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Secondary hardening ultra-high-strength steel is widely utilized in aerospace and other advanced engineering, with the nanoscale M2C precipitates serving as the primary strengthening factor. Mo plays a crucial role in the forming of Mo2C secondary hardening phase, which can form composite M2C precipitates with elements such as Cr, V, and W, thereby modifying the composition and properties of Mo2C. To investigate the effects of V and W doping on Mo2C, first-principles calculations are used to analyze the formation enthalpy, electronic structure, and mechanical properties of the doped systems. The CASTEP module is utilized in this study, with the Perdew-Burke-Ernzerhof (PBE) functional adopted in the generalized gradient approximation (GGA) framework. The results indicate that V doping reduces the lattice parameters and the formation enthalpy, thereby enhancing structural stability. In contrast, W doping increases the lattice parameters and lowers the formation enthalpy but leads the structural stability to decrease. In terms of mechanical properties, V doping reduces toughness while increasing hardness, whereas W doping improves the strength-toughness balance by mitigating the rate of hardness reduction. Covalent bonds are formed within the system, with V and W doping changing their characteristics: compared with the C—Mo bond, the C—V bond exhibits weaker covalency, while the C—W bond displays stronger covalency. Additionally, V doping enhances the stability of Mo—C bonds, whereas W doping reduces their stability. Charge population analysis reveals that metal atoms (Mo, V, and W) act as electron donors, while carbon atoms act as electron acceptors.
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Keywords:
- first-principles /
- M2C /
- elastic properties /
- electronic structure
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图 1 晶体结构以及超胞示意图 (a) Mo2C; (b) Mo8C4; (c) Mo7VC4; (d) Mo6V2C4; (e) Mo7WC4; (f) Mo6W2C4
Figure 1. Schematic diagram of crystal structure and the supercell: (a) Mo2C; (b) Mo8C4; (c) Mo7VC4; (d) Mo6V2C4; (e) Mo7WC4; (f) Mo6W2C4.
图 2 不同构型的弹性常数 (a) 弹性模量(B, G, E); (b) 模量比(G/B)和泊松比(ν)
Figure 2. Elastic constants of different configurations: (a) Elastic moduli (B, G, E); (b) modulus ratio (G/B) and Poisson's ratio (ν).
图 3 五种构型的态密度及各原子分波态密度图 (a) Mo8C4; (b) Mo7VC4; (c) Mo6V2C4; (d) Mo7WC4; (e) Mo6W2C4. (f) 四种掺杂构型的总态密度图
Figure 3. Density of states for five configurations and the partial density of states for each atom: (a) Mo8C4; (b) Mo7VC4; (c) Mo6V2C4; (d) Mo7WC4; (e) Mo6W2C4. (f) Total density of states for four doped configurations.
图 4 不同构型的态密度图 (a) (Mo, V)8C4; (b) (Mo, W)8C4
Figure 4. Density of states for different configurations: (a) (Mo, V)8C4; (b) (Mo, W)8C4.
图 5 未掺杂构型 (a) 差分片段; (b) 差分电荷密度图
Figure 5. Undoped configuration: (a) Differential fragment; (b) differential charge density map.
图 6 差分电荷密度图 (a) Mo7VC4; (b) Mo6V2C4; (c) Mo7WC4; (d) Mo6W2C4
Figure 6. Differential charge density maps: (a) Mo7VC4; (b) Mo6V2C4; (c) Mo7WC4; (d) Mo6W2C4.
表 1 V, W掺杂前后的晶格参数a, b, c(单位Å), V(Å3)以及α, β, γ
Table 1. Lattice parameters a, b, c (in Å), volume V (in ų), and angles α, β, γ before and after doping with V and W.
Configuration a b c V α β γ Volume expansion rate/% Mo2C 3.059 3.059 4.665 37.794 90.00 90.00 120.00 — (Mo4C2)[12] 3.056 3.056 9.331 75.476 — — — — (Mo2C)[13] 3.054 3.054 4.652 37.58 — — — — (Mo2C)[15] 3.051 3.051 4.624 37.3114 — — — — Mo8C4 6.108 3.054 9.346 150.996 90.06 89.98 120.00 — Mo7VC4 6.083 3.041 9.282 148.737 90.04 90.02 119.98 –1.50% Mo6V2C4 6.051 3.026 9.215 146.201 89.84 90.34 119.96 –3.18% Mo7WC4 6.109 3.055 9.350 151.139 90.07 89.96 119.99 0.09% Mo6W2C4 6.112 3.054 9.355 151.272 90.09 89.97 119.98 0.18% (V2C)[20] 3.045 3.045 4.409 35.4 — — — — (V2C)[21] 2.89 — — — — — — — (W2C)[20] 3.19 3.19 4.626 40.77 — — — — (W2C)[20] 3.060 3.060 4.703 — — — — — 
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表 2 不同构型的形成焓
Table 2. Enthalpy of formation for different configurations.
Configuration ΔH/(eV·atom–1) Mo8C4 –0.131 (Mo2C)[13] –0.113 Mo7VC4 –0.192 Mo6V2C4 –0.264 Mo7WC4 –0.121 Mo6W2C4 –0.111
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表 3 不同构型的单晶弹性常数
Table 3. Single crystal elastic constants of different configurations.
Configuration C11/GPa C12/GPa C13/GPa C33/GPa C44/GPa C66/GPa Mo8C4 475.26 119.88 180.55 451.77 137.69 178.17 Mo7VC4 481.10 117.69 177.65 466.54 134.82 178.91 Mo6V2C4 473.96 119.00 166.23 461.58 141.35 176.42 Mo7WC4 478.11 125.91 187.67 459.93 137.32 177.95 Mo6W2C4 483.65 131.41 196.03 468.71 137.92 178.09
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表 4 不同构型的维氏硬度HV, Hardness和硬度H
Table 4. Hardness of different configurations.
Configuration HV/GPa Hardness/GPa H/GPa Mo8C4 16.64 16.57 42.79 Mo7VC4 16.84 17.46 43.00 Mo6V2C4 18.12 18.13 41.71 Mo7WC4 15.98 16.00 43.92 Mo6W2C4 15.43 15.59 45.25
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表 5 不同构型的弹性各向异性指数(AU, AB, AG)
Table 5. Elastic anisotropy indices (AU, AB, AG) of different configurations.
Configuration BV/GPa GV/GPa BR/GPa GR/GPa AU AB/% AG/% Mo8C4 262.69 152.04 262.30 149.30 0.0931 0.0753 0.9073 Mo7VC4 263.86 153.98 263.30 149.99 0.1352 0.1059 1.3137 Mo6V2C4 256.93 155.90 256.62 153.64 0.0750 0.0617 0.7319 Mo7WC4 268.74 151.14 268.20 148.90 0.0773 0.1017 0.7471 Mo6W2C4 275.88 151.23 275.17 148.98 0.0779 0.1295 0.7472
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表 6 不同构型的键布居
Table 6. Different configurations of bond population.
Configuration Bond Number Population Length/Å Mo8C4 C—Mo 8 0.67 2.1141 C—Mo 8 0.28 2.1143 C—Mo[15] — — 2.118[15] Mo7VC4 C—Mo 7 0.69 2.1116 C—Mo 7 0.28 2.1131 C4—V1 1 0.17 2.0367 C3—V1 1 0.60 2.0380 Mo6V2C4 C—Mo 6 0.73 2.1057 C—Mo 6 0.28 2.1076 C—V 2 0.59 2.0420 C—V 2 0.17 2.0675 Mo7WC4 C—Mo 7 0.66 2.1152 C—Mo 7 0.27 2.1164 C4—W1 1 0.34 2.1173 C3—W1 1 0.79 2.1233 Mo6W2C4 C—Mo 6 0.65 2.1162 C—Mo 6 0.26 2.1187 C—W 2 0.31 2.1210 C—W 2 0.76 2.1238
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表 7 Mo8C4的电荷布居
Table 7. Charge distribution of Mo8C4.
Configuration Atom s p d f Total electron/e Muliken charge/e Mo8C4 Mo 2.20 6.64 4.86 0.00 13.70 0.30 C 1.44 3.16 0.00 0.00 4.60 –0.60
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表 9 W掺杂Mo8C4的电荷布居
Table 9. Charge distribution of W-doped Mo8C4.
Configuration Atom Total electron/e Muliken charge/e Configuration Atom Total electron/e Muliken charge/e Mo7WC4 Mo1 13.70 0.29 Mo6W2C4 Mo1 13.70 0.30 Mo2 13.70 0.31 Mo2 13.70 0.30 Mo3 13.70 0.29 Mo3 13.72 0.29 Mo4 13.70 0.30 Mo4 13.72 0.29 Mo5 13.70 0.30 Mo5 13.66 0.33 Mo6 13.68 0.32 Mo6 13.66 0.33 Mo7 13.66 0.33 W1 27.70 0.29 W1 27.84 0.27 W2 27.70 0.29 C1 4.60 –0.60 C1 4.60 –0.60 C2 4.60 –0.60 C2 4.60 –0.60 C3 4.62 –0.61 C3 4.62 –0.62 C4 4.60 –0.61 C4 4.62 –0.62
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表 8 V掺杂Mo8C4的电荷布居
Table 8. Charge distribution of V-doped Mo8C4.
Configuration Atom Total electron/e Muliken charge/e Configuration Atom Total electron/e Muliken charge/e Mo7VC4 Mo1 13.76 0.24 Mo6V2C4 Mo1 13.80 0.20 Mo2 13.70 0.30 Mo2 13.80 0.20 Mo3 13.76 0.24 Mo3 13.80 0.21 Mo4 13.72 0.29 Mo4 13.80 0.20 Mo5 13.74 0.26 Mo5 13.80 0.21 Mo6 13.78 0.22 Mo6 13.80 0.20 Mo7 13.76 0.25 V1 12.38 0.62 V1 12.38 0.63 V2 12.38 0.62 C1 4.60 –0.60 C1 4.62 –0.61 C2 4.60 –0.60 C2 4.62 –0.61 C3 4.62 –0.62 C3 4.62 –0.62 C4 4.62 –0.61 C4 4.62 –0.61
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