基于第一性原理的<bold>Mo<sub>2</sub>CoB<sub>2</sub> </bold>三元过渡金属硼化物掺杂效应研究

张宇瀚; 金娜; 刘颖; Zhang Yuhan; Jin Na; Liu Ying

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Abstract

The first-principles calculation was used to investigate the influence of doping fourth-period transition metal elements on the structural, mechanical, and thermal properties of Mo₂CoB₂. Through the calculation of cohesive energy and formation enthalpy as well as the calculation comparison between the obtained results and Born-Huang criterion, all doped compounds are thermodynamically and mechanically stable. Point defect theory was employed to determine the occupation sites and occupation preference of doped elements in the Mo₂CoB₂ crystal cell. Results show that Sc and Ti exhibit strong preference for Mo sites, and V has a weak preference for Mo sites. Additionally, Cr, Mn, Fe, Cu, and Zn have a weak preference for Co sites, and Ni has a strong preference for Co sites. Debye temperatures were obtained by the contrast calculation. The results reveal that except Mo₇TiCo₄B₈, Mo₇VCo₄B₈, and Mo₇CrCo₄B₈, the doped models all have lower Debye temperatures than the undoped model, suggesting that except Ti, V, and Cr elements, the addition of transition metal elements of large quantity into the Mo₂CoB₂ hard phase should be avoided. Furthermore, except that of the Cr-doped model, the hardness of the doped models is lower than that of the undoped model, and the models with doped elements at preferential sites normally exhibit higher hardness than those at non-preferred sites do. This research provides theoretical basis for the development of Mo₂CoB₂-Co cermet with improved properties.

Keywords

cermet; first-principles; mechanical properties; Mo₂CoB₂-Co

Cermet is a non-homogeneous composite material composed of metal or alloy with various ceramic phases^[

1–2], which not only has excellent metal properties, such as ductility and plasticity, but also presents good ceramic properties, such as high strength, great hardness, high wear resistance, and fine temperature resistance^{[Reference 3–4}3–4]. The cermet applications in tool materials and structural materials become more and more important, promoting the industrial development and improving the productivity.

WC-Co is one of the commonly used cermets due to its high hardness, toughness, and corrosion resistance^[

5], and it has been widely used as tool material, such as milling cutter, boring tool, turning tool, planer, and drill, to cut non-ferrous metals, cast iron, plastics, synthetic fibers, graphite, glass, stone, stainless steel, and high manganese steel. However, the raw material tungsten is a global strategic non-renewable resource^{[Reference 6

Baidu Scholar}6], and the fracture toughness of WC-Co is relatively low, resulting in the difficulty of W resource access and the easy occurrence of crack initiation under complex stress conditions^{[Reference 7

Baidu Scholar}7]. Therefore, tungsten-free cermet materials with high hardness and high toughness are urgently required.

Three generations of tungsten-free cermet materials have been developed^[

8]. The first-generation cermet is produced from Ni-doped TiC. The second-generation tungsten-free cermet is prepared by adding Mo element into the Ni-doped TiC phase, which improves the wettability and toughness. The third-generation tungsten-free cermet is prepared by adding nitrides into the hard phase to form a composite phase. The binder phase performance is enhanced by adding Co and other elements. Although the Ti(C,N)-based cermet exhibits good high-temperature hardness and excellent wear resistance, its lower toughness restricts the further application. Therefore, the tungsten-free cermet materials with excellent properties should be further developed.

Currently, the ternary transition metal boride with excellent abrasion resistance, good corrosion resistance, and relatively good mechanical properties at high temperatures attracts much attention in the research of tungsten-free cermet^[

9], which mainly involves the Mo₂FeB₂-Fe, Mo₂NiB₂-Ni, and WCoB-Co^{[Reference 10

Baidu Scholar}10]. Among them, the ternary boride Mo₂FeB₂-Fe^{[Reference 11

Baidu Scholar}11] has high wear resistance, Mo₂NiB₂-Ni^{[Reference 12

Baidu Scholar}12] has remarkable corrosion resis-tance, and WCoB-Co^{[Reference 13

Baidu Scholar}13] has exceedingly high-temperature resistance. In addition to these ternary borides, Mo₂CoB₂-Co is also worthy of attention: it is tungsten-free and its binder phase Co has higher corrosion resistance than Fe and Ni do. Although the ternary borides show great performance in various applications, the brittleness is still a problem^{[Reference 14

Baidu Scholar}14]. There-fore, the method of doping alloying elements is proposed to improve the cermet microstructure and to enhance the mechanical properties. Mo₂CoB₂-Co cermet is mainly composed of the hard phase Mo₂CoB₂ which directly affects the wear resistance and thermal properties of cermet. Therefore, the effects of transition metal elements doping on the mechanical and thermal properties of Mo₂CoB₂ hard phase should be thoroughly investigated.

The first-principles calculations have been widely employed to study the mechanical and thermal properties of materials. Zhang et al^[

15] used the first-principles method to investigate the elastic properties, thermodynamic properties, and electronic structure of AgAuPd medium entropy alloy under different pressures, providing theoretical basis for the experiments. Lin^{[Reference 16

Baidu Scholar}16] and Wang^{[Reference 17

Baidu Scholar}17] et al used the first-principles study to analyze the effects of Cr, Ni, and Mn elements on the crystal structure and mechanical properties of Mo₂FeB₂. It is found that Cr, Ni, and Mn tend to occupy the Fe sites in Mo₂FeB₂. Cr can significantly increase the shear modulus, Young's modulus, and Vickers hardness of Mo₂FeB₂, whereas Ni reduces the values of the abovementioned properties. However, Ni can improve the ductility of Mo₂FeB₂. Zhang et al^{[Reference 18–20}18–20] used the first-principles calculations to study the effects of doped Cr, Mn, V, and other transition metal elements on the crystal structure, mechanical properties, and electronic structure of WCoB. It is demonstrated that Cr, Mn, and V can reduce the shear modulus and bulk modulus of WCoB to a certain extent.

Therefore, in this research, the first-principles calculation was used to calculate the cohesive energy and enthalpy of Mo₂CoB₂ supercells doped with the fourth-period transition metal elements. The atomic sites of various doping elements in the Mo₂CoB₂ supercells were analyzed, and the effects of the doping elements on crystal structure, mechanical properties, and thermal properties of Mo₂CoB₂ were investigated.

1 First-Principles Calculation

Before constructing the transition metal element-doped Mo₂CoB₂ model, two issues should be considered: the doping concentration and doping site. Through the comprehensive consideration, a supercell model (20 atoms) of 1×2×1 of Mo₂CoB₂ crystal was selected in this research. Besides, the doped models were constructed with the transition metal (Sc, Ti, V, Cr, Mn, Fe, Ni, Cu, Zn) atoms occupying both Mo and Co sites, and the specific doping sites need to be determined based on the results of first-principles calculations. The schematic diagrams of doped models are shown in Fig.1a–1b, and the doped atoms occupying the Mo and Co sites are represented by X_Mo-site and X_Co-site, respectively. In addition, the anti-site models with a single Mo atom occupying Co site and a single Co atom occupying the Mo site were constructed, as well as vacancy models with vacancies at the Mo and Co sites, as shown in Fig.1c–1f.

Fig.1 Schematic diagrams of point defect models of Mo₂CoB₂ supercell: (a) Mo₇XCo₄B₈ with transition metal element X occupying Mo site; (b) Mo₈Co₃XB₈ with transition metal element X occupying Co site; (c) Mo₉Co₃B₈ with Mo element occupying Co site; (d) Mo₇Co₅B₈ with Co element occupying Mo site; (e) Mo₇Co₄B₈ with vacancy occupying Mo site; (f) Mo₈Co₃B₈ with vacancy occupying Co site (dashed circle indicates the site changes)

All first-principles calculations were conducted based on the density functional theory^[

21–22] with the Vienna Ab Initio Simulation Package^{[Reference 23–24}23–24]. The interactions between the atomic core and the valence electron were described by the Projector Augmented Wave potentials^{[Reference 25–26}25–26]. The exchange and cor-relation functions were evaluated through the Perdew-Burke-Ernzerh function with generalized gradient approximation^{[Reference 27

Baidu Scholar}27]. The cut-off energy of the plane wave was 500 eV. Monkhorst-Pack type k-points grid was used and the number of k-points was 6×6×6. The self-consistent field converged below 1×10^-5 eV/atom. The maximum force tolerance was below 1×10^-1 eV/nm.

2 Results and Discussion

2.1 Structural stability

In order to investigate the positions of doped elements, several criteria methods based on atomic radius^[

18], enthalpy of formation, and cohesive energy^{[Reference 28

Baidu Scholar}28] are proposed.

Table 1 presents the atomic radii of Mo and the fourth-period elements. According to the atomic radius-based occu-pancy criterion, the closer the distance between the transition metal element and Mo (or Co) element, the higher the occupancy probability of the Mo (or Co) site by the transition metal element^[

18]. The eigenvalue R_X-1/2(R_Mo+R_Co) is intro-duced to determine the critical distance between the transition metal element X and Mo/Co, where R_X is the atomic radius of the transition metal element X, R_Mo is the atomic radius of the Mo element, and R_Co is the atomic radius of Co element. If R_X-1/2(R_Mo+R_Co)>0, the atomic radius of X element is closer to Mo; otherwise, the atomic radius of X element is closer to that of Co. As shown in Table 1, Sc, Ti, V, and Zn elements tend to occupy the Mo sites, whereas Cr, Mn, Fe, Ni, and Cu elements tend to occupy the Co sites.

Table 1 Atomic radii, calculated eigenvalue, and preferential sites of the fourth-period elements and Mo element

Element	Atomic radius/nm	$R_{x} - 1 / 2 (R_{M o} + R_{C o})$	Preferential site
Mo	0.139	7	Mo
Sc	0.162	30	Mo
Ti	0.147	15	Mo
V	0.134	2	Mo
Cr	0.128	-4	Co
Mn	0.127	-5	Co
Fe	0.126	-6	Co
Co	0.125	-7	Co
Ni	0.124	-8	Co
Cu	0.128	-4	Co
Zn	0.134	2	Mo

The cohesive energy can also be used to determine the positions of doped elements. Lin et al^[

16] studied the doping positions of Cr, Ni, and Mn elements in Mo₂FeB₂ by cohesive energy. It is found that the cohesive energy of Cr, Ni, and Mn elements occupying the Fe sites is lower than that occupying the Mo sites, indicating that Cr, Ni, and Mn elements tend to occupy the Fe sites. Keränen et al^{[Reference 29

Baidu Scholar}29] proved that the Cr element does not only occupy the Fe sites in Mo₂FeB₂. Some Cr atoms also occupy the Mo sites. It is clear that the cohesive energy-based method is not accurate. Compared to the cohesive energy, the enthalpy of formation as the determination criterion of dopant elements is more reliable. Table 2 shows the lattice parameters, cohesive energies, and enthalpies of formation of undoped and doped transition metal models. In this research, the cohesive energy and formation enthalpy can be calculated by Eq.(1) and Eq.(2)^{[Reference 30

Baidu Scholar}30], respectively:

E_{c o h} (M o_{a} C o_{b} X_{c} B_{d}) = \frac{E_{t o t} (M o_{a} C o_{b} X_{c} B_{d}) - a E (M o) - b E (C o) - c E (X) - d E (B)}{a + b + c + d}

(1)

ΔH_r(Mo_aCo_bX_cB_d)=E_coh(Mo_aCo_bX_cB_d)-aE_coh(Mo)-bE_coh(Co)-cE_coh(X)-dE_coh(B)

(2)

Table 2 Lattice parameters, cohesive energies, and enthalpies of formation of Mo_aCo_bX_cB_d compounds with undoped, doped, and vacancy models

Compound	Lattice parameter/nm			Cell volume/×10^-3 nm^-3	Cohesive energy, E_coh/eV·atom^-1	Enthalpy of formation, ∆H_r/eV·molecule^-1
Compound	a	b	c	Cell volume/×10^-3 nm^-3	Cohesive energy, E_coh/eV·atom^-1	Enthalpy of formation, ∆H_r/eV·molecule^-1
Mo₈Co₄B₈	0.7113	0.9117	0.3163	205.12	-7.123	-9.137
Mo₇Co₄B₈	0.7073	0.9030	0.3141	200.63	-6.944	-6.190
Mo₈Co₃B₈	0.7043	0.9067	0.3146	200.90	-7.133	-7.536
Mo₇Co₅B₈	0.7064	0.9052	0.3150	201.43	-6.991	-8.758
Mo₉Co₃B₈	0.7214	0.9315	0.3156	212.11	-7.210	-8.624
Mo₇ScCo₄B₈	0.7161	0.9157	0.3178	208.40	-6.987	-9.865
Mo₈Co₃ScB₈	0.7136	0.9381	0.3210	214.88	-7.000	-7.867
Mo₇TiCo₄B₈	0.7110	0.9112	0.3165	205.06	-7.088	-10.724
Mo₈Co₃TiB₈	0.7119	0.9308	0.3186	211.15	-7.134	-9.397
Mo₇VCo₄B₈	0.7079	0.9084	0.3155	202.92	-7.067	-10.194
Mo₈Co₃VB₈	0.7133	0.9252	0.3161	208.59	-7.148	-9.570
Mo₇CrCo₄B₈	0.7067	0.9066	0.3149	201.75	-7.035	-9.398
Mo₈Co₃CrB₈	0.7146	0.9217	0.3142	206.92	-7.146	-9.371
Mo₇MnCo₄B₈	0.7064	0.9053	0.3147	201.27	-6.934	-9.098
Mo₈Co₃MnB₈	0.7128	0.9170	0.3154	206.17	-7.072	-9.611
Mo₇FeCo₄B₈	0.7056	0.9045	0.3153	201.22	-6.974	-8.792
Mo₈Co₃FeB₈	0.7116	0.9125	0.3162	205.33	-7.130	-9.654
Mo₇NiCo₄B₈	0.7075	0.9053	0.3150	201.77	-6.965	-8.011
Mo₈Co₃NiB₈	0.7112	0.9120	0.3170	205.59	-7.131	-9.762
Mo₇CuCo₄B₈	0.7093	0.9059	0.3161	203.10	-6.858	-7.928
Mo₈Co₃CuB₈	0.7116	0.9148	0.3177	206.81	-7.014	-8.795
Mo₇ZnCo₄B₈	0.7115	0.9063	0.3178	204.89	-6.732	-7.811
Mo₈Co₃ZnB₈	0.7109	0.9184	0.3195	208.58	-6.872	-8.353

where E_coh(Mo_aCo_bX_cB_d) and ΔH_r(Mo_aCo_bX_cB_d) are the cohesive energy and enthalpy of formation of Mo_aCo_bX_cB_d compounds (X indicates the doped transition metal element); E_tot(Mo_aCo_bX_cB_d) is the total energy of Mo_aCo_bX_cB_d molecule; E(Mo), E(Co), E(X), and E(B) are the energies of free Mo, Co, X, and B atoms, respectively; E_coh(Mo), E_coh(Co), E_coh(X), and E_coh(B) are the cohesive energies of Mo, Co, X, and B per atom, respectively. According to Table 2, all cohesive energy and enthalpy of formation are negative, indicating that all the compounds are thermodynamically stable. It can be seen that when Sc, Ti, V, and Cr elements occupy the Mo sites, their enthalpies of formation are smaller than those when they occupy the Co sites. When Mn, Fe, Ni, Cu, and Zn elements occupy the Co sites, their enthalpies of formation are smaller than those when they occupy the Mo sites. Hence, Sc, Ti, V, and Cr elements tend to occupy the Mo sites, whereas the Mn, Fe, Ni, Cu, and Zn elements tend to occupy the Co sites.

The abovementioned criteria only consider the single factor and do not take into account the vacancy movement and atomic migration in the material. Currently, the widely accepted theory is the point defect theory^[

31–32], whose predictions show good consistency with the experiments^{[Reference 33–35}33–35]. Therefore, the site occupancy by transition metal atoms in Mo₂CoB₂ crystal is further discussed based on this theory.

In this research, the point defects in Mo₂CoB₂ crystal can be divided into six types: (1) the transition metal atom X occu-pying the Mo site, as denoted by X_Mo; (2) the transition metal atom X occupying the Co site, as denoted by X_Co; (3) Mo atom enrichment in Mo₂CoB₂ crystal with Mo atom occupying the Co site, as denoted by Mo_Co; (4) Co atom enrichment in Mo₂CoB₂ crystal with Co atom occupying the Mo site, as denoted by Co_Mo; (5) a vacancy occupying the Mo site, as denoted by Va_Mo; (6) a vacancy occupying the Co site, as denoted by Va_Co. Thus, the enthalpy of formation of Mo₈Co₃XB₈ and Mo₇XCo₄B₈ compounds containing point defects can be represented as a linear function of defect atom concentrations, as follows:

Δ H = Δ H (M o_{8} C o_{4} B_{8}) + \sum_{d} H_{d} X_{d}

(3)

where X_d is the defect concentration of type d={X_Mo, X_Co, Mo_Co, Co_Mo, Va_Mo, Va_Co}, $Δ H (M o_{8} C o_{4} B_{8})$ is the enthalpy of formation for $M o_{8} C o_{4} B_{8}$ compound, and H_d is the enthalpy of formation for d type defect. Therefore, the enthalpy of formation for the Mo_8-_xCo_4-_yBX_x₊_y (x and y represent the numbers of missing Mo and Co atoms relative to the defect-free Mo₂CoB₂ crystal, respectively, and x+y is the total number of defect atoms) compound containing point defects can be calculated by Eq.(4), as follows:

Δ H (M o_{8 - x} C o_{4 - y} B_{8} X {}_{x + y}{) = E (M o_{8 - x} C o_{4 - y} B_{8} X {}_{x + y}) - (8 - x) E (M o) - (4 - y) E (C o) - 8 E (B) - (x + y) E (X)}

(4)

where E(Mo_8-_xCo_4-_yB₈X_x₊_y) is the total energy of Mo_8-_xCo_4-_yB₈X_x₊_y; E(Mo), E(Co), E(X), and E(B) are the ground state energies of the pure Mo, Co, X, and B elements, respectively.

Based on Eq.(3–4), the enthalpy of formation for models with point defects can be described:

H_{d} = \frac{Δ H (M o {}_{8 - x}C o_{4 - y} B_{8} X_{x + y}) - Δ H (M o_{8} C o_{4} B_{8})}{X_{d}}

(5)

The 1×2×1 Mo₂CoB₂ supercell was used. When the defect types are doping defects and anti-site defects, the X_d value is 20; when the vacancy defects occur, the X_d value is 19. The energy required to move the transition metal atom X from the Mo site to the Co site in the Mo₂CoB₂ supercell is denoted as $E_{X}^{M o \to C o}$ , and its expression is as follows:

\begin{array}{l} E_{X}^{M o \to C o} = E (M o_{8} C o_{3} X B_{8}) - E (M o_{7} X C o_{4} B_{8}) \\ + E (M o_{7} C o_{5} B_{8}) - E (M o_{8} C o_{4} B_{8}) \end{array}

(6)

If $E_{X}^{M o \to C o} < 0$ , E(Mo₇XCo₄B₈)+E(Mo₈Co₄B₈)>E(Mo₈Co₃XB₈) +E(Mo₇Co₅B₈). Thus, the energy of E(X_Co)+E(Co_Mo) is less than the energy of E(X_Mo)+E(Co_Co). This result indicates that when X moves from the Mo site to the Co site, the total energy of the system is reduced, and therefore the transition metal atom X prefers to occupy the Co site. If $E_{X}^{M o \to C o} < E (V a_{M o}) +$ E(Va_Co), the transition metal atom X prefers to occupy the Mo site, because the energy required for the X atom movement from Mo site to Co site is less than the anti-site exchange formation energy E^ant, where E^ant=E(Va_Mo)+E(Va_Co). In both cases, the atom shows a strong site preference. However, between 0 and E^ant, the weak site preference region also exists. Therefore, $E_{X}^{M o \to C o}$ can be normalized by E^ant, as follows:

{\tilde{E}}_{X}^{M o \to C o} = \frac{E_{X}^{M o \to C o}}{E^{a n t}}

(7)

Finally, the occupancy of transition metal atoms can be divided into four situations: (1) ${\tilde{E}}_{X}^{M o \to C o} < 0$ , strong Co site preference; (2) ${\tilde{E}}_{X}^{M o \to C o} > 1$ , strong Mo site preference; (3) $0 < {\tilde{E}}_{X}^{M o \to C o} < 0.5$ , weak Co site preference; (4) $0.5 < {\tilde{E}}_{X}^{M o \to C o}$ $< 1$ , weak Mo site preference.

Taking the data in Table 2 into the abovementioned formulas, ${\tilde{E}}_{X}^{M o \to C o}$ values can be calculated, and the results are shown in Fig.2. It can be seen that Sc and Ti exhibit a strong Mo site preference, whereas V exhibits a weak Mo site preference. Cr, Mn, Fe, Cu, and Zn show weak Co site preference, and Ni shows a strong Co site preference. The larger the ${\tilde{E}}_{X}^{M o \to C o}$ value, the stronger the preference of transition metal element occupying the Mo site. The smaller the ${\tilde{E}}_{X}^{M o \to C o}$ value, the stronger the preference of transition metal element occupying the Co sit. Additionally, with increasing the atomic number, the ${\tilde{E}}_{X}^{M o \to C o}$ value of the fourth-period transition metal element X is decreased firstly and then increased. The lowest ${\tilde{E}}_{X}^{M o \to C o}$ value appears when X is Ni element. Therefore, the Ni element has the strongest preference to occupy the Co site, and the Sc element has the strongest preference to occupy the Mo site.

Fig.2 Site preference distributions of doped transition element X of the fourth-period in Mo₂CoB₂ supercells

It can be seen that the atomic radius-based criterion cannot quantitatively present the strength of atomic site preference and cannot comprehensively consider the influencing factors, such as doping element concentration and temperature. It can only roughly predict the specific atomic site preference. In most cases, the results obtained by the atomic radius-based criterion are consistent with the predictions obtained by point defect theory, except for the site preference of Zn element. The cohesive energy-based criterion is theoretically inconsistent with experiment conditions, so it cannot be used as the criterion. Moreover, the enthalpy of formation criterion also predicts different results for Cr element.

2.2 Mechanical properties

Before the mechanical property analysis, the traditional mechanical stability conditions should be considered. The elastic constants C_ij of Mo_aCo_bX_cB_d compounds with undoped, doped, and vacancy models are shown in Table 3. For orthorhombic crystals with nine independent elastic stiffness constants, according to Born-Huang's theory^[

36], the criterion can be expressed as follows:

\{\begin{array}{l} C_{11} > 0, C_{44} > 0, C_{55} > 0, C_{66} > 0, C_{11} C_{22} > C_{12}^{2} \\ C_{11} C_{22} C_{33} + 2 C_{12} C_{13} C_{23} - C_{11} C_{23}^{2} - C_{22} C_{13}^{2} - C_{33} C_{12}^{2} \end{array}

(8)

Table 3 Elastic constants C_ij of Mo_aCo_bX_cB_d compounds with undoped, doped, and vacancy models

Compound	C₁₁	C₂₂	C₃₃	C₄₄	C₅₅	C₆₆	C₁₂	C₁₃	C₂₃
Mo₈Co₄B₈	551	560	532	226	226	205	155	226	214
Mo₇ScCo₄B₈	507	485	526	212	186	206	205	150	197
Mo₈Co₃ScB₈	476	498	480	209	199	26	130	200	196
Mo₇TiCo₄B₈	542	505	545	221	198	214	217	151	207
Mo₈Co₃TiB₈	498	515	521	86	218	207	220	205	154
Mo₇VCo₄B₈	542	552	512	227	218	200	149	220	216
Mo₈Co₃VB₈	530	549	500	227	208	137	158	229	218
Mo₇CrCo₄B₈	545	521	552	226	191	219	229	152	211
Mo₈Co₃CrB₈	542	561	522	232	215	178	160	230	222
Mo₇MnCo₄B₈	528	537	499	223	216	185	151	226	209
Mo₈Co₃MnB₈	553	545	485	231	218	192	152	223	207
Mo₇FeCo₄B₈	530	535	523	214	210	172	152	240	215
Mo₈Co₃FeB₈	557	527	556	233	201	219	228	156	220
Mo₇NiCo₄B₈	504	513	497	211	201	168	156	224	208
Mo₈Co₃NiB₈	539	551	528	216	227	194	156	230	206
Mo₇CuCo₄B₈	505	503	492	209	193	161	157	217	202
Mo₈Co₃CuB₈	523	538	516	208	227	181	151	230	205
Mo₇ZnCo₄B₈	486	506	494	205	197	153	161	212	207
Mo₈Co₃ZnB₈	512	523	509	204	223	167	151	225	195

By substituting the elastic constants C_ij in Table 3 into Eq.(8), it can be seen that all the compounds satisfy the criterion, indicating that these crystals are all mechanically stable. For orthorhombic crystals, the related parameters can be calculated by Eq.(9–12)^[

37–39], as follows:

B_{V} = \frac{1}{9} (C_{11} + C_{22} + C_{33}) + \frac{2}{9} (C_{12} + C_{23} + C_{13})

(9)

\frac{1}{B_{R}} = (S_{11} + S_{22} + S_{33}) + 2 (S_{12} + S_{23} + S_{13})

(10)

\begin{array}{l} G_{V} = \frac{1}{15} (C_{11} + C_{22} + C_{33}) - \frac{1}{15} (C_{12} + C_{23} + C_{13}) \\ + \frac{1}{5} (C_{44} + C_{55} + C_{66}) \end{array}

(11)

\begin{array}{l} \frac{1}{G_{R}} = \frac{4}{15} (S_{11} + S_{22} + S_{33}) - \frac{4}{15} (S_{12} + S_{23} + S_{13}) \\ + \frac{1}{5} (S_{44} + S_{55} + S_{66}) \end{array}

(12)

where S_ij is the elastic compliant coefficient which can be converted from the corresponding C_ij matrix; B is bulk modulus; G is shear modulus; the subscripts V and R indicate the Voigt constraint and Reuss constraint, respectively. The bulk modulus B and shear modulus G can be calculated by Voigt-Reuss-Hill approximations^[

40], namely B_VRH and G_VRH, respectively:

B_{V R H} = \frac{1}{2} (B_{V} + B_{R})

(13)

G_{V R H} = \frac{1}{2} (G_{V} + G_{R})

(14)

According to the calculated elastic modulus, Young's modulus E and Poisson's ratio υ can be calculated from the bulk modulus and shear modulus^[

41]. Moreover, the Vickers hardness H_V can be deduced by the empirical formula^{[Reference 42

Baidu Scholar}42], as follows:

E = \frac{9 B_{V R H} G_{V R H}}{3 B_{V R H} + G_{V R H}}

(15)

υ = \frac{3 B_{V R H} - 2 G_{V R H}}{2 (3 B_{V R H} + G_{V R H})}

(16)

H_{V} = 2 (K^{2} G_{V R H})^{0.585} - 3

(17)

K = \frac{G_{V R H}}{B_{V R H}}

(18)

where K is the Pugh's modulus ratio. Table 4 shows the results of bulk modulus B, shear modulus G, Young's modulus E, Poisson's ratio υ, B/G value, and hardness H_V of Mo_aCo_bX_cB_d compounds with undoped, doped, and vacancy models. The B/G value is commonly used as a criterion to evaluate the brittleness and ductility of material: the larger the B/G value, the better the material ductility^[

43]. Usually, the materials with the B/G value greater than 1.75 are considered as ductile materials.

Table 4 Bulk modulus B, shear modulus G, Young's modulus E, Poisson's ratio υ, B/G value, and Vickers hardness H_V of Mo_aCo_bX_cB_d compounds with undoped, doped, and vacancy models

Compound	B_V	B_R	B_VRH	G_V	G_R	G_VRH	E/GPa	υ	B/G	H_V/GPa
Mo₈Co₄B₈	314.8	314.3	314.6	201.2	196.5	198.9	492.7	0.239	1.58	22.9
Mo₇ScCo₄B₈	291.3	291.2	291.2	185.4	181.2	183.3	454.6	0.240	1.59	21.5
Mo₈Co₃ScB₈	278.4	277.3	277.8	148.8	81.4	115.1	303.4	0.318	2.41	9.1
Mo₇TiCo₄B₈	304.8	304.6	304.7	194.6	189.9	192.3	476.5	0.239	1.59	22.3
Mo₈Co₃TiB₈	299.0	298.5	298.7	166.1	147.6	156.8	400.4	0.277	1.91	15.4
Mo₇VCo₄B₈	308.1	307.8	308.0	197.1	191.6	194.3	481.7	0.239	1.59	22.5
Mo₈Co₃VB₈	309.9	309.7	309.8	179.3	171.0	175.2	442.1	0.262	1.77	18.3
Mo₇CrCo₄B₈	311.5	311.0	311.2	195.5	190.3	192.9	479.6	0.243	1.61	21.9
Mo₈Co₃CrB₈	316.6	316.3	316.5	192.4	186.8	189.6	474.1	0.25	1.67	20.7
Mo₇MnCo₄B₈	303.8	303.4	303.6	190.0	184.0	187.0	465.4	0.245	1.62	21.3
Mo₈Co₃MnB₈	305.2	305.2	305.2	194.9	188.1	191.5	475.1	0.241	1.59	22.1
Mo₇FeCo₄B₈	311.3	310.1	310.7	184.5	179.3	181.9	456.6	0.255	1.71	19.6
Mo₈Co₃FeB₈	316.3	315.9	316.1	199.6	194.3	197.0	489.3	0.242	1.61	22.3
Mo₇NiCo₄B₈	298.9	298.3	298.6	177.8	172.9	175.4	440.0	0.254	1.70	19.2
Mo₈Co₃NiB₈	311.5	311.0	311.2	195.8	191.2	193.5	480.8	0.243	1.61	22.0
Mo₇CuCo₄B₈	294.5	294.0	294.3	174.5	170.4	172.5	432.9	0.255	1.71	18.9
Mo₈Co₃CuB₈	305.4	304.7	305.1	189.4	184.2	186.8	465.4	0.246	1.63	21.1
Mo₇ZnCo₄B₈	293.9	293.2	293.6	169.5	165.8	167.6	422.5	0.26	1.75	17.9
Mo₈Co₃ZnB₈	298.5	297.8	298.1	183.5	178.5	181.0	451.7	0.248	1.65	20.4

Fig.3 shows B/G values of doped transition metal element X at different sites in the Mo₂CoB₂ supercell. It can be seen that when the transition metal elements occupy the Mo site, the B/G value of the crystal is generally increased with increasing the atomic number, indicating the increasing ductility. When the transition metal elements occupy the Co site, the B/G value of the crystal is decreased with the increasing the atomic number, presenting the decline in ductility. Moreover, the B/G values of all doped models are higher than those of the undoped model, suggesting that doping transition metal elements can improve the ductility of Mo₂CoB₂. It is worth noting that when the Sc, Ti, and V elements occupy the Co site, their B/G values are greater than 1.75. However, it is already found that Sc, Ti, and V tend to occupy the Mo site. Therefore, these three models cannot be used as reference.

Fig.3 B/G values of doped transition metal element X at different sites in Mo₂CoB₂ supercell (dashed line represents the B/G value of undoped model)

2.3 Debye temperature

Debye temperature Θ_D is related to many physical pro-perties of crystal, such as lattice vibration, thermal conduc-tivity, thermal expansion coefficient, and heat capacity. With previously calculated bulk modulus B and shear modulus G, the longitudinal sound velocity v_l, the transverse sound veloc-ity v_t, the average sound velocity v_m, and Debye temperature Θ_D of the doped models can be calculated by Eq.(19–22)^[

44–45], as follows:

v_{l} = \sqrt[]{(B_{V R H} + \frac{4}{3} G_{V R H}) \frac{1}{ρ}}

(19)

v_{t} = \sqrt[]{(G_{V R H}) \frac{1}{ρ}}

(20)

v_{m} = {[\frac{1}{3} (\frac{1}{v_{l}^{3}} + \frac{1}{v_{t}^{3}})]}^{- \frac{1}{3}}

(21)

Θ_{D} = \frac{h}{k_{B}} {[\frac{3 n}{4 π} (\frac{n_{A} ρ}{M})]}^{\frac{1}{3}} v_{m}

(22)

where h is Planck's constant, k_B is Boltzmann's constant, n is the number of atoms per formula unit, n_A is Avogadro's number, ρ is the density, and M is the molecular weight. The calculated results of these parameters are shown in Table 5, and the Debye temperatures of doped transition metal element X at different sites in the Mo₂CoB₂ supercell are shown in Fig.4.

Table 5 Longitudinal phonon velocity v_l, transverse phonon velocity v_t, average phonon velocity v_m, and Debye temperature Θ_D of Mo_aCo_bX_cB_d compounds with undoped, doped, and vacancy models

Compound	v_l/m·s^-1	v_t/m·s^-1	v_m/m·s^-1	Θ_D/K
Mo₈Co₄B₈	8106	4748	5264	721
Mo₇ScCo₄B₈	8045	4706	5219	711
Mo₈Co₃ScB₈	7203	3721	4166	562
Mo₇TiCo₄B₈	8155	4774	5293	725
Mo₈Co₃TiB₈	7737	4300	4789	650
Mo₇VCo₄B₈	8144	4767	5286	727
Mo₈Co₃VB₈	7943	4510	5014	683
Mo₇CrCo₄B₈	8126	4734	5251	724
Mo₈Co₃CrB₈	8093	4671	5186	709
Mo₇MnCo₄B₈	7994	4649	5158	711
Mo₈Co₃MnB₈	8006	4679	5189	710
Mo₇FeCo₄B₈	7992	4582	5090	702
Mo₈Co₃FeB₈	8115	4734	5251	719
Mo₇NiCo₄B₈	7840	4500	4998	689
Mo₈Co₃NiB₈	8042	4689	5201	712
Mo₇CuCo₄B₈	7787	4467	4962	682
Mo₈Co₃CuB₈	7941	4611	5116	699
Mo₇ZnCo₄B₈	7761	4419	4912	673
Mo₈Co₃ZnB₈	7863	4554	5055	689

Fig.4 Debye temperatures of doped transition metal element X at different sites in Mo₂CoB₂ supercell (dashed line represents Debye temperature value of undoped model)

According to Fig.4, the Debye temperature of each doped model is firstly increased and then decreased with increasing the atomic number. When the transition metal element occupies the Mo site, the maximum Debye temperature is obtained for the V element; when the transition metal element occupies the Co site, the maximum Debye temperature is obtained for the Fe element. It can be seen that all transition metal elements tend to have a higher Debye temperature when they occupy their preferred sites. For example, the Debye temperatures of Sc (strong), Ti (strong), and V (weak) elements occupying the Mo sites are much higher than those occupying the Co sites. In contrast, the Debye temperatures of Fe (weak), Ni (strong), Cu (weak), and Zn (weak) elements occupying the Co sites are much higher than those occupying the Mo sites. Moreover, the Debye temperatures of Mo₇TiCo₄B₈, Mo₇VCo₄B₈, and Mo₇CrCo₄B₈ are higher than those of the undoped model, and the Debye temperatures of other doped models show lower Debye temperatures.

2.4 Hardness

Hardness is a key factor influencing the wear resistance of cermet materials. The hardness of doped Mo₂CoB₂ supercells can be predicted by Eq.(17). Although the obtained hardness results have significant difference compared with those obtained by the bond order model, it does not affect the analysis of hardness variation in Mo₂CoB₂ crystals after doping with transition metal elements. The hardness prediction results are shown in Table 4, and Fig.5 shows the hardness of doped transition metal element X at different sites in the Mo₂CoB₂ supercell.

Fig.5 Hardness of doped transition metal element X at different sites in Mo₂CoB₂ supercell (dashed line represents the hardness value of undoped model)

The hardness values obtained from Eq.(17) only consider the intrinsic hardness of the doped models and do not consider the effects of lattice distortion caused by doping on defects, such as dislocations in the material. As shown in Fig.5, without considering the influence of dislocations in the material, the intrinsic hardness of all doped models is lower than that of the undoped model. Moreover, the hardness variation in doped models is basically consistent with the variation of Debye temperature. With increasing the atomic number, the hardness of doped elements occupying Mo sites is firstly increased and then decreased. The highest hardness value is obtained when the doped element is V element. Similarly, when the transition metal element occupies the Co site, the maximum hardness is achieved for the Fe element. All transition metal elements tend to have higher hardness when they occupy their preferred atomic sites. For example, the hardness of Sc (strong), Ti (strong), and V (weak)

elements occupying Mo sites is much higher than that occupying the Co sites. In contrast, the hardness of Mn (weak), Fe (weak), Ni (strong), Cu (weak), and Zn (weak) elements occupying the Co sites is much higher than that occupying the Mo sites. This may be related to the stability of the crystal structure and the strength of chemical bonds. When the transition metal elements occupy their preferred sites, the stability is better, resulting in stronger chemical bonds.

However, it should be noticed that when Cr occupies its preferential Co site, the hardness is lower than that occupying the Mo site. There are two reasons for this phenomenon. Firstly, the ${\tilde{E}}_{X}^{M o \to C o}$ value of Cr calculated by Eq.(7) is 0.481, which is close to 0.5. $E_{C r}^{M o \to C o}$ and $E_{C r}^{C o \to M o}$ values are basically the same, suggesting that the energy for Cr movement from Mo sites to Co sites is extremely close to that from Co sites to Mo sites. In this case, Cr element can occupy both sites. Hu et al^[

46] studied the effects of Cr element doping into Mo₂FeB₂-Fe ternary borides, and the results showed that Cr element can appear at both binder phase Fe and hard phase Mo₂FeB₂. Moreover, Yang et al^{[Reference 8

Baidu Scholar}8] found that Cr can also appear at binder phase Co and hard phase MoCoB in the MoCoB-Co ternary borides. In this case, it can be deduced that Cr tends to occupy both sites. Additionally, when Cr element is doped into the MoCoB-Co ternary borides, the volume fraction of binder phase is decreased with increasing the Cr addition, and there is no big difference for the hard phase. It can be noticed that Cr element tends to form a third phase with Co element, which presents low hardness and low fracture toughness. Hence, it can be inferred that Mo₂CoB₂ shows low hardness when Cr occupies the Co site.

3 Conclusions

1) Through the cohesive energy and formation enthalpy, all doped MoCoBX compounds are thermodynamically stable. Moreover, all doped models are mechanically stable with the consideration of Born-Huang's criterion.

2) The point defect theory is the optimal criterion to determine the occupation of dopant elements in the Mo₂CoB₂ crystal cell. Sc and Ti elements have strong Mo site preference, and V element has a weak Mo site preference. Cr, Mn, Fe, Cu, and Zn elements have a weak Co site preference, and Ni element has a strong Co site preference.

3) The Debye temperature of Mo₇TiCo₄B₈, Mo₇VCo₄B₈, and Mo₇CrCo₄B₈ models is higher than that of the undoped model, whereas that of other doped models are lower than that of the undoped model. Therefore, in the preparation of Mo₂CoB₂-based cermet, except Ti, V, and Cr elements, the addition of transition metal elements of large quantity into the Mo₂CoB₂ hard phase should be avoided.

4) With the consideration of crystal structure, when the transition metal elements occupy their preferential sites, they exhibit higher hardness, except for the Cr element. Moreover, the hardness of all doped models is lower than that of the undoped model.

References

Tinklepaugh J R, Crandall W B. Cermet[M]. Shanghai: Shanghai Scientific & Technical Publisher, 1964: 8 [Baidu Scholar]

Liu Futian, Huang Weiling, Li Wenhu et al. Materials Science and Technology[J], 2005, 13(5): 355 (in Chinese) [Baidu Scholar]

Shepeleva L, Medres B, Kaplan W D et al. Surface and Coatings Technology[J]. 2000, 125(1–3): 45 [Baidu Scholar]

Sailer B, Pjetursson E, Zwahlen M et al. Clinical Oral Implants Research[J], 2007, 18(3): 86 [Baidu Scholar]

Zhang Liehua, Fang Zhigang Zak, Xu Lin et al. Heat Treatment of Metals[J], 2016, 41(2): 84 (in Chinese) [Baidu Scholar]

Cao Fei, Yang Huipeng, Wang Wei et al. Conservation and Utilization of Mineral Resources[J], 2018, 38(2): 145 (in Chinese) [Baidu Scholar]

Tong Chuangchuang. Effects of Cr, Ni, and Mn on Mechanical Properties of Mo2FeB2 Based on First Principles[D]. Chengdu: Southwest Petroleum University, 2017 (in Chinese) [Baidu Scholar]

Yang Guoqiang. Design and Preparation of High Hardness Ternary Boride Basic Ceramics[D]. Beijing: University of Science and Technology Beijing, 2020 [Baidu Scholar]

Takagi K I. Journal of Solid State Chemistry[J], 2006, 179(9): 2809 [Baidu Scholar]

Sáez A, Arenas F, Vidal E. International Journal of Refractory Metals and Hard Materials[J], 2003, 21(1–2): 13 [Baidu Scholar]

Andreas L J, Helmuth K, Peter R et al. Journal of the Japan Institute of Metals[J], 2000, 64(2): 154 [Baidu Scholar]

Hirata H, Iwanaga K, Yamazaki Y et al. Journal of the Japan Society of Powder and Powder Metallurgy[J], 2006, 53(5): 447 [Baidu Scholar]

Pan Yingjun, Xu Ming, Hu Bing et al. Journal of Wuhan University of Science and Technology[J], 2011, 34(2): 96 (in Chinese) [Baidu Scholar]

Jian Y X, Huang Z F, Liu X T et al. Results in Physics[J], 2019, 15: 102 698 [Baidu Scholar]

Zhang Shunmeng, Xiong Kai, Jin Chengchen et al. Rare Metal Materials and Engineering[J], 2022, 51(12): 4533 (in Chinese) [Baidu Scholar]

Lin Y H, Tong C C, Pan Y et al. Modern Physics Letters B[J], 2017, 31(12): 1 750 138 [Baidu Scholar]

Wang S L, Pan Y, Lin Y H et al. Computational Materials Science[J], 2018, 146: 18 [Baidu Scholar]

Zhang T, Yin H Q, Zhang C et al. Modern Physics Letters B[J], 2018, 32(21): 1 850 240 [Baidu Scholar]

Zhang T, Yin H Q, Zhang C et al. Materials[J], 2019, 12(6): 967 [Baidu Scholar]

Zhang T, Yin H Q, Zhang C et al. Chinese Physics B[J], 2018, [Baidu Scholar]

27(10): 107 101 [Baidu Scholar]

Hohenberg P, Kohn W. Physics Review[J], 1964, 136(3B): B864 [Baidu Scholar]

Kohn W, Sham L J. Physics Review[J], 1965, 140(4A): A1133 [Baidu Scholar]

Kresse G, Furthmüller J. Physical Review B[J], 1996, 54(16): 11 169 [Baidu Scholar]

Kresse G, Furthmüller J. Computational Materials Science[J], 1996, 6(1): 15 [Baidu Scholar]

BlÖchl P E. Physical Review B[J], 1994, 50(24): 17 953 [Baidu Scholar]

Kresse G, Joubert D. Physical Review B[J], 1999, 59(3): 1758 [Baidu Scholar]

Perdew J P, Burke K, Ernzerhof M. Physical Review Letters[J], 1996, 77(18–28): 3865 [Baidu Scholar]

Chen M, Wang C Y. Scripta Materialia[J], 2009, 60(8): 659 [Baidu Scholar]

Keränen J, Stenberg T, Mantyla T et al. Surface and Coatings Technology[J], 1996, 82(1–2): 29 [Baidu Scholar]

Li Jian, Zhang Ming. Journal of Alloys and Compounds[J], 2013, 556: 214 [Baidu Scholar]

Ruban A V, Skriver H L. Solid State Communications[J], 1996, 99(11): 813 [Baidu Scholar]

Ruban A V, Skriver H L. Physical Review B[J], 1997, 55(2): 856 [Baidu Scholar]

Jiang C, Gleeson B. Scripta Materialia[J], 2006, 55(5): 433 [Baidu Scholar]

Kim D E, Shang S L, Liu Z K. Intermetallics[J], 2010, 18(6): 1163 [Baidu Scholar]

Xu W W, Wang Y, Wang C P et al. Scripta Materialia[J], 2015, 100: 5 [Baidu Scholar]

Mouhat F, Coudert F X. Physical Review B[J], 2014, 90(22): 224 104 [Baidu Scholar]

Jian Y X, Huang Z F, Xing J D et al. Materials Chemistry and Physics[J], 2019, 221: 311 [Baidu Scholar]

Ravindran P, Fast L, Korzhavyi P A et al. Journal of Applied Physics[J], 1998, 84(9): 4891 [Baidu Scholar]

Liang Y F, Shangetal S L. Intermetallics[J], 2011, 19(10): 1374 [Baidu Scholar]

Hill R. Proceedings of the Physical Society, Section A[J], 1952, 65(5): 349 [Baidu Scholar]

Pan Y. Journal of Alloys and Compounds[J], 2019, 779: 813 [Baidu Scholar]

Chen X Q, Niu H Y, Li D Z et al. Intermetallics[J], 2011, 19(9): 1275 [Baidu Scholar]

Pugh S F. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science[J], 1954, 45(367): 823 [Baidu Scholar]

Haque E, Hossain M A. Journal of Alloys and Compounds[J], 2018, 730: 279 [Baidu Scholar]

Lv Z L, Cui H L, Huang H M et al. Journal of Alloys and Compounds[J], 2017, 692: 440 [Baidu Scholar]

Hu Bing, Pan Yingjun, Wang Qingfang et al. Heat Treatment of Metals[J], 2011, 36(6): 29 (in Chinese) [Baidu Scholar]

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Doping Effects of Mo₂CoB₂ Ternary Transition Metal Boride: a First-Principles Study PDF

Abstract

Keywords

1 First-Principles Calculation