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三元Hf-C-N体系的空位有序结构及其力学性质和电子性质的第一性原理研究

彭军辉 TikhonovEvgenii

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三元Hf-C-N体系的空位有序结构及其力学性质和电子性质的第一性原理研究

彭军辉, TikhonovEvgenii

First-principles study of vacancy ordered structures, mechanical properties and electronic properties of ternary Hf-C-N system

Peng Jun-Hui, Tikhonov Evgenii
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  • 采用第一性原理方法, 研究了三元Hf-C-N体系的空位有序结构及其力学性质和电子性质. 首先采用第一性原理和进化算法, 预测得到8种可能存在的热力学稳定的Hf-C-N空位有序结构; 这些结构都具有岩盐结构, 与实验发现的无序固溶体的结构类型一致. 本文的预测结果证明了Hf-C-N空位化合物能够以有序结构形式存在, 空位与C, N原子都位于[Hf6]八面体间隙, 这一结构特点与HfCx的相同. 然后采用第一性原理方法, 计算了Hf-C-N空位有序结构的力学性质, 发现除C∶N = 1∶4外, 相同C/N下, 随着空位浓度的增大, Hf-C-N的体模量、剪切模量、弹性模量、Pugh比、维氏硬度等降低; 而Hf6CN4 (空位浓度为1/6)的维氏硬度高于Hf5CN4 (无空位), 表现出空位硬化现象. 最后, 计算了Hf-C-N空位有序结构的态密度和晶体轨道哈密顿分布, 发现其具有强共价性和金属性; 且随着空位浓度增大, 总体键强减弱, 因而模量减小.
    The thermal-mechanical properties of transition metal carbonitrides can be affected by the concentration and ordering of vacancies besides the C/N atomic ratio. However, there are few reports on the vacancy ordered structure of ternary transition metal carbonitrides. In the present paper, the first-principles method is used to study the vacancy ordered structures, mechanical properties, electronic properties and the effect of vacancies on the ternary Hf-C-N system. Firstly, the crystal structures of Hf-C-N system is examined by the first-principles and evolutionary algorithms implemented in USPEX under ambient pressure, and eight thermodynamical stable vacancy ordered structures are found, each of which has a rock-salt structure, and is also dynamical and mechanical stable, which are verified by the calculations of their phonon dispersion curves and elastic constants. The vacancies are occupied at the [Hf6] octahedral interstices, which replace the positions of non-metal atoms. Their crystallographic data such as space group, lattice constants are also predicted. To the best of our knowledge, there is no report on the Hf-C-N vacancy ordered structures and these structures investigated here in this work are all found for the first time. Then their mechanical properties are calculated. The Hf-C-N vacancy ordered structures all have very high bulk, shear and elastic modulus and hardness. It is found that except for C∶N = 1∶4, for the Hf-C-N system with the same C/N ratio the moduli, Vickers hardness values, and Pugh’s ratios decrease with the increase of the concentration of vacancy. However, the Vickers hardness of Hf6CN4 (the concentration of vacancy is equal to 1/6) is higher than that of Hf5CN4 (no vacancy), that is so-called vacancy hardening. Finally, the electronic density of states and the crystal orbital Hamilton populations are calculated. The chemical bonding of Hf-C-N vacancy ordered structure is analyzed, which is a mixture of covalence and metallic and is similar to that of binary transition metal carbides and nitrides. With the increase of the concentration of vacancy, the total bond strength decreases, and then the modulus decreases for Hf-C-N compound.
      通信作者: TikhonovEvgenii, tikhonov.e@nwpu.edu.cn
    • 基金项目: 外国人才引进与学术交流项目(批准号: B08040)资助的课题
      Corresponding author: Tikhonov Evgenii, tikhonov.e@nwpu.edu.cn
    • Funds: Project supported by the Foreign Talents Introduction and Academic Exchange Program of China (Grant No. B08040)
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  • 图 1  (a) 常压下, 三元Hf-HfC-HfN体系的能量凸包图, 黑色球表示热力学稳定结构, 其他为亚稳结构; (b) Hf-C-N空位有序结构的X射线衍射模拟图谱, 衍射源为Cu Kα射线

    Fig. 1.  (a) Enthalpy convex-hull of ternary Hf-HfC-HfN system at ambient pressure. The black sphere indicates stable structure, and others are metastable structure. (b) The simulated X-ray diffractions of Hf-C-N vacancy ordered structures with a copper Kα X-ray source.

    图 2  Hf-C-N空位有序结构在某一晶面上的空位分布 (a) Hf6C4N-$C2 $/m (0 0 1); (b) Hf6C3N-$C2 $ (1 0 0); (c) Hf6C3N2-$C2 $/m (1 0 0); (d) Hf3CN-$C2 $ (1 0 0); (e) Hf6C2N3-$C2 $ (1 0 0); (f) Hf4CN2-Cmmm (0 0 1); (g) Hf6CN3-$C2 $/m (1 0 0); (h) Hf6CN4-$C2 $/m (0 0 1)

    Fig. 2.  Vacancies on the crystallographic plane: (a) Hf6C4N-$C2 $/m (0 0 1); (b) Hf6C3N-$C2 $ (1 0 0); (c) Hf6C3N2-$C2 $/m (1 0 0); (d) Hf3CN-$C2 $ (1 0 0); (e) Hf6C2N3-$C2 $ (1 0 0); (f) Hf4CN2-Cmmm (0 0 1); (g) Hf6CN3-$C2 $/m (1 0 0); (h) Hf6CN4-$C2 $/m (0 0 1).

    图 3  Hf-C-N空位有序结构的声子谱曲线 (a) Hf6C4N-$C2 $/m; (b) Hf6C3N-$C2 $; (c) Hf6C3N2-$C2 $/m; (d) Hf3CN-$C2 $; (e) Hf6C2N3-$C2 $; (f) Hf4CN2-Cmmm; (g) Hf6CN3-$C2 $/m; (h) Hf6CN4-$C2 $/m

    Fig. 3.  Phonon dispersion curves of (a) Hf6C4N-$C2 $/m, (b) Hf6C3N-$C2 $, (c) Hf6C3N2-$C2 $/m, (d) Hf3CN-$C2 $, (e) Hf6C2N3-$C2 $, (f) Hf4CN2-Cmmm, (g) Hf6CN3-$C2 $/m, (h) Hf6CN4-$C2 $/m. They are all dynamical stable because no imaginary frequencies were found in Brillouin zone.

    图 4  三元Hf-HfC-HfN体系的力学性质-组分相图 (a) 体模量(B); (b) 剪切模量(G ); (c) 弹性模量(E ); (d) 维氏硬度(HV); (e) Pugh比(G/B); (f) 泊松比(μ)

    Fig. 4.  Mechanical properties-composition diagrams of ternary Hf-HfC-HfN system: (a) Bulk modulus (B); (b) shear modulus (G ); (c) elastic modulus (E ); (d) Vickers hardness (HV); (e) Pugh’s ratio (G/B); (f) Poisson’s ratio (μ).

    图 5  (a) Hf6C4N-$C2 $/m, (b) Hf6C3N-$C2 $, (c) Hf6C3N2-$C2 $/m, (d) Hf3CN-$C2 $, (e) Hf6C2N3-$C2 $, (f) Hf4CN2-Cmmm, (g) Hf6CN3-$C2 $/m和(h) Hf6CN4-$C2 $/m的态密度和分态密度; (i) Hf3CN和Hf2CN的总态密度对比; 其中Fermi能级位于0 eV

    Fig. 5.  Density of state (DOS) and partial density of state (PDOS) normalized by per HfCxNy of (a) Hf6C4N-$C2 $/m, (b) Hf6C3N-$C2 $, (c) Hf6C3N2-$C2 $/m, (d) Hf3CN-$C2 $, (e) Hf6C2N3-$C2 $, (f) Hf4CN2-Cmmm, (g) Hf6CN3-$C2 $/m and (h) Hf6CN4-$C2 $/m; (i) the total DOS of Hf3CN and Hf2CN normalized by per HfCxNy. The Fermi level is at 0 eV.

    图 6  Hf-C-N化合物的晶体轨道哈密顿分布(–COHP), Fermi能级位于0 eV

    Fig. 6.  Crystal orbital Hamilton populations (–COHP) of Hf-C-N compounds. The Fermi level is at 0 eV.

    表 1  Hf-C-N空位有序结构的空间群、晶格常数、反应焓ΔH (eV/atom)、Hf原子的配位数(CN) 和空位浓度(CV)

    Table 1.  Space group, lattice constants, the enthalpy of reaction ΔH (eV/atom), coordination number (CN) of Hf and the concentration of vacancy (CV) of Hf-C-N vacancy ordered structures.

    CompoundSpace groupLattice constants/ÅΔH/(eV·atom–1)CNCV
    Hf6C4N$C2 $/ma = 5.679, b = 9.799, c = 5.671, β = 70.6o–0.089951/6
    Hf6C3N$C2 $a = 5.658, b = 9.763, c = 9.262, β = 144.8o–0.098041/3
    Hf6C3N2$C2 $ma = 5.660, b = 9.783, c = 5.619, β = 109.6o–0.103851/6
    Hf3CN$C2 $a = 5.632, b = 9.705, c = 5.625, β = 109.8o–0.110741/3
    Hf6C2N3$C2 $a = 5.624, b = 9.725, c = 5.602, β = 109.6o–0.104751/6
    Hf4CN2Cmmma = 6.427, b = 9.147, c = 3.235–0.10824/51/4
    Hf6CN3$C2 $/ma = 5.592, b = 9.658, c = 6.455, β = 125.3o–0.089441/3
    Hf6CN4$C2 $/ma = 5.580, b = 9.681, c = 5.587, β = 70.3o–0.081551/6
    下载: 导出CSV

    表 2  Hf-C-N空位有序结构的弹性常数Cij (单位: GPa)

    Table 2.  Calculated elastic constants Cij (in GPa) of Hf-C-N vacancy ordered structures.

    CompoundsC11C22C33C44C55C66C12C13C23
    Hf6C4N-$C2 $/m414.3406.6415.6158.0170.6148.794.1116.1104.5
    Hf6C3N-$C2 $358.5362.8352.2100.0114.3132.387.598.391.6
    Hf6C3N2-$C2 $/m414.6417.4407.6152.2157.6147.8111.9115.0116.3
    Hf3CN-$C2 $354.5363.5348.790.6103.6128.7102.1109.5101.3
    Hf6C2N3-$C2 $409.7418.7418.1149.6160.2148.9123.4122.9126.5
    Hf4CN2-Cmmm373.4368.8406.8142.2133.1135.8146.4112.0124.4
    Hf6CN3-$C2 $/m361.1358.4351.784.999.8124.9108.1121.7114.2
    Hf6CN4-$C2 $/m401.2414.1403.5146.5157.2139.8134.0139.9147.8
    下载: 导出CSV

    表 3  Hf-C-N空位有序结构和HfC1–xNx[19]的力学性质—体模量(B)、剪切模量(G )、弹性模量(E )、泊松比(μ)、Pugh比(G/B)、维氏硬度(HV)等

    Table 3.  Mechanical properties—bulk modulus (B), shear modulus (G ), elastic modulus (E ), Poisson’s ratio (μ), Pugh’s ratio (G/B), Vickers hardness (HV) of Hf-C-N vacancy ordered structures and HfC1–xNx[19].

    CompoundB /GPaG /GPaE /GPaμG/BHV /GPa
    Hf6C4N229.0140.8350.60.24490.614817.5
    Hf5C4N[19]260.6201.3480.30.19280.772729.9
    Hf6C3N180.9121.5297.90.22560.671717.8
    Hf4C3N[19]262.2202.1482.40.19340.770729.9
    Hf6C3N2214.0151.1366.90.21430.705922.1
    Hf3CN188.0113.4283.30.24890.603114.6
    Hf2CN[19]268.1198.5477.60.20310.740328.1
    Hf6C2N3221.3149.7366.60.22390.676620.7
    Hf4CN2212.7132.8329.70.24170.624217.1
    Hf3CN2[19]272.8185.1452.80.22330.678623.9
    Hf6CN3195.4108.9275.60.26500.557412.7
    Hf4CN3[19]276.2179.6442.80.23280.650422.2
    Hf6CN4207.2156.1374.40.19890.753524.6
    Hf5CN4[19]279.0171.5427.00.24490.614720.0
    下载: 导出CSV

    表 4  Hf-C-N化合物的晶体轨道哈密顿分布的积分值(–ICOHP)

    Table 4.  Integrated crystal orbital Hamilton populations (–ICOHP) of Hf-C-N compounds.

    Compound–ICOHPCompound–ICOHP
    Hf—CHf—NHf—HfHf—CHf—NHf—Hf
    Hf6C4N3.3733.5670.529Hf6C3N23.1812.9900.571
    Hf5C4N3.3733.0330.459Hf4CN23.4743.1110.650
    Hf6C3N3.3503.0670.718Hf3CN23.5513.0910.541
    Hf4C3N3.3193.0290.454Hf6CN33.4083.2110.570
    Hf6C3N23.6073.1030.530Hf4CN33.3213.1590.520
    Hf3CN3.2773.2110.737Hf6CN43.6753.1790.591
    Hf2CN3.4832.8020.490Hf5CN43.3193.0170.500
    下载: 导出CSV
  • [1]

    Squire T, Marschall J 2010 J. Eur. Ceram. Soc. 30 2239Google Scholar

    [2]

    Opeka M M, Talmy I G, Zaykosk J A 2004 J. Mater. Sci. 39 5887Google Scholar

    [3]

    Levine S R, Opila E J, Halbig M C, Kiser J D, Singh M, Salem J A 2002 J. Eur. Ceram. Soc. 22 2757Google Scholar

    [4]

    Ushakov SV, Navrotsky A 2012 J. Am. Ceram. Soc. 95 1463Google Scholar

    [5]

    Grill A, Aron P R 1983 Thin Solid Films 108 173Google Scholar

    [6]

    Helmersson U, Todorova S, Barnett S A, Sundgren J E, Markert L C, Greene J E 1987 J. Appl. Phys. 62 481Google Scholar

    [7]

    Mirkarimi P B, Hultman L, Barnett S A 1990 Appl. Phys. Lett. 57 2654Google Scholar

    [8]

    Veprek S, Veprek-Heijman M G J, Karvankova P, Prochazka J 2005 Thin Solid Films 476 1Google Scholar

    [9]

    Hultman L, Bareno J, Flink A, Soderberg H, Larsson K, Petrova V, Oden M, Greene J E, Petrov I 2007 Phys. Rev. B 75 155437Google Scholar

    [10]

    Shin C S, Gall D, Hellgren N, Patscheider J, Petrov I, Greene J E 2003 J. Appl. Phys. 93 6025Google Scholar

    [11]

    Jhi S H, Louie S G, Cohen M L, Ihm J 2001 Phys. Rev. Lett. 86 3348Google Scholar

    [12]

    Shin C S, Rudenja S, Gall D, Hellgren N, Lee T Y, Petrov I, Greene J E 2004 J. Appl. Phys. 95 356Google Scholar

    [13]

    Lee T, Ohmori K, Shin C S, Cahill D G, Petrov I, Greene J E 2005 Phys. Rev. B 71 144106Google Scholar

    [14]

    Holleck H 1986 J. Vac. Sci. Technol., A 4 2661Google Scholar

    [15]

    Yang Q, Lengauer W, Koch T, Scheerer M, Smid I 2000 J. Alloys Compd. 309 L5Google Scholar

    [16]

    Jhi S H, Ihm J, Louie S G, Cohen M L 1999 Nature 399 132Google Scholar

    [17]

    Feng W, Cui S, Hu H, Zhang G, Lü Z 2011 Physica B 406 3631Google Scholar

    [18]

    Balasubramanian K, Khare S V, Gall D 2018 Acta Mater. 152 175Google Scholar

    [19]

    Peng J, Tikhonov E 2021 Comput. Mater. Sci. 195 110464Google Scholar

    [20]

    Gusev A I, Rempel A A, Magerl A J 2001 Disorder and Order in Strongly Nonstoichiometric Compounds (Berlin Heidelberg: Springer) pp179−243

    [21]

    Gusev A I 1991 Physical Chemistry of Non stoichiometric Refractory Compounds (Moscow: Nauka) (in Russian)

    [22]

    Rudy E 1965 Ternary Phase Equilibria in Transition Metal-boron-carbon-silicon Systems. Part II. Ternary Systems. Vol. I. Ta-Hf-C system (Air Force Materials Laboratory, Wright-Patterson Air Force Base) pp38−60

    [23]

    Lipatnikov V N, Lengauer W, Ettmayer P, Keil E, Groboth G, Kny E 1997 J. Alloys Compd. 261 192Google Scholar

    [24]

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出版历程
  • 收稿日期:  2021-02-01
  • 修回日期:  2021-06-28
  • 上网日期:  2021-08-15
  • 刊出日期:  2021-11-05

三元Hf-C-N体系的空位有序结构及其力学性质和电子性质的第一性原理研究

  • 1. 西北工业大学材料学院, 材料发现国际中心, 西安 710072
  • 2. 太原工业学院材料工程系, 太原 030008
  • 通信作者: TikhonovEvgenii, tikhonov.e@nwpu.edu.cn
    基金项目: 外国人才引进与学术交流项目(批准号: B08040)资助的课题

摘要: 采用第一性原理方法, 研究了三元Hf-C-N体系的空位有序结构及其力学性质和电子性质. 首先采用第一性原理和进化算法, 预测得到8种可能存在的热力学稳定的Hf-C-N空位有序结构; 这些结构都具有岩盐结构, 与实验发现的无序固溶体的结构类型一致. 本文的预测结果证明了Hf-C-N空位化合物能够以有序结构形式存在, 空位与C, N原子都位于[Hf6]八面体间隙, 这一结构特点与HfCx的相同. 然后采用第一性原理方法, 计算了Hf-C-N空位有序结构的力学性质, 发现除C∶N = 1∶4外, 相同C/N下, 随着空位浓度的增大, Hf-C-N的体模量、剪切模量、弹性模量、Pugh比、维氏硬度等降低; 而Hf6CN4 (空位浓度为1/6)的维氏硬度高于Hf5CN4 (无空位), 表现出空位硬化现象. 最后, 计算了Hf-C-N空位有序结构的态密度和晶体轨道哈密顿分布, 发现其具有强共价性和金属性; 且随着空位浓度增大, 总体键强减弱, 因而模量减小.

English Abstract

    • 碳化铪和氮化铪作为典型的超高温陶瓷材料, 具有非常优良的综合性能, 如非常高的熔点、高强度、高硬度、良好的导电导热性和化学稳定性等, 因而可用于飞行器鼻锥、机翼前缘、热结构防护、发动机热端和推进系统等的关键部位或部件[1-4]. 另外, 由于优异的抗腐蚀性、黏附性和高硬度等, 氮化铪可用作刀具防护的耐磨涂层[5]. 为了提高过渡金属碳、氮化合物的性质, 以满足对材料性能越来越高的要求, 研究者们常制备碳/氮化合物的超晶格结构[6,7]和纳米尺度复合材料[8,9]等, 以及采用空位强化[10-13]和合金化[14-19]等方法.

      常温常压下, HfC和HfN均为B1型岩盐结构, 铪原子形成面心立方结构, 碳、氮原子处于[Hf6]八面体间隙. HfC和HfN可形成三元无限固溶体, 且实验和理论上发现三元Hf-C-N体系具有更好的力学性质[14-19]. 例如, Holleck[14]研究了HfC1–xNx的微观硬度与组分之间的关系, 发现HfN的含量x = N/(C + N) = 0.4时, 微观硬度最大. Yang等[15]研究得到x = 0.2—0.6时, HfC1–xNx的剪切模量、弹性模量、体模量最大, 而纳米硬度和显微硬度随着HfN含量的增大而减小. 理论上, Jhi等[16]采用第一性原理方法, 研究得到x = 0.4时, HfC1–xNx的剪切模量最大. Feng等[17]和Balasubramanian等[18]采用第一性原理方法, 计算了HfC1–xNx的力学性能和电子性质随组分变化的关系, 发现x = 0.25时, HfCxN1–x的弹性模量和硬度最大. Peng和Tikhonov[19]采用第一性原理和进化算法, 研究了HfC1–xNx的结构、力学性质等随组分的变化, 发现x = 0.25—0.3时, HfC1–xNx的剪切模量、弹性模量、维氏硬度等最大, 断裂韧性也得到了改善. 因此, 相比于二元化合物, 三元Hf-C-N体系的模量、硬度和韧性等都提高了; 合金化是改善力学性能的一种有效方法.

      实际上, 二元及以上过渡金属碳/氮和碳氮化合物都存在一定浓度的空位, 被称为强非化学计量比化合物. Gusev等[20]发现在一定条件下, 空位会重新分布, 晶体结构发生改变, 即形成了空位有序结构. 强非化学计量比化合物的性质将受到空位浓度及其分布(有序无序性)等影响[11,14,20-23]. Rudy[22]研究发现Hf-C体系中, HfC0.94的熔点最高. Holleck[14]研究发现随着空位浓度的减小, TaNx, TaCx, NbNx和NbCx的微观硬度先增加后减小; 而TiCx, ZrCx, HfCx, TiNx和ZrNx的微观硬度随着空位浓度的减小而增大. Jhi等[11]采用第一性原理方法, 分别研究了Ti-N和Nb-C体系的剪切性质和电子性质, 发现空位对Ti-N和Nb-C体系的力学性质产生了完全不同的影响, 即空位硬化和软化, 且与Holleck[14]的实验结果一致. Gusev等[20]在著作中总结了二元强非化学计量比化合物的有序结构, 并分析了空位有序性对性质的影响.

      近年来研究人员采用第一性原理结合晶体结构预测等方法, 对二元过渡金属碳/氮化合物的有序结构进行了大量研究[24-38]. 基于第一性原理和进化算法, Yu等[24-26]和Xie等[27,28]分别采用晶体结构预测软件USPEX[39-41], 对过渡金属碳化物(TM = Ti, Zr, Hf, V, Nb, Ta等)的结构进行了预测, 发现了一系列含阴离子空位的结构, 包括TM2C, TM3C2, TM4C3, TM5C4, TM6C5, TM7C6和TM8C7等, 并研究了空位对结构和力学性质的影响. Zhang等[29]和Gunda等[30]分别采用CE方法[31]预测了ZrCx和TiCx的结构, Weinberger和Thompson[32]基于有序参数函数方法[33]预测了第IVB, VB族过渡金属碳化物的稳定结构. 基于第一性原理和进化算法, Yu等[34,35]和Fan等[36]分别预测了TiNx, ZrNx和HfNx的结构, 除了含阴离子空位的化合物外, 还发现含过渡金属空位的结构如Zr15N16, Zr7N8, Zr4N5, Hf4N5和Hf5N6等, 并研究空位对力学性质等的影响. 采用USPEX[39-41], Zhao等[37]和Li等[38]分别预测了Nb-N和Ta-N体系的稳定结构.

      然而, 目前关于三元过渡金属碳/氮化合物空位有序结构及性质的报道还很少. Rudy[42]采用X射线粉末图谱法研究了Ta2VC2的结构, 空间群为$ R \overline {3} m$, 然而XRD无法判断碳原子和碳原子空位的具体分布情况. Erniraliev等[43]在TixV1–xC0.5 (0.2 < x < 0.3)固溶体中, 发现了一种anti-CaCl2型的有序结构. Karimov等[44]和Em等[45]采用中子衍射方法研究了MCxNy (M = Ti, Zr)体系, 发现空位浓度较小时, 不存在有序结构; x + y < 0.74时, 发现有序结构$Fd \overline {3} m$. Em和Tashmetov[46]采用中子衍射方法, 发现TiCxNy (x + y ≈ 0.63)中存在两种有序结构-立方晶系(空间群为$Fd \overline {3} m$)和三方晶系(空间群为$ R \overline {3} m$P3121). Rudy[22]研究了高温下Hf-Ta-C体系的结构和性质, 发现许多含空位的三元化合物, 均为具有岩盐结构或Ta2C型的无序固溶体, 但没有发现有序结构. Binder等[47]研究了1423 K时, Ti-C-N, Zr-C-N和Hf-C-N体系的相平衡区间, Ti(CxN1–x)1–y, Zr(CxN1–x)1–y, Hf(CxN1–x)1–y均是B1型无序结构. Yang等[15]研究了Ti(CxN1–x)0.81的微观硬度、纳米硬度、体模量、剪切模量、弹性模量等, 但没有给出Ti(CxN1–x)0.81的结构类型. Hong和van de Walle等[48]采用从头算分子动力学方法, 模拟了含阴离子空位的Hf-C-N和Hf-Ta-C体系的熔点, 使用的均是B1型超胞结构. Buinevich等[49]合成了非化学计量比化合物HfC0.5N0.35, 为B1型无序结构, 并测得其熔点大于HfC0.98的, 维氏硬度为21.3 GPa, 断裂韧性为4.7 MPa·m1/2. 因而, 实验上要研究三元过渡金属碳/氮化合物的有序结构非常困难, 第一性原理结合晶体结构预测是一种有效方法.

      综上所述, 三元过渡金属碳氮化合物的性质除了受到化学组成的影响外, 还将受到空位的浓度及其分布等的影响. 为了研究三元Hf-C-N体系, 建立组分-结构-性质的关系, 必须考虑空位对结构及性质的影响. 然而, 目前实验和理论上, 很少关于三元Hf-C-N空位有序结构、力学性质及其关系的研究报道. 本文基于第一性原理和进化算法, 采用晶体结构预测软件USPEX[39-41]—已成功应用于二元过渡金属碳/氮化合物及许多新结构的预测[24-28,34-41], 搜索了三元Hf-C-N的空位有序结构, 研究了空位、化学组成等对力学性质的影响. 为该材料的实验合成、制备和应用等提供了理论指导和依据, 也为其他三元过渡金属碳氮化合物的研究提供了借鉴.

    • 晶体结构预测软件USPEX[39-41]预测材料结构时, 模仿生物进化机制, 基于进化算法, 采用随机生成、遗传、变异等操作产生结构; 然后, 基于能量、硬度或体积等, 搜索得到满足目标函数(如能量最低)的晶体结构. USPEX[39-41]可以直接从材料的化学元素组成出发, 无需其他实验数据, 即可预测得到其结构. 根据无偏差测试[50], USPEX[39-41]在计算效率和可靠性上都优于其他晶体结构预测方法.

      采用USPEX[39-41]分别搜索了Hf6C4N, Hf6C3N, Hf6C3N2, Hf4C2N, Hf3CN, Hf5C2N2, Hf4CN2, Hf6C2N3, HfCN3和Hf6CN4等组分的结构, 单胞中原子数分别为22, 20, 22, 14, 10/20, 22, 14, 22, 20和22. 初始结构(共60个)由USPEX[39-41]随机产生, 从第2代开始, 每代结构(共50个)分别由遗传(40%)、软模变异(20%)、晶格变异(10%)、原子位置交换(10%)、随机(20%)等进化操作产生. 对USPEX[39-41]产生的每个结构, 采用VASP软件[51]计算其能量, 筛选出能量最低的结构. 当连续20代能量最低的结构相同, 或搜索了30代结构时, 计算停止.

      然后, 采用VASP软件[51]对Hf-C-N各组分能量最低结构进一步进行结构优化和性质计算. 结构优化时, 离子实与价电子之间的相互作用, 采用投影缀加波方法[52]进行描述, 截断能为600 eV, 电子与电子间交换关联能采用GGA-PBE[53]方法进行处理. 布里渊区高对称点间距为2π × 0.018 Å–1. 能量收敛判据为: 能量差为10–8 eV/atom, 压力差为10–3 eV/Å. 基于“凸包结构”判断得到热力学稳定结构, 即某一结构分解为任意其他结构时, 分解能为正, 则该结构为热力学稳定结构. 然后, 采用VASP软件[51]计算弹性常数, 根据经验公式[54-58]计算得到体模量、剪切模量、弹性模量、泊松比、维氏硬度、Pugh比等. 最后, 基于密度泛函微扰理论, 采用Phonopy软件[59]计算Hf-C-N空位有序结构的声子谱曲线, 判断晶格动力学稳定性. 采用VESTA软件[60]画出其晶体结构和模拟X射线衍射图谱. 采用LOBSTER软件[61]计算Hf-C-N空位有序结构的晶体轨道哈密顿分布(–COHP).

    • 由于Hf-C-N体系的性质受到空位浓度的影响, 首先采用空位调控的方法, 对三元Hf-C-N体系的组分进行设计, 采用晶体结构预测软件USPEX[39-41]结合VASP[51], 搜索了三元空位有序结构. 图1(a)为三元Hf-HfC-HfN体系的能量凸包图, 反应焓ΔH (eV/atom)的计算公式如下:

      图  1  (a) 常压下, 三元Hf-HfC-HfN体系的能量凸包图, 黑色球表示热力学稳定结构, 其他为亚稳结构; (b) Hf-C-N空位有序结构的X射线衍射模拟图谱, 衍射源为Cu Kα射线

      Figure 1.  (a) Enthalpy convex-hull of ternary Hf-HfC-HfN system at ambient pressure. The black sphere indicates stable structure, and others are metastable structure. (b) The simulated X-ray diffractions of Hf-C-N vacancy ordered structures with a copper Kα X-ray source.

      $ \begin{split} &\Delta H\left( {{\rm{Hf}}{{\rm{C}}_x}{{\rm{N}}_y}} \right) = H\left( {{\rm{Hf}}{{\rm{C}}_x}{{\rm{N}}_y}} \right) -[ xH\left( {{\rm{HfC}}} \right) \\ &\quad+ yH\left( {{\rm{HfN}}} \right) + \left( {1- x- y} \right)H\left( {{\rm{Hf}}} \right)]. \end{split}$

      根据“凸包结构”判据, 如图1(a)所示, 除文献[19]中报道的不含空位的HfC1–xNx外, 本文还预测得到了8种可能存在的热力学稳定的空位有序结构, 其化学式、反应焓ΔH (eV/atom)和空位浓度等, 如表1所列. 就我们所知, 目前还没有关于三元Hf-C-N空位有序结构的报道, 本文预测得到的这些稳定结构都是第一次被发现的. 且本文预测得到了Hf-C-N空位有序结构的晶体学数据, 如空间群、晶格常数等, 如表1所列.

      CompoundSpace groupLattice constants/ÅΔH/(eV·atom–1)CNCV
      Hf6C4N$C2 $/ma = 5.679, b = 9.799, c = 5.671, β = 70.6o–0.089951/6
      Hf6C3N$C2 $a = 5.658, b = 9.763, c = 9.262, β = 144.8o–0.098041/3
      Hf6C3N2$C2 $ma = 5.660, b = 9.783, c = 5.619, β = 109.6o–0.103851/6
      Hf3CN$C2 $a = 5.632, b = 9.705, c = 5.625, β = 109.8o–0.110741/3
      Hf6C2N3$C2 $a = 5.624, b = 9.725, c = 5.602, β = 109.6o–0.104751/6
      Hf4CN2Cmmma = 6.427, b = 9.147, c = 3.235–0.10824/51/4
      Hf6CN3$C2 $/ma = 5.592, b = 9.658, c = 6.455, β = 125.3o–0.089441/3
      Hf6CN4$C2 $/ma = 5.580, b = 9.681, c = 5.587, β = 70.3o–0.081551/6

      表 1  Hf-C-N空位有序结构的空间群、晶格常数、反应焓ΔH (eV/atom)、Hf原子的配位数(CN) 和空位浓度(CV)

      Table 1.  Space group, lattice constants, the enthalpy of reaction ΔH (eV/atom), coordination number (CN) of Hf and the concentration of vacancy (CV) of Hf-C-N vacancy ordered structures.

      图1(b)为Hf-C-N空位有序结构的X射线衍射模拟图谱, 三元化合物的衍射峰形状与HfC, HfN的相同, 且衍射角位于HfC, HfN的衍射角之间; 即结构与HfC和HfN的结构相同, 都具有岩盐结构. 此结果与Binder等[47]和Buinevich等[49]发现的Hf-C-N无序固溶体的结构类型一致. 本文的理论预测结果证明了Hf-C-N空位化合物能够以有序结构的形式存在.

      图2为Hf-C-N空位有序结构某一晶面上的空位分布, 黑色方框表示空位, 取代的是部分C和N原子的位置; 即对于Hf-C-N化合物, 原子数比Hf/(C + N)小于化学计量比1∶1时, 空位占据剩余C和N原子的晶格位置. 则C和N原子的配位数均为6, 而Hf的配位数小于6, Hf原子的配位数如表1所列. 与二元HfCx[24-28]的结构相似, 过渡金属原子形成[Hf6]八面体, 八面体共棱连接, C, N原子和空位都处于八面体间隙位置. 图3为Hf-C-N空位有序结构的声子谱曲线, 在布里渊区(高对称点路径采用SeeK-path软件[62]得到), 都不存在虚频, 证明这些结构都是晶格动力学稳定的.

      图  2  Hf-C-N空位有序结构在某一晶面上的空位分布 (a) Hf6C4N-$C2 $/m (0 0 1); (b) Hf6C3N-$C2 $ (1 0 0); (c) Hf6C3N2-$C2 $/m (1 0 0); (d) Hf3CN-$C2 $ (1 0 0); (e) Hf6C2N3-$C2 $ (1 0 0); (f) Hf4CN2-Cmmm (0 0 1); (g) Hf6CN3-$C2 $/m (1 0 0); (h) Hf6CN4-$C2 $/m (0 0 1)

      Figure 2.  Vacancies on the crystallographic plane: (a) Hf6C4N-$C2 $/m (0 0 1); (b) Hf6C3N-$C2 $ (1 0 0); (c) Hf6C3N2-$C2 $/m (1 0 0); (d) Hf3CN-$C2 $ (1 0 0); (e) Hf6C2N3-$C2 $ (1 0 0); (f) Hf4CN2-Cmmm (0 0 1); (g) Hf6CN3-$C2 $/m (1 0 0); (h) Hf6CN4-$C2 $/m (0 0 1).

      图  3  Hf-C-N空位有序结构的声子谱曲线 (a) Hf6C4N-$C2 $/m; (b) Hf6C3N-$C2 $; (c) Hf6C3N2-$C2 $/m; (d) Hf3CN-$C2 $; (e) Hf6C2N3-$C2 $; (f) Hf4CN2-Cmmm; (g) Hf6CN3-$C2 $/m; (h) Hf6CN4-$C2 $/m

      Figure 3.  Phonon dispersion curves of (a) Hf6C4N-$C2 $/m, (b) Hf6C3N-$C2 $, (c) Hf6C3N2-$C2 $/m, (d) Hf3CN-$C2 $, (e) Hf6C2N3-$C2 $, (f) Hf4CN2-Cmmm, (g) Hf6CN3-$C2 $/m, (h) Hf6CN4-$C2 $/m. They are all dynamical stable because no imaginary frequencies were found in Brillouin zone.

    • 首先采用VASP软件[51]计算了Hf-C-N空位有序结构的弹性常数, 如表2所列, 这些结构的弹性常数都满足Born判据[63], 即都是力学稳定的. 再根据计算的弹性常数, 基于Voigt-Reuss-Hill近似[54-56], 得到体模量B、剪切模量G、弹性模量E、泊松比等. 采用Chen-Niu模型[57], 计算Hf-C-N空位有序结构的维氏硬度HV, 计算公式如下:

      CompoundsC11C22C33C44C55C66C12C13C23
      Hf6C4N-$C2 $/m414.3406.6415.6158.0170.6148.794.1116.1104.5
      Hf6C3N-$C2 $358.5362.8352.2100.0114.3132.387.598.391.6
      Hf6C3N2-$C2 $/m414.6417.4407.6152.2157.6147.8111.9115.0116.3
      Hf3CN-$C2 $354.5363.5348.790.6103.6128.7102.1109.5101.3
      Hf6C2N3-$C2 $409.7418.7418.1149.6160.2148.9123.4122.9126.5
      Hf4CN2-Cmmm373.4368.8406.8142.2133.1135.8146.4112.0124.4
      Hf6CN3-$C2 $/m361.1358.4351.784.999.8124.9108.1121.7114.2
      Hf6CN4-$C2 $/m401.2414.1403.5146.5157.2139.8134.0139.9147.8

      表 2  Hf-C-N空位有序结构的弹性常数Cij (单位: GPa)

      Table 2.  Calculated elastic constants Cij (in GPa) of Hf-C-N vacancy ordered structures.

      ${H_{\rm{V}}} = 2{\left( {{k^2}G} \right)^{0.585}} - 3, $

      其中Pugh比[58]k = G/B.

      Hf-C-N空位有序结构的力学性质如表3所列, 可以看到这些结构都具有非常高的体模量、剪切模量、弹性模量和维氏硬度等. 为了对比, HfC1–xNx的力学性质也在表3中列出, 尽管这些结构的力学性质已经在文献[19]中被报道了.

      CompoundB /GPaG /GPaE /GPaμG/BHV /GPa
      Hf6C4N229.0140.8350.60.24490.614817.5
      Hf5C4N[19]260.6201.3480.30.19280.772729.9
      Hf6C3N180.9121.5297.90.22560.671717.8
      Hf4C3N[19]262.2202.1482.40.19340.770729.9
      Hf6C3N2214.0151.1366.90.21430.705922.1
      Hf3CN188.0113.4283.30.24890.603114.6
      Hf2CN[19]268.1198.5477.60.20310.740328.1
      Hf6C2N3221.3149.7366.60.22390.676620.7
      Hf4CN2212.7132.8329.70.24170.624217.1
      Hf3CN2[19]272.8185.1452.80.22330.678623.9
      Hf6CN3195.4108.9275.60.26500.557412.7
      Hf4CN3[19]276.2179.6442.80.23280.650422.2
      Hf6CN4207.2156.1374.40.19890.753524.6
      Hf5CN4[19]279.0171.5427.00.24490.614720.0

      表 3  Hf-C-N空位有序结构和HfC1–xNx[19]的力学性质—体模量(B)、剪切模量(G )、弹性模量(E )、泊松比(μ)、Pugh比(G/B)、维氏硬度(HV)等

      Table 3.  Mechanical properties—bulk modulus (B), shear modulus (G ), elastic modulus (E ), Poisson’s ratio (μ), Pugh’s ratio (G/B), Vickers hardness (HV) of Hf-C-N vacancy ordered structures and HfC1–xNx[19].

      图4为三元Hf-HfC-HfN体系的力学性质-组分相图, 其中包括三元空位有序结构, Hf-C, Hf-N体系和HfC1–xNx等的性质[19,25,36]. 从图4可以看到: 相同C/N比下, 随着空位浓度的增大, 体模量、剪切模量、弹性模量等减小; 如图5图6所示, 空位浓度增大, Hf-C-N化合物的有效价电子浓度及总体键强减弱, 材料抵抗外力的能力减小, 则Hf-C-N化合物的模量减小. 然而, Hf6CN4 (空位浓度为1/6)的维氏硬度和Pugh比大于Hf5CN4(无空位)的维氏硬度和Pugh比, 表现出空位硬化现象; 而其他组分下, C/N比相同时, 维氏硬度和Pugh比等随空位浓度的增大而减小. 如图4(f)所示, 泊松比的变化规律与Pugh比的正好相反.

      图  4  三元Hf-HfC-HfN体系的力学性质-组分相图 (a) 体模量(B); (b) 剪切模量(G ); (c) 弹性模量(E ); (d) 维氏硬度(HV); (e) Pugh比(G/B); (f) 泊松比(μ)

      Figure 4.  Mechanical properties-composition diagrams of ternary Hf-HfC-HfN system: (a) Bulk modulus (B); (b) shear modulus (G ); (c) elastic modulus (E ); (d) Vickers hardness (HV); (e) Pugh’s ratio (G/B); (f) Poisson’s ratio (μ).

      图  5  (a) Hf6C4N-$C2 $/m, (b) Hf6C3N-$C2 $, (c) Hf6C3N2-$C2 $/m, (d) Hf3CN-$C2 $, (e) Hf6C2N3-$C2 $, (f) Hf4CN2-Cmmm, (g) Hf6CN3-$C2 $/m和(h) Hf6CN4-$C2 $/m的态密度和分态密度; (i) Hf3CN和Hf2CN的总态密度对比; 其中Fermi能级位于0 eV

      Figure 5.  Density of state (DOS) and partial density of state (PDOS) normalized by per HfCxNy of (a) Hf6C4N-$C2 $/m, (b) Hf6C3N-$C2 $, (c) Hf6C3N2-$C2 $/m, (d) Hf3CN-$C2 $, (e) Hf6C2N3-$C2 $, (f) Hf4CN2-Cmmm, (g) Hf6CN3-$C2 $/m and (h) Hf6CN4-$C2 $/m; (i) the total DOS of Hf3CN and Hf2CN normalized by per HfCxNy. The Fermi level is at 0 eV.

      图  6  Hf-C-N化合物的晶体轨道哈密顿分布(–COHP), Fermi能级位于0 eV

      Figure 6.  Crystal orbital Hamilton populations (–COHP) of Hf-C-N compounds. The Fermi level is at 0 eV.

    • 为了分析Hf-C-N空位有序结构的成键特性和空位对电子性质的影响, 对其态密度、分态密度和晶体轨道哈密顿分布进行了计算. 图5(a)(h)为Hf-C-N空位有序结构的态密度和分态密度图, 在能级为–8至–2 eV, Hf-d轨道与C-p和N-p轨道之间存在大量重叠, 即存在强的杂化作用, 则Hf—C和Hf—N键存在强的共价性. 这是Hf-C-N空位有序结构具有非常高的模量和硬度的原因. 同时, 在Fermi面上存在自由电子, 证明其具有金属性. 这些成键特点和二元过渡金属碳、氮化合物[24-28]及HfC1–xNx[19]的相同. 对比Hf2CN (不含空位, 该结构及其电子性质已在文献[19]中报道)和Hf3CN (空位浓度为1/3)的总态密度, 分析空位对态密度的影响, 如图5(i)所示: 可以看到两者价电子的能级排布基本不变, 然而Hf3CN各能级处对应的价电子状态数小于Hf2CN的; 这是由于空位存在, Hf3CN的有效价电子浓度(VEC为7)小于Hf2CN的(VEC为8.5). 而空位对其他Hf-C-N空位有序结构的态密度也存在相似的影响.

      图6为Hf-C-N空位有序结构的晶体轨道哈密顿分布(–COHP), 同时对比了HfC1–xNx的–COHP. 图6中, 正值表示成键态, 负值表示反键态. 从图6可以看到, Hf-C-N空位有序结构与HfC1–xNx的–COHP相似, 杂化能级(–8 — –2 eV)范围内, 存在宽的成键区域, 即具有强的共价性, 与态密度的计算结果一致. –COHP的积分用ICOHP表示, ICOHP反映了化学键的强弱. 表4为Hf-C-N化合物的Hf—C, Hf—N和Hf—Hf键的–ICOHP的大小, 可以看到: 相同C/N比下, 含空位化合物(如Hf3CN)的Hf—C键和Hf—N键的键强要比不含空位的(如Hf2CN)更强, 且Hf—Hf金属键也更强. 然而, 随着空位浓度增大, Hf—C键和Hf—N键的总键数目减小(如表1所列Hf的配位数小于6), 则总体键强减弱, 因此抵抗外力的能力减弱, Hf-C-N化合物的体模量、剪切模量和弹性模量随之减小.

      Compound–ICOHPCompound–ICOHP
      Hf—CHf—NHf—HfHf—CHf—NHf—Hf
      Hf6C4N3.3733.5670.529Hf6C3N23.1812.9900.571
      Hf5C4N3.3733.0330.459Hf4CN23.4743.1110.650
      Hf6C3N3.3503.0670.718Hf3CN23.5513.0910.541
      Hf4C3N3.3193.0290.454Hf6CN33.4083.2110.570
      Hf6C3N23.6073.1030.530Hf4CN33.3213.1590.520
      Hf3CN3.2773.2110.737Hf6CN43.6753.1790.591
      Hf2CN3.4832.8020.490Hf5CN43.3193.0170.500

      表 4  Hf-C-N化合物的晶体轨道哈密顿分布的积分值(–ICOHP)

      Table 4.  Integrated crystal orbital Hamilton populations (–ICOHP) of Hf-C-N compounds.

    • 本文采用空位调控方法, 对三元Hf-C-N体系的组分进行了设计; 采用第一性原理方法, 研究了Hf-C-N的空位有序结构及其力学性质和电子性质:

      1) 搜索发现了8种新的Hf-C-N空位有序结构, 且都具有岩盐结构; 空位有序地分布在[Hf6]八面体间隙, 证明了Hf-C-N空位化合物可以以有序结构的形式稳定存在;

      2) Hf-C-N空位有序结构具有非常高的模量和硬度; 相同C/N比时, 随着空位浓度的增大, 体模量、剪切模量、弹性模量减小;

      3) 发现了一个空位硬化现象, Hf6CN4 (空位浓度为1/6)的硬度大于Hf5CN4(无空位)的硬度;

      4) Hf-C-N空位有序结构的化学键具有强共价性和金属性. 随着空位浓度的增加, Hf—C和Hf—N共价键键强增大, 金属性增强, 但总体键强减弱, 使其模量减小.

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