First-principles study on structures and physical properties of multicomponent V1–x FexC carbides in steel

ZHANG Dong; SUN Yihua; YIN Chaochao

doi:10.7498/aps.74.20250713

Abstract

Vanadium carbides commonly serve as strengthening phases in metallic materials, where their elastic and ductile-brittle characteristics are critical for mechanical performance. This work systematically investigates the structural stability, electronic properties, mechanical behaviors, and thermal characteristics of multi-component V_1–x Fe_xC carbides by using first-principles calculations, aiming to elucidate the influence of Fe content on their physical properties and provide a theoretical basis for the design and application of carbides in high-performance steels. The calculations are performed using the Vienna ab initio simulation package (VASP) based on density functional theory (DFT). Special quasirandom structures (SQS) are employed to construct five carbide models with varying Fe/V ratios (from V_0.125Fe_0.875C to V_0.875Fe_0.125C). Key parameters including formation enthalpy, electronic density of states, elastic constants, Debye temperature, and thermal conductivity are computed. The results indicate that as the Fe content decreases, the formation enthalpy shifts from positive to negative, reflecting a significant improvement in thermodynamic stability. Electronic structure analyses reveal metallic behavior of all compositions, with stronger covalent bonding in V–C than that in Fe–C. The V_0.875Fe_0.125C carbide exhibits the highest elastic modulus (C₁₁ = 615.80 GPa) and Vickers hardness (21.06 GPa), which is attributed to its strong covalent interactions, though it also shows increased brittleness. The Debye temperature rises with the decrease of Fe content, further confirming superior mechanical strength at elevated temperatures. Calculations of the thermal conductivity for V_0.875Fe_0.125C yield values of 9.427 W·m^–¹·K^–¹ at 300 K and 2.357 W·m^–¹·K^–¹ at 1300 K. Its minimum lattice thermal conductivity (2.001 W·m^–¹·K^–¹) is comparable to that of typical thermal barrier coating materials, demonstrating high potential for high-temperature thermal insulation. This study reveals the structure-property relationships in V_1–x Fe_xC carbides on an atomic scale, indicating that low-Fe compositions are advantageous for high-temperature and high-strength applications. These findings provide important theoretical support for the development of novel heat-resistant coatings and high-strength steels.

Keywords:

Authors and contacts

1.
Hubei Key Laboratory of Hydropower Machinery Design and Maintenance, China Three Gorges University, Yichang 443002, China
2.
Collaborative Innovation Center for Advanced Steels, Wuhan University of Science and Technology, Wuhan 430081, China
3.
School of Intelligent Manufacturing, Wenzhou Polytechnic, Wenzhou 325035, China

Corresponding author: SUN Yihua, sunny.hust@ctgu.edu.cn ; YIN Chaochao, ycc0125@126.com

Funds: Project supported by the National Key Research and Development Program, China (Grant No. 2023YFB3812200), the Natural Science Foundation Innovation Development Joint Fund of Hubei Province, China (Grant No. 2025AFD412), the Open Fund of the Hubei Key Laboratory of Hydroelectric Mechanical Equipment Design and Maintenance (China Three Gorges University), China (Grant No. 2025KJX07), the Open Fund of the Hubei Key Laboratory of System Science for Metallurgical Industry Processes (Wuhan University of Science and Technology), China (Grant No. Y202301), and the Scientific Research Start Fund of Three Gorges University, China (Grant No. 2023RCKJ0031).

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References

[1]	Williams W S 1971 Prog. Solid State Chem. 6 57 Google Scholar
[2]	Chen Y, Ye C, Chen X, Hu H 2024 Metals 14 175 Google Scholar
[3]	康俊雨, 孙新军, 李昭东, 雍岐龙 2015 钢铁 50 64 Google Scholar Kang J L, Sun X J, Li Z D, Yong Q L 2015 Iron and Steel 50 64 Google Scholar
[4]	Giang N A, Kuna M, Hütter G 2017 Theor. Appl. Fract. Mech. 92 89 Google Scholar
[5]	Weinberger C R, Thompson G B 2018 J. Am. Ceram. Soc. 101 4401 Google Scholar
[6]	Jang J H, Lee C H, Heo Y U, Suh D W 2012 Acta Mater. 60 208 Google Scholar
[7]	Li X T, Zhang X Y, Qin J Q, Zhang S H, Ning J L, Jing R, Ma M Z, Liu R P 2014 J. Phys. Chem. Solids 75 1234 Google Scholar
[8]	Zhang D, Tang X H, Humphries E, Li D Y 2023 Wear 523 204808 Google Scholar
[9]	Sun C C, Zheng Y, Chen L L, Fang F, Zhou X F, Jiang J Q 2022 J. Alloys Compd. 895 162649 Google Scholar
[10]	Zhang D, Hou T P, Quan X L, Zhou J, Yin C C, Lin H F, Lu Z H, Wu K M 2023 J. Mater. Res. Technol. 25 210 Google Scholar
[11]	Kohn W, Becke A D, Parr R G 1996 J. Phys. Chem. 100 12974 Google Scholar
[12]	Blöchl P E 1994 Phys. Rev. B 50 17953 Google Scholar
[13]	van de Walle A, Tiwary P, de Jong M, Asta M, Dick A, Shin D, Wang Y, Chen L Q, Liu Z K 2013 Calphad 42 13 Google Scholar
[14]	Yu R, Zhu J, Ye H Q 2010 Comput. Phys. Commun. 181 671 Google Scholar
[15]	张梅玲, 陈玉红, 张材荣, 李公平 2019 68 087101 Google Scholar Zhang M L, Chen Y H, Zhang C R, Li G P 2019 Acta Phys. Sin. 68 087101 Google Scholar
[16]	Zhang D, Xiang R, Sun Y 2025 Mol. Phys. 123 e2379994 Google Scholar
[17]	Kobayashi S, Ikuhara Y, Mizoguchi T 2018 Phy. Rev. B 98 134114 Google Scholar
[18]	Mishra S, Ganguli B 2013 J. Solid State Chem. 200 279 Google Scholar
[19]	Yamada K, Yosida K, Hanzawa K 1992 Prog. Theor. Phys. Suppl. 108 141 Google Scholar
[20]	Bader R F W 1985 Acc. Chem. Res. 18 9 Google Scholar
[21]	Guo L Q, Tang Y Q, Cui J, Li J Q, Yang J R, Li D Y 2021 Scr. Mater. 190 168 Google Scholar
[22]	Wu Y, Ma L S, Zhou X L, Duan Y H, Shen L, Peng M J 2022 Int. J. Refract. Met. Hard Mater 109 105985 Google Scholar
[23]	Zhang D, Hou T P, Liang X, Zheng P, Zheng Y H, Lin H F, Wu K M 2022 Vacuum 203 111175 Google Scholar
[24]	Soni P, Pagare G, Sanyal S P 2011 J. Phys. Chem. Solids 72 810 Google Scholar
[25]	Zuo L, Humbert M, Esling C 1992 J. Appl. Crystallogr. 25 751 Google Scholar
[26]	Pugh S F 1954 Lond. Edinb. Phil. Mag. 45 823 Google Scholar
[27]	Pettifor D G 1992 Mater. Sci. Technol. 8 345 Google Scholar
[28]	Niu H, Niu S, Oganov A R 2019 J. Appl. Phys. 125 065105 Google Scholar
[29]	Munro R G, Freiman S W, Baker T L 1998 Natl. Inst. Stand. Technol. 158 6153 Google Scholar
[30]	王坤, 徐鹤嫣, 郑雄, 张海丰 2025 74 137101 Google Scholar Wang K, Xu H Y, Zheng X, Zhang H F 2025 Acta Phys. Sin. 74 137101 Google Scholar
[31]	Yang J, Shahid M, Wan C, Jing F, Pan W 2017 J. Eur. Ceram. Soc. 37 689 Google Scholar
[32]	Shindé S L, Goela J 2006 High Thermal Conductivity Materials (New York: Springer) pp111–123
[33]	Arab F, Sahraoui F A, Haddadi K, Bouhemadou A, Louail L 2016 Phase Transit. 89 480 Google Scholar
[34]	Ahmed T, Roknuzzaman M, Sultana A, Biswas A, Safin A M, Saiduzzaman M, Hossain K M 2021 Mater. Today Commun. 29 102973 Google Scholar
[35]	Vagge S T, Ghogare S 2022 Mater. Today 56 1201 Google Scholar
[36]	Feng J, Xiao B, Wan C L, Qu Z X, Huang Z C, Chen J C, Zhou R, Pan W 2011 Acta Mater. 59 1742 Google Scholar
[37]	Feng J, Xiao B, Zhou R, Pan W, Clarke D R 2012 Acta Mater. 60 3380 Google Scholar

Cited By

图 1 试样钢微观组织以及定性表征元素分布　(a)微观组织形貌; (b)高角环形暗场像; (c) HRTEM-EDS能谱面扫和(d)线扫图

Figure 1. Microstructure of sample steel and distribution of qualitatively characterised elements: (a) Microstructural morphology; (b) high-angle annular dark-field image; (c) HRTEM-EDS energy spectrum area scan and (d) line scan image.

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图 2 V_1–x Fe_xC碳化物晶体结构　(a) VC; (b) V_0.125Fe_0.875C; (c) V_0.25Fe_0.75C; (d) V_0.5Fe_0.5C; (e) V_0.75Fe_0.25C; (f) V_0.875Fe_0.125C

Figure 2. Crystal structure of V_1–x Fe_xC carbides: (a) VC; (b) V_0.125Fe_0.875C; (c) V_0.25Fe_0.75C; (d) V_0.5Fe_0.5C; (e) V_0.75Fe_0.25C; (f) V_0.875Fe_0.125C

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图 3 V_1–x Fe_xC碳化物的TDOS和PDOS　(a) V_0.125Fe_0.875C; (b) V_0.25Fe_0.75C; (c) V_0.5Fe_0.5C; (d) V_0.75Fe_0.25C; (e) V_0.875Fe_0.125C

Figure 3. TDOS and PDOS of V_1–x Fe_xC carbides: (a) V_0.125Fe_0.875C; (b) V_0.25Fe_0.75C; (c) V_0.5Fe_0.5C; (d) V_0.75Fe_0.25C; (e) V_0.875Fe_0.125C

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图 4 V_1–x Fe_xC碳化物的声速(纵波v_l、剪切波v_s和平均声速v_m)以及德拜温度($ \theta_{\text{D}} $)

Figure 4. Calculated sound velocities (long wave v_l, shear wave v_s, and average sound velocity v_m), and Debye temperature ($ \theta_{\text{D}} $) of V_1–x Fe_xC carbides.

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表 1 X射线能谱分析下的碳化物元素含量

Table 1. Elemental content of carbides after energy-dispersive X-ray spectroscopy.

元素	原子百分比/%	质量分数/%
C	4.47	1.04
Fe	60.96	66.11
V	27.35	27.06
Cr	1.16	1.17
Mn	2.29	2.44
Other elements	3.77	2.18

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表 2 V_1–x Fe_xC碳化物结构晶胞参数、形成焓(ΔH_f)、金属性(f_m)和Bader电荷

Table 2. Calculated structural cell parameters, formation energy (ΔH_f), metallicness (f_m) and Bader charge of V_1–x Fe_xC carbides

	a/Å	b/Å	c/Å	$\alpha $/(°)	$\beta $/(°)	γ/(°)	ΔH_f/( meV·atom^–1)	f_m	Bader
V_0.125Fe_0.875C	8.061	8.065	8.063	90.08	89.99	89.95	0.443	0.548	0.840
V_0.25Fe_0.75C	8.106	8.107	8.107	90.00	90.00	90.01	0.321	0.622	0.914
V_0.5Fe_0.5C	8.203	8.188	8.195	90.00	90.00	90.00	0.042	0.844	1.052
V_0.75Fe_0.25C	8.262	8.261	8.258	89.92	90.08	89.99	–0.230	0.971	1.146
V_0.875Fe_0.125C	8.287	8.287	8.288	89.93	90.00	89.97	–0.439	1.19	1.213

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表 3 V_1–x Fe_xC碳化物的弹性常数C_ij、体积模量B_H、剪切模量G_H、杨氏模量E、泊松比$\nu $、普格模量比B_H/G_H、硬度H_V、断裂韧性K_IC以及脆性指数$ {{M}}_{{x}} $

Table 3. Calculated elastic constants C_ij, bulk modulus B_H, shear modulus G_H, Young’s modulus E, Poisson’s ratio $\nu $, Pugh modulus ratio B_H/G_H, hardness H_V, fracture toughness K_IC, and brittleness index $ {{M}}_{{x}} $ of V_1–x Fe_xC carbides.

碳化物	C₁₁/GPa	C₁₂/GPa	C₄₄/GPa	B_H/GPa	G_H/GPa	B_H/G_H	E	$\nu $	H_V/GPa	K_IC/(MPa·m^1/2)	M_x/μm^–1/2
VC	668.78	138.75	200.00	315.43	223.89	1.41	543.15	0.345	28.73	3.83
VC	648.24^[24]	156.88^[24]	209.99^[24]	318^[9]	213^[9]	1.49^[9]	521^[9]	0.356^[9]	25.8^[9]
V_0.125Fe_0.875C	552.84	165.68	75.32	294.40	110.79	2.657	442.11	0.333	8.49	2.564	3.310
V_0.25Fe_0.75C	553.84	155.42	86.486	288.10	121.67	2.368	319.96	0.315	10.34	2.665	3.878
V_0.5Fe_0.5C	563.87	152.52	113.41	289.61	144.23	2.008	371.09	0.286	14.06	2.925	4.808
V_0.75Fe_0.25C	584.01	148.76	149.97	293.96	174.39	1.686	436.80	0.252	19.63	3.254	6.034
V_0.875Fe_0.125C	615.80	154.92	162.02	308.46	186.60	1.650	465.87	0.248	21.06	3.450	6.104

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表 4 V_1–x Fe_xC碳化物沿[100], [110]和[111]方向的声速(m/s)

Table 4. Calculated sound velocities (m/s) along [100], [110], and [111] directions of V_1–x Fe_xC carbides.

		V_0.125Fe_0.875C	V_0.25Fe_0.75C	V_0.5Fe_0.5C	V_0.75Fe_0.25C	V_0.875Fe_0.125C
[100]	[100]v_l	9003.42	9126.02	9446.38	9820.16	10182.71
	[010]v_s1	3323.25	3606.30	4236.44	4976.31	5223.03
	[001]v_s2	3323.25	3606.30	4236.44	4976.31	5223.03
[110]	[110]v_l	7982.57	8144.52	8639.03	9233.81	9600.33
	$[1{\bar 1}0] $v_s1	5327.68	5473.24	5705.14	5994.67	6229.07
	[001]v_s2	3323.25	3606.30	4236.44	4976.31	5223.03
[111]	[111]v_l	7611.92	7789.92	8352.592	9029.91	9398.19
	$ [11{\bar 2}] $v_s1	4754.38	4930.12	5261.33	5675.55	5912.77
	$[11{\bar 2}] $v_s2	4754.38	4930.12	5261.33	5675.55	5912.77

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表 5 V_1–x Fe_xC碳化物的格林艾森参数γ, A_γ, $\delta $, M_av, 晶格热导率k_ph以及最小晶格热导率k_min

Table 5. Calculated Grüneisen parameter γ, A_γ, $ \delta$, M_av, lattice thermal conductivity k_ph, and minimum lattice thermal conductivity k_min of V_1–x Fe_xC carbides.

碳化物	γ	A_γ /(10^–8)	V	$\delta $/Å	M_av/(kg·mol^–1)	n	k_ph(300)/(W·m^–1·K^–1)	k_ph(1300)/(W·m^–¹·K^–1)	k_min/(W·m^–1·K^–1)
V_0.125Fe_0.875C	1.996	3.039	524.18	2.016	22.413	64	2.139	0.535	1.536
V_0.25Fe_0.75C	1.869	3.075	532.81	2.027	22.208	64	2.873	0.718	1.609
V_0.5Fe_0.5C	1.692	3.132	550.38	2.049	21.800	64	4.729	1.182	1.752
V_0.75Fe_0.25C	1.511	3.199	563.61	2.065	21.392	64	8.183	2.046	1.929
V_0.875Fe_0.125C	1.492	3.206	569.15	2.072	21.188	64	9.427	2.357	2.001

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map

[1]	Williams W S 1971 Prog. Solid State Chem. 6 57 Google Scholar
[2]	Chen Y, Ye C, Chen X, Hu H 2024 Metals 14 175 Google Scholar
[3]	康俊雨, 孙新军, 李昭东, 雍岐龙 2015 钢铁 50 64 Google Scholar Kang J L, Sun X J, Li Z D, Yong Q L 2015 Iron and Steel 50 64 Google Scholar
[4]	Giang N A, Kuna M, Hütter G 2017 Theor. Appl. Fract. Mech. 92 89 Google Scholar
[5]	Weinberger C R, Thompson G B 2018 J. Am. Ceram. Soc. 101 4401 Google Scholar
[6]	Jang J H, Lee C H, Heo Y U, Suh D W 2012 Acta Mater. 60 208 Google Scholar
[7]	Li X T, Zhang X Y, Qin J Q, Zhang S H, Ning J L, Jing R, Ma M Z, Liu R P 2014 J. Phys. Chem. Solids 75 1234 Google Scholar
[8]	Zhang D, Tang X H, Humphries E, Li D Y 2023 Wear 523 204808 Google Scholar
[9]	Sun C C, Zheng Y, Chen L L, Fang F, Zhou X F, Jiang J Q 2022 J. Alloys Compd. 895 162649 Google Scholar
[10]	Zhang D, Hou T P, Quan X L, Zhou J, Yin C C, Lin H F, Lu Z H, Wu K M 2023 J. Mater. Res. Technol. 25 210 Google Scholar
[11]	Kohn W, Becke A D, Parr R G 1996 J. Phys. Chem. 100 12974 Google Scholar
[12]	Blöchl P E 1994 Phys. Rev. B 50 17953 Google Scholar
[13]	van de Walle A, Tiwary P, de Jong M, Asta M, Dick A, Shin D, Wang Y, Chen L Q, Liu Z K 2013 Calphad 42 13 Google Scholar
[14]	Yu R, Zhu J, Ye H Q 2010 Comput. Phys. Commun. 181 671 Google Scholar
[15]	张梅玲, 陈玉红, 张材荣, 李公平 2019 68 087101 Google Scholar Zhang M L, Chen Y H, Zhang C R, Li G P 2019 Acta Phys. Sin. 68 087101 Google Scholar
[16]	Zhang D, Xiang R, Sun Y 2025 Mol. Phys. 123 e2379994 Google Scholar
[17]	Kobayashi S, Ikuhara Y, Mizoguchi T 2018 Phy. Rev. B 98 134114 Google Scholar
[18]	Mishra S, Ganguli B 2013 J. Solid State Chem. 200 279 Google Scholar
[19]	Yamada K, Yosida K, Hanzawa K 1992 Prog. Theor. Phys. Suppl. 108 141 Google Scholar
[20]	Bader R F W 1985 Acc. Chem. Res. 18 9 Google Scholar
[21]	Guo L Q, Tang Y Q, Cui J, Li J Q, Yang J R, Li D Y 2021 Scr. Mater. 190 168 Google Scholar
[22]	Wu Y, Ma L S, Zhou X L, Duan Y H, Shen L, Peng M J 2022 Int. J. Refract. Met. Hard Mater 109 105985 Google Scholar
[23]	Zhang D, Hou T P, Liang X, Zheng P, Zheng Y H, Lin H F, Wu K M 2022 Vacuum 203 111175 Google Scholar
[24]	Soni P, Pagare G, Sanyal S P 2011 J. Phys. Chem. Solids 72 810 Google Scholar
[25]	Zuo L, Humbert M, Esling C 1992 J. Appl. Crystallogr. 25 751 Google Scholar
[26]	Pugh S F 1954 Lond. Edinb. Phil. Mag. 45 823 Google Scholar
[27]	Pettifor D G 1992 Mater. Sci. Technol. 8 345 Google Scholar
[28]	Niu H, Niu S, Oganov A R 2019 J. Appl. Phys. 125 065105 Google Scholar
[29]	Munro R G, Freiman S W, Baker T L 1998 Natl. Inst. Stand. Technol. 158 6153 Google Scholar
[30]	王坤, 徐鹤嫣, 郑雄, 张海丰 2025 74 137101 Google Scholar Wang K, Xu H Y, Zheng X, Zhang H F 2025 Acta Phys. Sin. 74 137101 Google Scholar
[31]	Yang J, Shahid M, Wan C, Jing F, Pan W 2017 J. Eur. Ceram. Soc. 37 689 Google Scholar
[32]	Shindé S L, Goela J 2006 High Thermal Conductivity Materials (New York: Springer) pp111–123
[33]	Arab F, Sahraoui F A, Haddadi K, Bouhemadou A, Louail L 2016 Phase Transit. 89 480 Google Scholar
[34]	Ahmed T, Roknuzzaman M, Sultana A, Biswas A, Safin A M, Saiduzzaman M, Hossain K M 2021 Mater. Today Commun. 29 102973 Google Scholar
[35]	Vagge S T, Ghogare S 2022 Mater. Today 56 1201 Google Scholar
[36]	Feng J, Xiao B, Wan C L, Qu Z X, Huang Z C, Chen J C, Zhou R, Pan W 2011 Acta Mater. 59 1742 Google Scholar
[37]	Feng J, Xiao B, Zhou R, Pan W, Clarke D R 2012 Acta Mater. 60 3380 Google Scholar

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Vol. 74, Issue 21 - 2025-11-05

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