than ScAl3C3. According to the theoretical model proposed by He et al. [24,25], the bond stiffness k can be obtained by fitting the relationship between the relative bond lengths d/d0 (d0 is the bond length at 0 GPa) and the hydrostatic pressure P to a form of polynomial function (d/d0 = C0 + C1P + C2P2, where Ci (i = 0, 1, 2) are the quadratic fitting coefficients) (Fig. 4). And then the bond stiffness k can be expressed as,
dðd=d0 Þ 1 ¼ jC 1 þ 2C 2 Pj1 k ¼ dP
ð2Þ
Using this model, we calculated the quadratic fitting coefficients and bond stiffness of different bonds in ScAl3C3 and UAl3C3 at 0 GPa, and listed them in Table 1. It can be seen that the U-C bonds overall have higher stiffness than Al-C bonds for UAl3C3. But for ScAl3C3, an interesting phenomenon is that the Al2-C2 bonds (667GPa) possess the highest bond stiffness. Another obvious phenomenon is the Al1-C1, Al2-C2 and Al3C3 bonds is much stronger than Al1-C2 and Al3-C2 bonds in Al-C slab for both ScAl3C3 and UAl3C3. Similar results are obtained for (MC)nAl3C2 (M = Hf and Zr) [26,27]. In addition, the ratios of the bond stiffness of the weakest bond to that of the strongest bond for both ScAl3C3 and UAl3C3 are larger than 1/2. According to previous studies [26,28], it means these two compounds are ceramics with low damage tolerance and fracture toughness. 3.4. Elastic stiffness and hardness In above sections, we have revealed the chemical bonding nature of ternary ceramic ScAl3C3 and UAl3C3. It is known that the bonding properties have a close connection with elastic modulus. To disclose the relationship between chemical bonding and elastic stiffness, we calculated and listed the second-order elastic
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Fig. 2. Band structures of ScAl3C3 (a) and UAl3C3 (b); total and partial electronic density of states of ScAl3C3 (c) and UAl3C3 (d).
Fig. 3. Normalized lattice constants of (a) ScAl3C3 (b) UAl3C3.
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Fig. 4. Normalized bond length as function of pressure for (a) ScAl3C3 (b) UAl3C3.
Table 1 The quadratic fitting coefficients and bond stiffness (GPa) of different bonds in (a) ScAl3C3 (b) UAl3C3. ScAl3C3
C1 (1 0 3)
C2 (1 0 6)
k
UAl3C3
C1 (1 0 3)
C2 (1 0 6)
k
Sc-C1 Sc-C3 Al1-C1 Al1-C2 Al2-C2 Al3-C2 Al3-C3
1.73 1.68 1.53 2.02 1.50 2.02 1.58
7.43 6.98 6.18 9.43 6.33 9.44 6.70
578 595 654 495 667 495 633
U-C1 U-C3 Al1-C1 Al1-C2 Al2-C2 Al3-C2 Al3-C3
1.32 1.33 1.60 1.99 1.59 1.91 1.74
5.70 5.68 5.66 9.43 6.17 8.44 7.52
758 752 625 503 629 524 575
constants, Pugh’s ratio, Poisson’s ratio m and elastic stiffness of ScAl3C3 and UAl3C3 in Table 2. Table 2 presents the calculated elastic constants of ScAl3C3 and UAl3C3 as well as some Hf- and Zr-containing (MC)nAl3C2. The estimated B of ScAl3C3 and UAl3C3 are slightly smaller than those we obtained by fitting Eq(1) of the EOS, but the error is within the allowable range. From Table 2, it can be seen that C11 of ScAl3C3 is much higher than that of C33, which is consistent with Hf- and Zr-containing (MC)nAl3C2 as well as other MAX phases [24]. While it is notice that the reverse situation has occurred on UAl3C3. Its C11 is lower than C33, which indicates that the resistance to tensile deformation along the axis is lower than that of the c axis. In addition, the bulk and shear modulus of ScAl3C3 and UAl3C3 are much lower that their counterpart of Hf- and Zr-containing (MC)nAl3C2, which may be due to the different crystal structure and chemical bonding of M-C (M = Hf, Zr, U, Sc) bonds. The relative bond strength order of these four bonds is Zr-C > Hf-C > Sc-C > U-C. Compared to Hf- and Zr-containing (MC)nAl3C2, the lower bond strength of Sc-C and U-C bonds leads to the lower bulk and shear moduli for ScAl3C3 and UAl3C3. Pugh’s ratio, shear to bulk modulus, G/B is commonly used as a criterion to distinguish the ductility or brittleness of materials. From Table 2, the Pugh’s ratios of ScAl3C3, UAl3C3 and other Hfand Zr-containing (MC)nAl3C2 are higher than 0.571 implying their brittle nature. But UAl3C3 possess a much lower G/B ratio than other compounds, suggesting the toughness of UAl3C3 is better than them. In addition, the higher G of ScAl3C3, Hf- and Zrcontaining (MC)nAl3C2 reflect they have a better shear deformation resistance than UAl3C3.
Poisson ratio value m is an important parameter to evaluate the degree of the covalent bonding. For covalent crystals, Poisson ratio is small (m = 0.1), whereas it is about 0.25 for ionic compounds [32]. From Table 2, we can see the Poisson ratios of ScAl3C3, UAl3C3 and other Hf- and Zr-containing (MC)nAl3C2 are between 0.1 and 0.25, which suggests that these two ceramics possess partial ionic and covalent component. Hardness is considered as an important parameter to quantify the crystal structures’ resistance ability to various kinds of deformation. And it is related to the chemical bond types in crystals. Chen et al. proposed an empirical theoretical model to predict the hardness of metals, intermetallics and ceramics [33] by shear and bulk modulus. In this model, the Vickers hardness can be calculated by following formula,
0:585 2 Hv ¼ 2 k G 3
ð3Þ
where k = G/B is the Pugh’s modulus ratio. Using this model, we obtained the calculated hardness of ScAl3C3 and UAl3C3 are 26.7 and 17.3 GPa, respectively. High intrinsic hardness indicates that these two ceramics are relatively ‘hard’ materials. 3.5. Thermophysical properties Debye temperature is an important thermal quantity. It can be used to predict the melting temperature of the material. The Debye temperature, HD, can be calculated from the average sound velocity of a polycrystalline material as follow,
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Table 2 Calculated elastic constants cij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus modulus E (GPa), G/B, Poisson’s ratio m and hardness of ScAl3C3 and UAl3C3. Compound
C11
C12
HfC Hf2Al3C4 Hf3Al3C5 Zr2Al3C4 Zr3Al3C5 YAl3C3 UC UAl3C3
525 413 425 420 429 359 169 304 302 368
108 116 120 109 110 80 146 115 110 94
ScAl3C3
C13
C33
95 99 88 93 63
347 362 366 378 291
97 100 68
326 282 293
C44 160 171 175 171 179 143 47 154 147 147
C66 148 152 156 160 139 95 96 137
B
G
E
G/B
m
247 198 205 196 202 168 154 172 167 164
180 156 160 161 166 138 61
489 371 381 379 391 325 162
0.73 0.79 0.78 0.82 0.82 0.82
0.17 0.19 0.19 0.18 0.18 0.18 0.34
114 139
279 325
0.68 0.85
0.22 0.17
Hv
12.6
17.3 26.7
Ref [26] [26] [26] [29] [29,30] [14] [31] [17] This work This work
Table 3 Our calculated density; longitudinal, transverse sound, mean sound velocities, and the Debye temperatures obtained from the mean sound velocities of ScAl3C3 and UAl3C3; and corresponding data for YAl3C3, Zr2Al3C4 and Zr3Al3C5 obtained from previous studies. Compounds
B (GPa)
G (GPa)
q (g/cm3)
vl (m/s)
vt (m/s)
va (m/s)
HD (K)
Ref
YAl3C3 Zr2Al3C4 Zr3Al3C5 ScAl3C3 UAl3C3
158
139
164 167
139 114
3.90 4.80 5.28 3.29 6.62
9382 9249 8954 10,311 6943
5970 5792 5607 6505 4151
6563 6379 6175 7158 4593
837 830 806 938.8 593
[13] [32] [32] This work This work
HD ¼
1 h 3n NA q 3 va kB 4p M
ð4Þ
where h is the Plank’s constant, kB is the Boltzmann’s constant, NA is Avogadro’s number, q is the mass density, M is the molecular weight, and n is the number of atoms in the molecule, va is the average sound velocity of a material. While the average sound velocity is defined as,
va ¼
13 1 2 1 þ 3 v 3t v 3l
ð5Þ
where vt and vl are the transverse and longitudinal sound velocities, respectively. The longitudinal and transverse sound velocities can be calculated from the shear and bulk moduli, and the mass density (q) by following equations,
vl ¼ vt ¼
1 3B þ 4G 2 3q
12 G
q
ð6Þ
ð7Þ
According to above Debye model, the longitudinal sound vl, transverse sound vt, mean sound velocities va and the Debye temperatures HD of ScAl3C3 and UAl3C3 are calculated and presented in Table 3. The Debye temperatures for YAl3C3, Zr2Al3C4 and Zr3Al3C5 obtained from previous studies are also listed for comparison. It is obvious that the Debye temperature of ScAl3C3 is even higher than YAl3C3, Zr2Al3C4 and Zr3Al3C5. According to the Debye-Lindemann theory, the relationship between the melting temperature Tm and the Debye temperature HD can be expressed as Tm = AH2D (A depends on the density and the atomic mass) [34], which means ScAl3C3 may even possess a higher melting point than YAl3C3, Zr2Al3C4 and Zr3Al3C5. The Debye temperatures of UAl3C3 is lower than YAl3C3, Zr2Al3C4 and Zr3Al3C5, but it’s still a higher value, so the melting temperature Tm of UAl3C3 is high. This means UAl3C3 also exhibits a relatively good high temperature resistance.
4. Conclusion In this paper, we investigated the related physical properties of ScAl3C3 and UAl3C3 to discuss their potential applications. Some meaningful results are summarized as follows, 1) Calculated structural parameters are in agreement with the experimental results. Both of these two compounds satisfy the mechanical stability criterion. 2) Both ScAl3C3 and UAl3C3 presents metallic-like nature. For ScAl3C3, Sc d states, Al p states and C p states mainly contribute to its metallic behavior. For UAl3C3, the f and d orbitals of U atoms are mainly contributions for the electronic conduction behavior. In addition, the chemical bonds in ScAl3C3 are stronger than in UAl3C3. 3) The study of bond stiffness, elastic modulus, G/B ratio and hardness show that both ScAl3C3 and UAl3C3 are ceramics with low damage tolerance, low fracture toughness and high hardness. 4) The calculations of Debye temperature show that these two ceramics have high melting point, they exhibit relatively good high temperature resistance. To sum up, the results of this work offer a theoretical guidance for the potential industrial applications of ternary ceramic ScAl3C3 and UAl3C3. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments Supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (No. 18A140036) and the National Natural Science Foundation of China (Grant No. 61571403).
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