Optimized Design of Quinary High-Entropy Transition Metal Carbide Ceramics Based on First Principles

Abstract

In this paper, we developed models for 21 quinary high-entropy transition metal carbide ceramics (HETMCCs), composed of carbon and the transition metals Ti, Zr, Mo, V, Nb, W, and Ta, employing the Special Quasirandom Structures (SQS) method. We investigated how the transition metal elements influence lattice distortion, mixing enthalpy, Gibbs free energy of mixing, and the electronic structure of the systems through first-principles calculations. The calculations show that 21 systems can form a stable single phase, among which (TiMoVNbTa)C₅, (ZrMoNbWTa)C₅, and (MoVNbWTa)C₅ exhibit superior stability. The formation energy and migration energy of carbon vacancies in systems with strong single-phase stability were calculated to predict their radiation resistance. The formation energy of carbon vacancies is closely related to the types of surrounding transition metal elements, with values ranging between the maximum and minimum formation energies observed in binary transition metal carbides (TMCs). The range of migration energy for carbon vacancies is wider than that observed in TMCs, which can hinder their long-range migration and enhance the radiation resistance of the materials.

1. Introduction

With the growing demand for energy, nuclear energy has emerged as a prominent clean energy alternative. However, the safety of nuclear power plants remains a paramount concern, which requires that the structural materials employed in nuclear reactors possess the ability to endure prolonged exposure to elevated temperatures, intense neutron radiation, and highly corrosive environments [1,2]. TMCs have high melting points, high thermal conductivity, good thermal stability, and excellent irradiation resistance, rendering them promising candidates as structural materials in high-temperature nuclear reactors. Notably, ZrC, TiC, and the MAX phase (Ti₃SiC₂) are the most representative materials [3,4]. However, under conditions characterized by high temperature and substantial irradiation doses, these materials are susceptible to phase transitions and elemental segregation. The formation of numerous point defects can lead to the development of dislocation loops and stacking faults through migration, resulting in noticeable damage to macroscopic properties. Recent studies have shown that the high entropy effect, lattice distortion effect, sluggish diffusion effect, and cocktail effect in high-entropy alloys (HEAs) greatly minimize the occurrence of irradiation defects. Additionally, these effects significantly lower the mobility of defects, prevent their expansion, and facilitate the recombination of irradiated defects via intricate diffusion pathways within the materials. [5,6,7,8,9,10,11]. All these factors suggest that incorporating the high-entropy effect into TMCs is likely to further improve their resistance to radiation.

Currently, studies on HETMCCs mainly concentrate on aspects such as single-phase stability, methods of preparation, mechanical characteristics, antioxidant abilities, and thermal stability. There are only a few publications that discuss experimental research on radiation resistance [12,13,14,15,16,17,18,19,20,21]. Wang et al. [22] synthesized (ZrTaNbTi)C₄ by spark plasma sintering and discovered that the material maintained excellent phase stability with no phase changes, void formation, or irradiation-induced segregation when irradiated with Zr ions at 3 MeV for 20 displacements per atom (dpa) at temperatures of 25, 300, and 500 °C. Zhu et al. [23] synthesized (WTiVNbTa)C₅ by spark plasma sintering and irradiated it with C ions at an energy of 1.0 MeV at room temperature and 650 °C. Their findings indicated that black spot defects led to lattice expansion and microstrain, and none of the samples showed signs of amorphization and porosity after irradiation. The remarkable phase stability of HETMCCs enhances their resistance to radiation. However, there are various kinds of HETMCCs, and not all demonstrate strong phase stability. Therefore, large-scale screening using first-principles calculation methods is a useful strategy. Due to the similarity of high-entropy ceramics (HECs) and HEAs, several of the criteria for the single-phase stability of HECs are also applicable to HEAs. Zhang et al. suggested that according to thermodynamics and the Hume-Rothery rule, a system is likely to create random solid solutions when the mixing enthalpy (ΔH_mix) falls between −15 and 5 kJ/mol, and the difference in atomic radii (δ_r) is between 1% and 5% [24,25]. Ye et al. [26] discovered that the sintering temperature required for (ZrNbTiV)C₄ to form a single phase in their experiment was much higher than the 959 K predicted by first-principles calculations. Therefore, there remains a discrepancy between the outcomes of first-principles calculations conducted at 0 K and the experimental findings observed at higher temperatures. Wang et al. [27] calculated δ_r and the lattice constant difference (δ_a) of (Hf_0.2Zr_0.2Ta_0.2M_0.2Ti_0.2)B₂ (M = Nb, Mo, Cr) by first principles and contrasted their results with experimental data. They found that δ_a is a more suitable descriptor for lattice distortion. In 2018, Sarker P et al. [28] developed the parameter EFA (Entropy Forming Ability) descriptor for the first time, which successfully forecasts the single-phase stability of HETMCCs. However, the calculation parameters only account for HECs with an equal molar ratio and exclude Cr element. At present, there is no complete framework for the theoretical analysis of the single-phase stability of HECs. As a result, it is crucial to create criteria for assessing the single-phase stability of HECs using simulation calculations and experimental methods.

Due to the diverse component ratios of HETMCCs, the generation and evolution of irradiation defects are crucial for regulating radiation resistance, and the material has radiation after neutron irradiation. Therefore, in this paper, five elements are randomly selected from Ti, Zr, Mo, V, Nb, W, and Ta to form 21 different types of quinary HETMCCs in equal molar ratios. The assessment of HETMCCs, characterized by enhanced single-phase stability, is conducted through analyses of lattice distortion and thermodynamic behavior, employing first-principles computational methods. The investigation includes calculations of the formation and migration energy of carbon vacancies within HETMCCs, with the aim of predicting their radiation resistance, promoting the development of cost-effective components, and providing a theoretical foundation for empirical validation. In Section 3.1, we explore methodologies for predicting the single-phase stability of HETMCCs, taking into account factors such as lattice distortion, thermodynamic parameters, and bond length distributions among adjacent atoms. In Section 3.2, we delve into the assessment of single-phase stability based on their electronic structure, while Section 3.3 addresses the prediction of anti-irradiation performance through the analysis of vacancy formation energy and migration energy.

2. Materials and Methods

The HETMCCs are modeled using the Special Quasirandom Structures (SQS) method from the “mcsqs” module in the Alloy Theoretic Automated Toolkit (ATAT) [29]. The basic idea of this method is to adjust the arrangement of the atoms in small supercells so that the association function of the cluster can match that of the actual crystal in a completely disordered state. The numbering of 21 HETMCCs and their phase composition observed in the experiment are shown in

Table 1. Numbering and phase composition of 21 HETMCCs.

Chemical Formula	System	Phase Composition
(TiZrMoVNb)C5	HETMCC-1	-
(TiZrMoVW)C5	HETMCC-2	M [30]
(TiZrMoVTa)C5	HETMCC-3	-
(TiZrMoNbW)C5	HETMCC-4	-
(TiZrMoNbTa)C5	HETMCC-5	S [31]
(TiZrMoWTa)C5	HETMCC-6	-
(TiZrVNbW)C5	HETMCC-7	-
(TiZrVNbTa)C5	HETMCC-8	S [31]
(TiZrVWTa)C5	HETMCC-9	-
(TiZrNbWTa)C5	HETMCC-10	S [31]
(TiMoVNbW)C5	HETMCC-11	-
(TiMoVNbTa)C5	HETMCC-12	S [31,32]
(TiMoVWTa)C5	HETMCC-13	S [33]
(TiMoNbWTa)C5	HETMCC-14	S [30]
(TiVNbWTa)C5	HETMCC-15	S [32]
(ZrMoVNbW)C5	HETMCC-16	-
(ZrMoVNbTa)C5	HETMCC-17	S [30]
(ZrMoVWTa)C5	HETMCC-18	-
(ZrMoNbWTa)C5	HETMCC-19	S [34]
(ZrVNbWTa)C5	HETMCC-20	S [30]
(MoVNbWTa)C5	HETMCC-21	S [32]

. First-principles density functional theory (DFT) calculations were carried out with the castep module of Materials Studio 8.0 software. The exchange association functionals used the Perfectly Matched Bases Energies (PBE) form under generalized gradient approximation (GGA). The choice of pseudopotential was “ultrasoft”. The plane wave energy cutoff was set to 500 eV, the fine option was selected for K points in the Brillouin zone, and the total energy convergence criterion was 10⁻⁵ eV.

The δ_r can be calculated by using the following equation [26]:

$${\delta }_r=100\sqrt{{\sum }_{i=1}^n{c_i(1-{\displaystyle \frac{r_i}{{\sum }_{j=1}^n{c}_j{r}_j}})}^2}$$

(1)

where n is the number of components of HETMCCs, c_i is the percentage of the ith transition metal atoms in the total metal atoms, c_j is the percentage of the jth transition metal atoms in the total metal atoms, r_i is the ith transition metal atomic radius, and r_j is the jth transition metal atomic radius.

The δ_a can be calculated by using the following equation [26]:

$${\delta }_a=100\sqrt{{\sum }_{i=1}^n{c_i(1-{\displaystyle \frac{a_i}{{\sum }_{j=1}^n{c}_j{a}_j}})}^2}$$

(2)

where a_i is the lattice constant of the ith TMCs, and a_j is the lattice constant of the jth TMCs.

The rate of volume change ${\omega }_V$ between the generated HETMCC and the reactants can be calculated by using the following equations:

$$∆V=V_H-V$$

(3)

$${\omega }_V(\%)=\left|{\displaystyle \frac{∆V}{V}}\right|\times 100\%$$

(4)

where V is the total volume of multi-component TMCs corresponding to HETMCCs, V_H is the volume of HETMCCs, and ΔV is the difference between the total volume of TMCs and the volume of HETMCCs.

The thermodynamic stability of materials is determined by the Gibbs free energy of mixing (ΔG_mix). It can be calculated by using the following equation [26]:

$$∆G_{mix}=∆H_{mix}-T∆S_{mix}$$

(5)

where ΔH_mix is the mixing enthalpy of HETMCCs, and ΔS_mix is the mixing entropy of HETMCCs. ΔH_mix is not sensitive to temperature, and its value at 0 K can be used to estimate its value at other temperatures.

The ΔH_mix of HETMCCs at 0 K and 0 Pa can be calculated by using the following formula [26]:

$$∆H_{mix}^{0K}=E_{HETMCC}-{\sum }_{i=1}^5{c}_i{E}_i$$

(6)

where E_HETMCC is the energy of HETMCCs after relaxation at 0 K and 0 Pa. $c_i$ is the percentage of the ith transition metal atoms in the total transition metal atoms. E_i is the energy of the ith TMC.

The lattice structure of HETMCCs consists of a carbon sublattice and a transition metal sublattice. h represents the carbon sublattice with X position points, and k represents the metal sublattice with Y position points. Then, mixing entropy can be defined using following formula [26]:

$$∆S_{mix}=-R\left\{{\displaystyle \frac{X}{X+Y}}{\sum }_{i=1}^{N_h}{x}_i^{h}ln\left(x_i^h\right)+\right.\left.{\displaystyle \frac{X}{X+Y}}{\sum }_{i=1}^{N_k}{x}_i^{k}ln\left(x_i^k\right)\right\}$$

(7)

where R is the ideal gas constant, N_h and N_k are the types of elements in the sublattice h and k, respectively, and $x_i^h$ and $x_i^k$ are the mole fractions of component i in the sublattice h and k, respectively. The calculated ΔS_mix of five-component HETMCCs per mole is about 0.805 R.

The vacancy defect formation energy E_f can be calculated by using the following equation [35]:

$$E_f{=E}_{\mathit{def}}{-E}_{\mathit{udef}}+N{\mu }_i$$

(8)

where E_def is the total energy of the defective cell, E_udef is the total energy of the perfect cell, N is the number of atoms removed, and ${\mu }_i$ is the chemical potential of an atom removed in the crystal. Since ${\mu }_i$ of an atom in a system is difficult to calculate accurately, it is often replaced by the energy of a single atom in its own pure matter.

The vacancy defect migration energy E_m can be calculated by using the following equation:

$$E_m=E_{mid}-E_0$$

(9)

where E_mid is the total energy of the cell at the highest point of the potential barrier during the migration process, and E₀ is the total energy of the cell at the equilibrium point before the migration.

3. Results and Discussion

3.1. HETMCC Single-Phase Stability Calculations

Models with NaCl-type structures featuring two sets of face-centered cubic (FCC) lattice phase nesting were established for the seven chosen TMCs: TiC, ZrC, MoC, VC, NbC, WC, and TaC. Figure 1 shows the TiC model. The computed values for the lattice constant, volume, and energy are listed in

Table 2. Lattice constant and energy calculation results of TMCs.

System	a (Å)	V (Å3)	Ei (eV)
TiC	4.33	81.26	−7039.42
ZrC	4.71	104.22	−5751.55
MoC	4.36	82.90	−8366.56
VC	4.16	71.80	−8529.70
NbC	4.48	89.90	−6831.44
WC	4.37	83.66	−8353.95
TaC	4.56	95.03	−1171.19

.

A plethora of experiments have shown that HETMCCs maintain the same FCC sublattice structure as TMCs upon the formation of a solid solution, one FCC sublattice is occupied by carbon atoms, while metal atoms are randomly distributed in the other FCC sublattice [13,30,31,32,33,34,35,36,37,38]. In this paper, we constructed 21 distinct HETMCC models, each comprising 80 atoms (40 metal atoms and 40 carbon atoms). A schematic illustration of the crystal cell model is presented in Figure 2.

The elevated stability of the single-phase structure is advantageous for maintaining the integrity of the single-phase solid solution and reducing the likelihood of phase transformation and amorphization of HETMCCs after irradiation. Presently, the criteria for assessing single-phase stability lack consistency. This work aims to explore the phase stability of HETMCCs through an examination of three critical factors: structural distortion, thermodynamic properties, and the distribution of bond lengths among neighboring atoms. Three parameters were selected to analyze the structural distortion of HETMCCs: the difference in the radius of metal atoms (δ_r), the difference in lattice constant (δ_a), and the volume change rate (ω_v). Two parameters were selected to analyze the thermodynamics of HETMCCs: the mixing enthalpy (ΔH_mix) and the mixing of Gibbs free energy (ΔG_mix).

The calculations of δ_r, δ_a, and ω_v for 21 HETMCC models are presented in

Table 3. Calculated values of δ_r, δ_a and ω_v of HETMCCs.

System	VH (Å3)	δr (%)	δa (%)	ωv (%)
HETMCC-1	861.85	6.95	4.13	0.20
HETMCC-2	846.63	7.25	4.07	0.12
HETMCC-3	874.78	6.01	4.33	0.50
HETMCC-4	880.71	6.01	3.09	0.36
HETMCC-5	904.41	5.44	3.06	0.25
HETMCC-6	892.11	6.01	3.24	0.23
HETMCC-7	861.56	6.85	4.11	0.01
HETMCC-8	887.38	6.43	4.26	0.33
HETMCC-9	873.89	6.85	4.31	0.22
HETMCC-10	905.10	5.27	3.00	0.34
HETMCC-11	821.16	4.04	2.41	0.26
HETMCC-12	847.55	4.09	3.17	0.68
HETMCC-13	833.57	4.04	2.97	0.51
HETMCC-14	866.32	2.92	1.96	0.09
HETMCC-15	847.27	3.99	3.16	0.46
HETMCC-16	864.61	7.11	4.06	−0.04
HETMCC-17	891.80	6.88	4.18	0.46
HETMCC-18	881.40	7.11	4.24	0.70
HETMCC-19	907.93	5.99	2.85	0.13
HETMCC-20	891.84	6.78	4.15	0.29
HETMCC-21	853.37	3.31	3.12	0.80

. To facilitate analysis, these data are plotted in Figure 3. It can be seen that the δ_a of 21 HETMCCs is less than 5%, indicating their theoretical capacity to exist as a single phase. The less structural distortion present, the lower the distortion energy, which is beneficial for the stability of the structure. The HETMCC-14 system demonstrates the lowest δ_r and δ_a values, along with the systems examined, along with a minimal ω_v, which suggests it has strong single-phase stability. Conversely, the HETMCC-3 system displays significant structural distortion across three distinct forms, resulting in diminished single-phase stability. In Figure 3, the values of δ_r and δ_a for the 21 HETMCCs are roughly consistent, whereas ω_v does not reveal a discernible trend. The existing literature has yet to definitively establish the reliability of ω_v as a predictor of the single-phase stability of HETMCCs. Therefore, when considering solely δ_r and δ_a, the HETMCC-10, HETMCC-11, and HETMCC-13 systems are identified as exhibiting good single-phase stability, while HETMCC-9, HETMCC-17, and HETMCC-18 systems are identified as exhibiting poor single-phase stability.

The calculated values of ΔH_mix and ΔG_mix for HETMCCs at room temperature (T = 298 K) are shown in

Table 4. Calculated values of ΔH_mix and ΔG_mix of HETMCCs at room temperature (T = 298 K).

System	ΔHmix (kJ/mol)	ΔGmix (kJ/mol)
HETMCC-1	1.78	−0.21
HETMCC-2	−3.49	−5.48
HETMCC-3	1.19	−0.81
HETMCC-4	−4.83	−6.82
HETMCC-5	−2.08	−4.08
HETMCC-6	−5.11	−7.10
HETMCC-7	−0.95	−2.95
HETMCC-8	3.14	1.14
HETMCC-9	−1.03	−3.03
HETMCC-10	−3.69	−5.69
HETMCC-11	−4.51	−6.50
HETMCC-12	−0.30	−2.30
HETMCC-13	−3.31	−5.31
HETMCC-14	−4.08	−6.07
HETMCC-15	−2.19	−4.18
HETMCC-16	−0.47	−2.46
HETMCC-17	2.87	0.88
HETMCC-18	−0.44	−2.44
HETMCC-19	−2.32	−4.32
HETMCC-20	0.72	−1.27
HETMCC-21	0.31	−1.68

. These data are illustrated in Figure 4. The ΔH_mix values across the 21 systems range from −15 kJ/mol to 5 kJ/mol. Theoretically, all systems have the potential to form single-phase solid solutions. Specifically, the ΔH_mix value approaching zero correlates with an increased likelihood of developing a disordered solid solution. Consequently, based solely on the mixing enthalpy, HETMCC-12, HETMCC-16, HETMCC-18, and HETMCC-21 show superior single-phase stability, whereas HETMCC-4, HETMCC-6, HETMCC-8, and HETMCC-17 display inadequate single-phase stability. The ΔG_mix values for HETMCC-8 and HETMCC-17 are both positive, indicating that the two systems are unstable and prone to phase decomposition at room temperature.

With the exception of the HETMCC-2 system, the phase compositions predicted by the computational analyses are in strong agreement with the experimental observations. Specifically, experiments revealed that HETMCC-2 synthesized a multiphase structure [30]. Based on the calculation results, we speculate that this phenomenon may be related to the larger lattice distortion and the mixing enthalpy with greater absolute value of HETMCC-2. A stable single-phase structure can be maintained in the system only when the structural distortion and thermodynamic properties both fulfill the criteria necessary for single-phase formation. Considering the lattice distortion and thermodynamic properties, it is posited that HETMCC-2, HETMCC-8, and HETMCC-17 exhibit limited stability as a single phase, with HETMCC-8 and HETMCC-17 being especially prone to phase decomposition at room temperature. The phase stability of the HETMCC-5, HETMCC-12, HETMCC-19, and HETMCC-21 systems is comparatively superior.

This work explores the connection between microstructure and the single-phase stability of HETMCCs. We have gathered the nearest neighbor atomic pair bond lengths for five systems that show strong single-phase stability (HETMCC-5, HETMCC-12, HETMCC-13, HETMCC-19, HETMCC-21) and five systems that exhibit weak single-phase stability (HETMCC-2, HETMCC-4, HETMCC-6, HETMCC-8, HETMCC-17), as depicted in Figure 5. The bond lengths of Zr-C and Ta-C are generally longer than those of other TMCs. In contrast, The V-C and W-C bond lengths are comparatively short, and the range of bond lengths is wider. It is evident that there is no clear relationship between the bond length distribution in HETMCCs and their single-phase stability.

3.2. Electronic Structure Analysis of HETMCCs

In this part, we investigate the microscopic mechanism that contributes to the single-phase stability of HETMCCs by analyzing their electronic structure. Figure 6 displays the total density of states (DOS) diagram for 21 systems, and

Table 5. The total number of electrons gained and lost by atoms in 21 HETMCCs.

System	Mulliken’s Charge
HETMCC-1	Ti	Zr	Mo	V	Nb	C
	0.79~0.85	0.89~1.02	0.28~0.38	0.62~0.72	0.58~0.68	−0.63~−0.76
HETMCC-2	Ti	Zr	Mo	V	W	C
	0.8~0.96	0.91~1.1	0.3~0.51	0.64~0.77	0.31~0.48	−0.62~−0.73
HETMCC-3	Ti	Zr	Mo	V	Ta	C
	0.83~1.00	1.07~1.24	0.36~0.59	0.67~0.87	−0.05~0.22	−0.59~−0.74
HETMCC-4	Ti	Zr	Mo	Nb	W	C
	0.79~0.96	0.94~1.08	0.35~0.53	0.6~0.82	0.34~0.47	−0.59~−0.74
HETMCC-5	Ti	Zr	Mo	Nb	Ta	C
	0.89~0.98	1.07~1.22	0.41~0.58	0.73~0.91	0~0.15	−0.62~−0.74
HETMCC-6	Ti	Zr	Mo	W	Ta	C
	0.87~1.13	1.14~1.33	0.48~0.75	0.48~0.56	0~0.36	−0.64~−0.75
HETMCC-7	Ti	Zr	V	Nb	W	C
	0.78~0.89	0.87~1.03	0.62 ~0.75	0.52~0.74	0.25~0.44	−0.65~−0.75
HETMCC-8	Ti	Zr	V	Nb	Ta	C
	0.82~1.02	0.92~1.23	0.66~0.76	0.72~0.87	−0.08~0.14	−0.62~−0.75
HETMCC-9	Ti	Zr	V	W	Ta	C
	0.87~1.06	1.06~1.32	0.67~0.87	0.4~0.54	0.03~0.19	−0.63~−0.75
HETMCC-10	Ti	Zr	Nb	W	Ta	C
	0.91~1.02	1.09~1.21	0.75~1.02	0.42~0.53	0~0.22	−0.69~−0.76
HETMCC-11	Ti	Mo	V	Nb	W	C
	0.87~1.02	0.38~0.51	0.69~0.89	0.68~0.85	0.36~0.52	−0.62~−0.71
HETMCC-12	Ti	Mo	V	Nb	Ta	C
	0.87~1.08	0.48~0.63	0.74~0.85	0.82~1.15	−0.09~0.21	−0.63~−0.71
HETMCC-13	Ti	Mo	V	W	Ta	C
	0.97~1.12	0.53~0.74	0.8~0.98	0.5~0.66	−0.04~0.35	−0.63~−0.70
HETMCC-14	Ti	Mo	Nb	W	Ta	C
	0.98~1.15	0.63~0.72	0.98~1.15	0.45~0.62	−0.02~0.19	−0.66~−0.72
HETMCC-15	Ti	V	Nb	W	Ta	C
	0.93~1.09	0.73~0.92	0.83~1.15	0.47~0.64	0~0.2	−0.65~−0.72
HETMCC-16	Zr	Mo	V	Nb	W	C
	0.99~1.11	0.34~0.46	0.7~0.85	0.67~0.83	0.36~0.46	−0.63~−0.74
HETMCC-17	Zr	Mo	V	Nb	Ta	C
	1.01~1.27	0.46~0.6	0.73~0.9	0.77~1.05	−0.02~0.17	−0.61~−0.75
HETMCC-18	Zr	Mo	V	W	Ta	C
	1.16~1.36	0.56~0.7	0.74~0.94	0.45~0.62	−0.04~0.38	−0.59~−0.72
HETMCC-19	Zr	Mo	Nb	W	Ta	C
	1.13~1.28	0.48~0.74	0.86~1.04	0.5~0.64	0.04~0.35	−0.65~−0.75
HETMCC-20	Zr	V	Nb	W	Ta	C
	1.08~1.36	0.69~0.9	0.79~1.06	0.41~0.53	0.03~0.25	−0.64~−0.75
HETMCC-21	Mo	V	Nb	W	Ta	C
	0.58~0.78	0.86~0.97	0.9~1.15	0.55~0.6	−0.01~0.28	−0.62~−0.71

provides a summary of the total number of electrons gained and lost by the atoms. It is clear that the DOS for the 21 systems is quite comparable, and the DOS at the Fermi level is above zero, suggesting that HETMCCs fundamentally exhibit metallic characteristics. A lower DOS at the Fermi level signifies enhanced stability of the system. The diagram shows that the DOS values for the six systems without the V element are the lowest, indicating that the inclusion of the V element adversely affects single-phase stability. The number of electrons obtained by carbon atoms in different systems varies between 0.59 e and 0.75 e, showing no major differences. In contrast, the number of electrons lost by the same metal element differs across various systems. This variation arises from the competitive interactions between different metal atoms. When a metal element with a greater propensity to lose electrons is present, the electron loss from other metal elements decreases accordingly. The electron loss capacity of the seven transition metal elements across all systems follows this order: Zr > Ti > Nb > V > Mo > W > Ta.

To examine how different metal elements affect the electronic structure, the partial density of states (PDOS) and differential charge density for HETMCC-14 and HETMCC-17 were computed, as shown in Figure 7 and Figure 8. It is clear that the d orbitals of the seven metal atoms, along with the s and p orbitals of the nearby carbon atoms, show significant peaks at identical energy levels. This observation suggests a propensity for the metal atoms to engage in bonding interactions with the surrounding carbon atoms. In the differential charge density diagram, the blue regions correspond to an increase in electron density, while the red regions signify a reduction in electron density. The charge density between the Ti, Zr, and V atoms and their neighboring C atoms both increases and decreases, suggesting that the Ti-C, Zr-C, and V-C bonds have both ionic and covalent properties. The charge density around Mo, Nb, W, and Ta atoms decreases, while the charge density around C atoms increases, suggesting that the Mo-C, Nb-C, W-C, and Ta-C bonds are mainly defined by ionic bonding. A significant number of delocalized electrons are spread out around the metal atoms, suggesting that metallic bonds exist within the system. According to Table 5, the number of electrons lost by Ta atoms is very low, with some values even showing negative numbers. However, there are still formant peaks observed between Ta and C atoms, suggesting that they interact with one another. As shown in Figure 8, when Zr, Ti, and V atoms, which possess a strong ability to lose electrons, are nearby Ta atoms, the non-localized electrons are no longer found in the space between the metals, leading to a reduction in the strength of the metallic bonds among them. Consequently, the negative value related to Ta atoms may be a result of the weak interaction with C atoms. In a system with multiple metal elements, the movement of free electrons is limited, making it easier for them to be drawn towards Ta atoms.

3.3. Calculation of Vacancy Defect Properties for HETMCCs

This study calculated the vacancy defect properties of TMCs and HETMCCs using first principles to predict their radiation resistance. TMC models are scaled up to 64 atoms, after which a carbon atom is randomly removed to form a 63-atom TMC model containing a carbon vacancy defect, as shown in Figure 9. The chemical potential of a carbon atom in TMCs corresponds to the energy of a carbon atom in graphite. The formation energies of carbon vacancy defects in TMCs are summarized in

Table 6. Calculation results of carbon vacancy defect formation energy of TMCs.

System	μi (eV)	Ef (eV)
		In This Work	Other Work
ZrC	−154.9	1.3	1.08 [39], 0.97 [35], 0.28 [40]
TiC	−154.9	0.9	0.8 [39], 0.59 [35]
VC	−154.9	−0.7	−0.93 [40]
NbC	−154.9	−0.7	−0.25 [35], −0.42 [40]
TaC	−154.9	−0.9	-
MoC	−154.9	−1.3	-
WC	−154.9	−1.3	-

. The data reveal that the formation energies of carbon vacancy defects in ZrC and TiC are positive, suggesting that such defects are unlikely to form spontaneously. This property may contribute to enhanced radiation resistance. The formation energy of carbon vacancy defects in other TMCs is negative, indicating that the existence of carbon vacancy defect enhances the stability of the system. The formation energy of carbon vacancy defects in MoC and WC is relatively low. When Mo and W elements form an FCC structure with carbon, the non-stoichiometric state is more stable than the stoichiometric state. Therefore, it is advisable to adjust the concentrations of Mo and W elements in HETMCCs to optimize the radiation resistance of materials.

In this study, we calculated the migration energy of a carbon vacancy defect to the closest carbon vacancy defect. We used twelve positional points that are linearly interpolated between the starting and ending states of the vacancy migration to illustrate the migration pathways. The LST/QST tool of the castep module was used to search for the transition state. The model used to calculate the formation energy of vacancy defects is the pre-migration model. The post-migration model is formed by exchanging the positions of the carbon vacancy defects in the pre-migration model with the nearest carbon atoms. An energy distribution diagram for the systems along the vacancy migration pathway is produced, taking the TiC system as a case study, as shown in Figure 10. It is clear that the saddle point is located at the center of the migration path.

Table 7. Carbon vacancy defects migration energy of TMCs.

System	Em (eV)
	In This Work	Other Work
ZrC	4.1	4.39 [39], 4.01 [35], 4.3 [40]
TiC	3.7	3.87 [39], 3.87 [35], 3.7 [40]
TaC	3.6	4.08 [35], 4.0 [40]
NbC	3.5	3.73 [35], 3.60 [40]
VC	3.3	3.0 [40]
MoC	2.8	-
WC	1.9	-

shows the calculated values for the migration energy of carbon vacancy defects. The results reveal that the migration energy for carbon vacancy defects across all systems is positive, indicating that a potential barrier must be overcome for migration to take place. The carbon vacancy defects in ZrC and TiC show the highest migration resistance, followed by those in TaC, NbC, and VC. Conversely, the carbon vacancy defects in MoC and WC show comparatively lower migration resistance.

Among 21 HETMCCs, five systems (HETMCC-4, HETMCC-6, HETMCC-10, HETMCC-11, and HETMCC-14) were selected to investigate the characteristics of carbon vacancy defects. The analysis revealed that ZrC and TiC exhibit the highest energies for the formation and migration of vacancy defects, while WC demonstrates the lowest energies. Therefore, three types of carbon vacancy defects located at various positions were considered. In the ideal crystal model, the carbon atom at the center, the carbon atom that is surrounded by a higher number of W atoms, and the carbon atom that is surrounded by a higher number of Zr (or Ti) atoms were removed. The calculated formation energies for carbon vacancy defects at these different positions are presented in

Table 8. The formation energy of carbon vacancy defects at three different positions of the center, the nearest neighbor atom with more Zr (or Ti) atoms and the nearest neighbor atom with more W atoms in HETMCCs.

The Position of the Vacancy Defect	System	The Type of Atoms Closest to the Vacancy Defect	Ef (eV)
In the center	HETMCC-4	2Ti1Zr1Mo2Nb	−0.2
	HETMCC-6	2Ti2Zr2Ta	0.4
	HETMCC-10	1Ti2Zr1Nb1W1Ta	0.1
	HETMCC-11	1Ti1Mo2V2Nb	−0.4
	HETMCC-14	2Ti1Mo2W1Ta	−0.4
Near the Zr (or Ti) atoms	HETMCC-4	1Ti2Zr2Mo1Nb	0.1
	HETMCC-6	2Zr2Mo1W1Ta	0.2
	HETMCC-10	1Ti3Zr1W1Ta	0.4
	HETMCC-11	4Ti1V1W	0.1
	HETMCC-14	4Ti1Mo1Ta	0.1
Near the W atoms	HETMCC-4	1Ti2Nb3W	−0.3
	HETMCC-6	1Zr1Mo3W1Ta	−0.7
	HETMCC-10	1Zr1Nb3W1Ta	−0.4
	HETMCC-11	2V1Nb3W	−0.7
	HETMCC-14	1Ti1Mo3W1Ta	−0.7

. The data indicate a clear trend: the formation energy of these defects increases with the number of surrounding Zr (or Ti) atoms, whereas it decreases with the number of surrounding W atoms. The process of forming vacancy defects inherently involves the disruption of atomic bonds; thus, the formation energy for carbon vacancy defects in HETMCCs is affected by the types of metal atoms that are adjacent to the carbon vacancy. Zhao [35] also suggests that the inclusion of Zr and Ti elements will enhance the vacancy formation energy. Due to the random distribution of metal atoms in HETMCCs, the arrangement of atoms around carbon vacancies is quite intricate. The formation energy of these vacancies is affected by the collective influence of all the nearest metal atoms. Consequently, the formation energy values for carbon vacancy defects show significant variation, ranging between the formation energies of carbon vacancies found in ZrC and WC. Matheus A. Tunes et al. [41] and Zhao [35] also found a larger distribution of vacancy formation energy in HETMCCs through first-principles calculations and thought that this could effectively prevent vacancy migration.

Table 9. The migration energy of carbon vacancy defects at three different positions of the center, the nearest neighbor atom with more Zr (or Ti) atoms and the nearest neighbor atom with more W atoms in HETMCCs.

The Position of the Vacancy Defect	System	The Type of Atoms Closest to the Vacancy Defect	The Type of Atoms near the Saddle Point	Em (eV)
In the center	HETMCC-4	2Ti1Zr1Mo2Nb	1Ti1Nb	2.9
	HETMCC-6	2Ti2Zr2Ta	1Zr1Ta	4.3
	HETMCC-10	1Ti2Zr1Nb1W1Ta	1Ti1W	2.8
	HETMCC-11	1Ti1Mo2V2Nb	1V1Nb	3.2
	HETMCC-14	2Ti1Mo2W1Ta	1Ti1W	2.6
Near the Zr (or Ti) atoms	HETMCC-4	1Ti2Zr2Mo1Nb	1Zr1Ti	3.9
	HETMCC-6	2Zr2Mo1W1Ta	1Mo1W	2.3
	HETMCC-10	1Ti3Zr1W1Ta	2Zr	4.9
	HETMCC-11	4Ti1V1W	1V1W	2.7
	HETMCC-14	4Ti1Mo1Ta	1Ti1Mo	2.6
Near the W atoms	HETMCC-4	1Ti2Nb3W	1Nb1W	3.0
	HETMCC-6	1Zr1Mo3W1Ta	2W	2.1
	HETMCC-10	1Zr1Nb3W1Ta	1W1Ta	1.8
	HETMCC-10	1Zr1Nb3W1Ta	1Zr1Ta	5.2
	HETMCC-11	2V1Nb3W	1V1W	2.9
	HETMCC-11	2V1Nb3W	1Nb1W	4.3
	HETMCC-14	1Ti1Mo3W1Ta	1Ti1Mo	3.5

shows the calculated migration energies for carbon vacancies in different positions. The random distribution of different metal elements in HETMCCs leads to a relatively uneven potential energy surface. Consequently, there is a wide range of carbon vacancy migration energy values, which allows for the diffusion of carbon vacancies in regions with lower energy barriers. However, the carbon vacancy migration energy of some locations even exceeds the maximum vacancy migration energies of seven TMCs. These locations can obstruct the long-range diffusion of carbon vacancy, significantly prevent the clustering of vacancies, and consequently improve the material’s resistance to radiation. The migration of carbon vacancy defects is affected by the energy barrier created by their nearest neighbor atoms. So, the type of nearest neighbor metal atoms during the migration process is pivotal in determining the migration energy. If the vacancy defect is adjacent to a higher number of W atoms when it begins to migrate and a larger number of neighboring Zr and Ti atoms at the saddle point, this will lead to the vacancy defect moving from a lower point on the potential energy surface, requiring a greater potential barrier to be overcome at the saddle point. As a result, the migration energy of this carbon vacancy defect will increase, possibly surpassing the highest vacancy migration energy found in TMCs. Zhao reached a similar conclusion, stating that the migration energy of HETMCCs is generally higher. In addition, the neighboring Zr atoms hinder vacancy migration [35]. Therefore, selecting elements like Zr and Ti elements, which increase the migration energy, along with W element, which decreases migration energy, could contribute to the development of HETMCCs with enhanced resistance to irradiation.

4. Conclusions

The lattice distortion and thermodynamic properties of 21 types of HETMCCs were calculated by using first-principles methods. Except for HETMCC-2, all systems were speculated to maintain a single phase at room temperature, with (TiMoVNbTa)C₅, (ZrMoNbWTa)C₅, and (MoVNbWTa)C₅ showing superior single-phase stability.

By analyzing the electronic structure of HETMCCs, we observed that the DOS at the Fermi level is positive across all systems, suggesting that they possess metallic properties. Notably, the systems devoid of the V element exhibit the lowest DOS values at the Fermi level, which implies that the inclusion of V enhances the single-phase stability. The electron loss ability of transition metals in all systems follows this order: Zr > Ti > Nb > V > Mo > W > Ta. The bonding in HETMCCs is characterized by a combination of ionic, covalent, and metallic bonds.

The formation energy of carbon vacancy defects in HETMCCs is situated between the highest and lowest values in TMCs. The vacancy formation energy is more closely relative to its composition. An increase in the number of Zr atoms adjacent to the carbon vacancy defect results in an elevation of the formation energy. Conversely, a higher concentration of W atoms surrounding the carbon vacancy defect is associated with a reduction in the formation energy.

The migration energy distribution for carbon vacancy defects in HETMCCs shows considerable variation, which allows carbon vacancy defects to diffuse through lower energy barriers. In contrast, higher energy barriers can hinder long-range diffusion and effectively prevent the clustering of vacancies, thus improving the radiation resistance of the materials. Choosing the HETMCCs that include the Ti and Zr elements, which can raise vacancy migration energy, along with the W element, which can decrease it, will create a rougher potential energy surface. This complicates the migration of carbon vacancies, leading to enhanced resistance to irradiation.