Mechanical Properties, Electronic Structures, and Debye Temperature of Ni_xB_y Compounds Obtained by the First Principles Calculations

Abstract

Mechanical properties, electronic properties, and Debye temperatures of Ni_xB_y (Ni₃B, Ni₂B, Ni₄B₃ and NiB) compounds were obtained by the first principles calculations based on the density functional theory (DFT). The results showed that the formation enthalpy of the Ni_xB_y compounds were stable with negative formation enthalpy. NiB had the largest B, G, and E, and the smallest υ; it also had the highest hardness (10.8 GPa) and Debye temperature (681.8 K). Ni₄B₃ had the strongest anisotropy. It was found that the valence bonds of the Ni_xB_y compounds studied were composed of both metal bond and covalent bond, and the mechanical properties and Debye temperature of the Ni_xB_y compounds increased with the increase of the B atomic ratio.

1. Introduction

Nickel-based alloy powders are widely adopted in surface modification due to their good wettability, superior high-temperature properties, and moderate price [1,2]. Nickel (Ni) and boron (B) can generate different Ni-B binary compounds, which have an important effect on the properties of the coatings [3]. Appropriate B can prevent liquid metal oxidation and reduce the inclusion in the coating [4]. However, excessive B will increase the crack susceptibility of the coatings and should be avoided [5]. Researchers have carried out a lot of experimental studies on the properties of Ni-B binary compounds [6,7,8,9]. Laser cladding is a new type of surface modification technology, in which some powders of special properties are cladded onto the surface of a substrate [10,11]. Compared with other traditional surface modification technologies, it can create coatings with better mechanical properties [12,13]. However, the studies are very limited on the effect of Ni-B binary compounds on the Ni-based laser cladding layer properties. It is very expensive to analyze the effects of different Ni-B contents by experimental method.

In recent years, computational materials science has been developing rapidly with the continuous development of computer technology. It can be used to predict the structure and properties of materials, improve efficiency, and reduce cost in production [14]. The first principles method based on density functional theory has been used to investigate the properties of Ni_xB_y compounds. Researchers have studied some Ni_xB_y compounds in different systems. Shein et al. [15] calculated the lattice constant, magnetic properties, and formation energy of Ni₃B, which indicated that the strong stability of Ni₃B was due to the strong Ni–B hybridization. Zhou et al. [16] researched the stability, electronic, and structural properties of X₂B (X = Cr, Mn, Fe, Co, Ni, Mo, and W) compounds. Zhou et al. [17] investigated the electronic structure and mechanical properties of NiB, which is predicted to be a promising interphase material for future ultrahigh-temperature ceramic fiber reinforced ceramic matrix (UHTCF/UHTC) composites. However, there is no literature that studied the effects of different ratios of Ni to B on the mechanical properties, electronic properties, and Debye temperature of Ni_xB_y compounds.

In our previous research work, Ni-based composite coating was obtained by Ni-based self-fluxing alloy powders [18], but there was no deep and comprehensive research on the properties of Ni_xB_y compounds. In this study, the mechanical properties, electronic properties, and Debye temperature of Ni_xB_y compounds were calculated by using the first-principles. The obtained results could provide guidance for the properties of Ni-based laser cladding layers.

2. Methods

In this paper, mechanical properties, electronic properties, and Debye temperature of Ni_xB_y compounds were calculated based on density functional theory (DFT) by CASTEP [19]. Kohn-Shan equation was calculated by Perthew-Burke-Ernzerhof (PBE) method in the generalized gradient approximation (GGA). The ultrasoft pseudopotential was used to describe the interactions between valence electrons and ionic cores [20]. The valence electrons of Ni and B are 3p⁶3d⁸4s² and 2s²2p¹, respectively. The crystals structures of these Ni_xB_y compounds are shown in Figure 1. The coordinates of Ni₃B are Ni (0.180, 0.063, 0.156), (0.028, 0.250, 0.633) and B (0.618, 0.750, 0.561). The coordinates of Ni₂B are Ni (0.170, 0.330, 0) and B (0.500, 0.500, 0.250). The coordinates of Ni₄B₃ are Ni (0.175, 0.064, 0.346), (0.019, 0.250, 0.869) and B (0.887, 0.250, 0.420). The coordinates of NiB are Ni (0, 0.147, 0.250) and B (0.5, 0.067, 0.750). To optimize equilibrium crystal parameters and the atomic position of these Ni_xB_y compounds, the Broyden–Fletcher–Goldfarb–Shannon (BFGS) algorithm was used. According to the convergence results, the maximum cut-off energy of the plane wave was taken as 500 eV. The Monkhorst–Pack was used to characterize energy integration in the first irreducible Brillouin zone (BZ), and the k point mesh was adopted 16 × 16 × 16. The total energy changes during the optimization process were finally converged to 2 × 10⁻⁶ eV, and the maximum force on the atoms was below 0.001 eV/Å.

The thermodynamic stability of the compounds is related to formation enthalpy. A compound is thermodynamically stable when the formation enthalpy of the compound is negative. The smaller the values of the formation enthalpy, the more stable is the compound. The formation enthalpy can be calculated by following equation [21]:

$${\Delta }_r{H}_m(N{i}_x{B}_y)=\frac{E_{total}(N{i}_{\mathrm{x}}{\mathrm{B}}_y)-x\cdot E_{bulk}(Ni)-y\cdot E_{bulk}(B)}{n}$$

(1)

where ${\Delta }_r{H}_m(N{i}_x{B}_y)$, $E_{total}$, $E_{bulk}$, and n are the formation enthalpy, the total energy, the chemical potential, and the sum of the number of atoms, respectively.

3. Results and Discussion

3.1. Stability

To ensure reliability of the calculation, the structure of crystal was first optimized and then the lattice parameters under 0 K were obtained. Optimized lattice parameters are listed in

Table 1. The optimized parameters of Ni_xB_y compounds.

Compound	Space Group	Crystal Type	Lattice Constants (Å)			Cell Angles (°)			Formation
			a	b	c	α	β	γ	Energy (kJ/mol)
Ni	Fm-3m	cubic	3.530	3.530	3.530	90.0	90.0	90.0
			3.520 a	3.520 a	3.520 a
Ni3B	Pnma	orthorhombic	4.417	6.671	5.199	90.0	90.0	90.0	−32.819
			4.429 b	6.659 b	5.108 b
Ni2B	I4/mcm	tetragonal	4.991	4.286	4.991	90.0	90.0	90.0	−35.848
			4.991 c	4.284 c	4.991 c
Ni4B3	Pnma	orthorhombic	3.005	6.603	12.011	90.0	90.0	90.0	−36.026
			2.981 d	6.568 d	11.954 d
NiB	Cmcm	orthorhombic	2.991	2.982	7.338	90.0	90.0	90.0	−32.547
			2.927 e	2.963 e	7.394 e
B	R-3m	rhombohedral	4.899	4.899	12.551	90.0	90.0	120.0
			4.924 f	4.924 f	12.609 f

^a Ref. Cal. [22]; ^b Ref. Cal. [23]; ^c Ref. Cal. [16]; ^d Ref. Cal. [24]; ^e Ref. Cal. [17]; ^f Ref. Cal. [25].

. The error between our results and predecessors calculation data was less than 3.0%, which proved that the parameters used in this research were reliable. The formation enthalpy of Ni_xB_y compounds calculated using Equation (1) is shown in Table 1. The formation enthalpies of Ni₃B, Ni₂B, Ni₄B₃, and NiB are −32.819 kJ/mol, −35.848 kJ/mol, −36.026 kJ/mol, and −32.547 kJ/mol, respectively. The formation enthalpies are all negative, which indicate that these Ni_xB_y compounds are thermodynamically stable. Ni₄B₃ has more thermodynamic stability than other compounds because its formation enthalpy is the lowest.

3.2. Mechanical Properties

3.2.1. Elastic Constant and Elastic Modulus

Mechanical properties are related to elastic constants C_ij, which are determined by the bond strength between the atoms, and are the scientific basis in designing and developing new materials [26]. The stress-strain method is used to evaluate the elastic constants of these Ni-B compounds. The strain-stress curve is described by the Hooke’s law, which is given by following equation [27]:

$${\sigma }_{\mathrm{ij}}=C_{ijkl}{\epsilon }_{kl}$$

(2)

where σ_ij is the stress tensor, ε_kl is the strain tensor, and C_ijkl is the elastic constant tensor which is a 6×6 matrix (36 elements in general cases). The elastic constants are calculated as follows [26]:

$$\left(\begin{array}{l}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\tau }_1\\ {\tau }_2\\ {\tau }_3\end{array}\right)=\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& C_{14}& C_{15}& C_{16}\\ & C_{22}& C_{23}& C_{24}& C_{25}& C_{26}\\ & & C_{33}& C_{34}& C_{35}& C_{36}\\ & & & C_{44}& C_{45}& C_{46}\\ & & & & C_{55}& C_{56}\\ & & & & & C_{66}\end{array}\right)\left(\begin{array}{l}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\gamma }_1\\ {\gamma }_2\\ {\gamma }_3\end{array}\right)$$

(3)

where σ_i and τ_i are the normal stress and the shear stress, respectively. C_ij is the elastic constant. ε_i and γ_i are the shearing strain and the normal strain, respectively.

Born–Huang lattice dynamical theory can judge the mechanical stability of the compounds and it needs to meet the following criteria [28,29,30]:

Orthorhombic phases (for Ni₃B, Ni₄B₃, and NiB)

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

(4)

Tetragonal phase (for Ni₂B)

$$\begin{array}{l}{C}_{11}>0,C_{22}>0,C_{33}>0,C_{44}>0,C_{55}>0,C_{66}>0,\\ C_{11}+C_{33}-2C_{13}>0,\\ 2(C_{11}+C_{12})+C_{33}+4C_{13}>0\end{array}$$

(5)

Table 2. The calculated elastic constants (in GPa) of Ni_xB_y compounds.

Compounds	Elastic Constants
	C11	C12	C13	C22	C23	C33	C44	C55	C66
Ni3B	348.6	184.1	161.7	340.8	184.4	363.4	132.6	96.7	112.4
Ni2B	405.7	207.6	165.5	408.6	167.1	416.0	102.9	102.8	138.9
Ni4B3	379.3	201.1	210.2	417.4	169.2	401.5	110.8	147.8	140.5
NiB	352.3	170.0	194.7	508.5	178.1	419.9	137.2	111.2	129.4

shows the elastic constants of Ni_xB_y compounds. Examining the data against Equations (2)–(3), it can be found that these Ni_xB_y compounds meet the criteria of mechanical stability. Therefore, Ni₃B, Ni₂B, Ni₄B₃, and NiB are mechanically stable structures.

Elastic properties are mainly determined by elastic modulus. Bulk modulus (B), Young’s modulus (E), shear modulus (G), and Poisson’s ratio (υ) can be calculated by Voigt-Reuss-Hill (VRH) approximation [31]. VRH approximation is mainly based on the symmetry of the crystal, which is the average value of the lower limit value of Voigt and the upper limit value of Reuss. VRH approximation provides methods to estimate the mechanical properties of compounds from elastic constants [32]. The equations are as follows [33,34]:

$$B=\frac{1}{2}(B_V+B_R)$$

(6)

$$G=\frac{1}{2}(G_V+G_R)$$

(7)

$$E=\frac{9B\cdot G}{3B+G}$$

(8)

$$\upsilon =\frac{3B-2G}{2(3B+G)}$$

(9)

where B_V, B_R, G_V, and G_R are the bulk modulus and the shear modulus calculated by Voigt and Reuss, respectively.

The calculation results are shown in

Table 3. Bulk modulus (B), Shear modulus (G), Young’s modulus (E), and Poisson’s ratio (υ) of the Ni_xB_y compounds.

Compounds	BV	BR	B	GV	GR	G	E	υ	B/G
Ni3B	234.8	234.7	234.8	103.2	99.8	101.5	266.2	0.311	2.313
Ni2B	256.7	256.4	256.6	114.9	112.7	113.8	297.5	0.307	2.254
Ni4B3	262.1	262.1	262.1	121.0	116.5	118.8	309.2	0.308	2.276
NiB	262.9	258.4	260.7	124.7	120.9	122.8	318.5	0.296	2.122

. Bulk modulus is a parameter that reflects the resistance of material to volume change, and it can characterize the ability of material to resist deformation. In general, the larger a bulk modulus, the higher is the hardness of the compound [35]. The bulk modulus of Ni₃B, Ni₂B, Ni₄B₃, and NiB are 234.8 GPa, 256.6 GPa, 262.1 GPa, and 260.7 GPa, respectively. With the increase of B atomic ratio, bulk modulus of Ni_xB_y increases. Bulk modulus of NiB is the largest, which implies that the NiB has the highest hardness. Shear modulus and Young’s modulus represent the resistance to elastic deformation (or stiffness) under shear and normal stresses. Shear modulus (122.8 GPa) and Young’s modulus (318.5 GPa) of NiB are also the highest, which indicate that NiB has the greatest resistance to elastic deformation. The ratio of bulk modulus and shear modulus (B/G) can be used to judge whether a compound is brittle or ductile. The compound is ductile when the B/G value is higher than 1.75; otherwise the compound is brittle [36]. The B/G values of these Ni_xB_y compounds are all greater than 1.75, so they are all ductile. Poisson’s ratio can also be used to characterize the brittleness or ductility of a material. A compound exhibits ductility (brittleness) when the Poisson’s ratio is larger than (smaller than) 0.26 [37]. The Poisson’s ratios are also shown in Table 3. Poisson’s ratios of these Ni_xB_y compounds are all larger than 0.26, which indicates that they have good ductility. From the results of the B/G values and the Poisson’s ratios, it can be found that the ductility of the Ni_xB_y compounds gradually decreases as the B atomic ratio increases.

3.2.2. Anisotropy

Anisotropy can help to understand mechanical properties of compounds [34]. Elastic anisotropy of Ni_xB_y compounds can be estimated through the elastic constants of anisotropic index. In this study, the universal anisotropic index (A^U), the percent anisotropy (A_B and A_G) are calculated to characterize the anisotropy. The equations are as follows [27]:

$$A^U=5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6\ge 0$$

(10)

$$A_B=\frac{B_V-B_R}{B_V+B_R}$$

(11)

$$A_G=\frac{G_V-G_R}{G_V+G_R}$$

(12)

where B_V, B_R, G_V, and G_R are the bulk moduli and shear moduli obtained by the Voigt and Reuss method, respectively.

In addition, shear anisotropy needs to be considered because these Ni_xB_y compounds can assume orthorhombic and tetragonal crystal structures. The shear anisotropic factors A₁, A₂, and A₃ are defined as follows [38]:

$$A_1=\frac{4C_{44}}{C_{11}+C_{33}-2C_{13}}$$

(13)

$$A_2=\frac{4C_{55}}{C_{22}+C_{33}-2C_{23}}$$

(14)

$$A_3=\frac{4C_{66}}{C_{11}+C_{22}-2C_{12}}$$

(15)

The results of A^U, A_B, A_G, A₁, A₂, and A₃ are shown in

Table 4. The calculated universal anisotropic index (A^U), bulk anisotropy (A_B and A_G), and shear anisotropic factors (A₁, A₂, A₃) of Ni_xB_y compounds.

Compounds	AU	AB	AG	A1	A2	A3
Ni3B	0.167	0.000	0.016	1.365	1.153	1.400
Ni2B	0.099	0.001	0.010	0.838	0.839	1.392
Ni4B3	0.196	0	0.019	1.230	1.231	1.425
NiB	0.174	0.009	0.015	1.433	0.777	0.994

. It can be seen from Table 4 that the values of A₁, A₂, and A₃ are not equal, which indicates that these Ni_xB_y compounds are shear anisotropic. The A_B values are very small, which indicate that these Ni_xB_y compounds have weak anisotropy in bulk modulus. A_G values of these compounds are greater than the A_B value, which suggests that the differences of shear modulus of Voigt and Reuss have a more significant impact on A^U value than bulk modulus. The larger the value of A^U, the greater is the anisotropy [39]. The A^U values of Ni₃B, Ni₂B, Ni₄B₃, and NiB are 0.167, 0.099, 0.196, and 0.174, respectively. Therefore, the order of the anisotropy for these Ni_xB_y compound is Ni₄B₃ > NiB > Ni₃B > Ni₂B. With the increase of the B atomic ratio, the anisotropy is strengthened.

To characterize the Young’s modulus anisotropy of these Ni_xB_y compounds more intuitively, the spherical orientation is used in different directions. The directional dependence of Young’s modulus for different type crystals is given by following equations [40,41]:

Orthorhombic crystal [40]

$$\begin{array}{l}1/E=l_1^4{S}_{11}+l_2^4{S}_{22}+l_3^4{S}_{33}+2l_1^2{l}_2^2{S}_{12}+2l_1^2{l}_3^2{S}_{13}+2l_2^2{l}_3^2{S}_{23}\\ +l_2^2{l}_3^2{S}_{44}+l_1^2{l}_3^2{S}_{55}+l_1^2{l}_2^2{S}_{66}\end{array}$$

(16)

Tetragonal crystal [41]

$$1/E=(l_1^4+l_2^4)S_{11}+l_3^4{S}_{33}+l_1^2{l}_2^2(2S_{12}+S_{66})+l_3^2(1-l_3^2)(2S_{13}+S_{44})+2l_1^{}{l}_2(l_1^2-l_2^2)S_{16}$$

(17)

where S_ij are the elastic compliance constants, l₁, l₂ and l₃ are the directional cosines in spherical coordinates with respect to θ and φ (l₁ = sin θcos φ, l₂ = sin θsin φ, l₃ = cos φ). The more irregular the sphere, the greater is the anisotropy [42]. It can be seen from Figure 2 that the shape of Ni₄B₃ is the most irregular and Ni₄B₃ is the most regular, which indicate that the anisotropy of Ni₄B₃ is the greatest. This is consistent with above results of the anisotropy index.

The projections on the X-Y plane, X-Z plane, and Y-Z plane show more details about the anisotropy of Young’s modulus as shown in Figure 3. It can be seen that the Young’s modulus of these Ni_xB_y compounds have great difference in different axles. The planar contour of Ni₄B₃ is the most irregular and Ni₂B is the most regular, which indicate that the anisotropy of Ni₄B₃ is the greatest and the anisotropy of Ni₂B is the least. This is also consistent with the result of the anisotropy index presented above.

3.2.3. Hardness

Hardness is an important mechanical index that reflects the resistance to localized plastic deformation of a material. In general, wear resistance of material is positively correlated with hardness, which depends on shear modulus and bulk modulus [43]. The hardness (H_V) of the Ni_xB_y compounds was calculated using the following equations [44]:

$$H_V=2{(k^{2}G)}^{0.585}-3$$

(18)

$$\mathrm{k}=\frac{G}{B}$$

(19)

where G and B are shear modulus and bulk modulus, respectively.

The hardness of the Ni_xB_y compounds is shown in Figure 4. The hardness values of Ni₃B, Ni₂B, Ni₄B_3, and NiB are 8.2 GPa, 9.3 GPa, 10.0 GPa, and 10.8 GPa, respectively. NiB has the highest hardness, which is consistent with bulk modulus discussed above. With the increase of B atomic ratio, the hardness increases, which also indicates the ductility of the Ni_xB_y compound decreases.

3.3. Electronic Structures

Electronic structure and characteristics of chemical bonds can be characterized by density of state (DOS). Figure 5 shows total density of states (TDOS) and partial electronic density of states (PDOS) of these Ni_xB_y compounds. Two main peaks are observed in the TDOS of Ni_xB_y compounds. The electronic structure are mainly determined by B-p band and Ni-d band. In addition, the fermi levels are also determined by Ni-d band and a small amount of B-p band, which indicates that these Ni_xB_y compounds exhibit p-d hybridization and the bond between the Ni and B atom is in the form of a covalent bond. Ni_xB_y compounds also have metal bonds because the TDOS values are greater than zero at the fermi level. Therefore, it can be concluded that the valence bonds of these Ni_xB_y compounds are composed of both metal bond and covalent bond.

The population analysis can provide more insightful information on chemical bonds of these Ni_xB_y compounds, the results are listed in

Table 5. Milliken population analysis results of Ni_xB_y, the unit of bond length is Å.

Species	Atoms	s	p	d	Total Electrons	Charge
Ni3B	B	1.12	2.46		3.58	−0.58
	Ni	0.35	0.73	8.72	9.81	0.19
Ni2B	B	1.07	2.45		3.52	−0.52
	Ni	0.30	0.72	8.72	9.74	0.26
Ni4B3	B	1.00	2.50		3.53	−0.53
	Ni	0.24	0.63	8.73	9.60	0.40
NiB	B	0.99	2.55		3.54	−0.54
	Ni	0.18	0.55	8.73	9.46	0.54
Species	Bond	$\overline{\mathbf{L}}\left(\mathbf{A}\mathbf{B}\right)$(Å)			$\overline{{\mathbf{n}}_{\mathbf{A}\mathbf{B}}}\left(\mathit{e}\mathit{l}\mathit{e}\mathit{c}\mathit{t}\mathit{r}\mathit{o}\mathit{n}\mathit{s}\right)$
Ni3B	B-Ni	2.11			0.28
	Ni-Ni	2.56			−0.01
Ni2B	B-B	2.14			0.65
	B-Ni	2.14			0.15
	Ni-Ni	2.59			−0.11
Ni4B3	B-B	2.33			0.76
	B-Ni	2.30			0.17
	Ni-Ni	2.93			−0.13
NiB	B-B	1.78			1.45
	B-Ni	2.15			0.28
	Ni-Ni	2.61			−0.14

. The Mulliken method is used to calculate the overlap population and the charge. Average bond strength ($\overline{L}(\mathrm{AB})$) and mean overlap population (${\overline{n}}_{AB}$) can be calculated by the following equations [16]:

$$\overline{L}(\mathrm{AB})=\frac{{{\displaystyle {\sum }_{i}L}}_i{N}_i}{{\displaystyle {\sum }_i{N}_i}}$$

(20)

$${\overline{n}}_{AB}=\frac{{\displaystyle {\sum }_i{n}_i^{AB}}{N}_i}{{\displaystyle {\sum }_i{N}_i}}$$

(21)

where N_i is the total number of i bond and the L_i is the bond length of i type.

As can be seen from Table 5, for all of the Ni_xB_y compounds, B atoms and Ni atoms carry the negative charge and the positive charge, respectively. The charge value of Ni atom varies from 0.19 (Ni₃B) to 0.54 (NiB). With the increase of B atomic ratio, the charge values of Ni in the Ni_xB_y compounds increase. There are two electron transfer paths in the Ni_xB_y compound: one is inside the Ni atom and the B atom, and the other is between the Ni atom and the B atom. The one refers to p–d hybridization covalent bond between B and Ni, and the other one is induced by the metal or weak covalent bonding among Ni atoms. In the former case, the electrons are transferred from Ni atom to B atom. The values of overlap population of Ni-Ni bond are negative, which indicate that there is an anti-bond state or strong electrostatic repulsion between Ni-Ni bond. The strengths of B-B and B-Ni covalent bonds are increased with the increase of B content. In summary, the valence bonds of these Ni_xB_y compounds are composed of the metal bond and the covalent bond.

3.4. Debye Temperature

During the laser cladding process, compound undergoes rapid heating and cooling, so studying the thermodynamic properties of the material is very important. Debye temperature and sound velocity are important parameters for thermodynamic properties, which are related to the chemical bonding characteristics and thermal properties of Ni_xB_y compounds [45]. Debye temperature and sound velocity can be calculated by following equations [46,47,48]:

$${\Theta }_D=\frac{\mathrm{h}}{k_B}[\frac{3\mathrm{n}}{4\pi }(\frac{N_A\rho }{M})]{}^{1/3}{v}_{\mathrm{m}}$$

(22)

$$v_m={[\frac{1}{3}(\frac{2}{v_s{}^3}+\frac{1}{v_1{}^3})]}^{-1/3}$$

(23)

$$v_1={[(B+\frac{4}{3}G)\frac{1}{\rho }]}^{1/2}$$

(24)

$$v_s={(G/\rho )}^{1/2}$$

(25)

where h, k_B, n, ρ, N_A, and M are, respectively, the Planck constant, Boltzmann constant, the number of atoms, the density, the Avogadro constant, and the molar mass of the compound. $v_m$, $v_1$, and $v_s$ are respectively the average speed of sound, the velocity of the longitudinal sound and transverse velocity.

The values of Debye temperatures and sound velocities of Ni₃B, Ni₂B, Ni₄B₃, and NiB are calculated and listed in

Table 6. Theoretically calculated thermal properties of the Ni_xB_y compounds, including v_s, v₁, v_m, and Θ_D.

Compounds	vs/m∙s−1	v1/m∙s−1	vm/m∙s−1	ΘD/K
Ni3B	3505.6	6693.7	3921.2	553.4
Ni2B	3663.8	6939.4	4095.9	600.7
Ni4B3	3958.1	7447.8	4422.9	645.1
NiB	4052.4	7532.9	4524.4	681.8

. The order of Debye temperatures for Ni_xB_y compounds are NiB > Ni₄B₃ > Ni₂B > Ni₃B. The Debye temperature of NiB is the highest (681.8 K), so the thermodynamic stability of NiB is superior to other Ni-B compounds. With the increase of the B atomic ratio, the Debye temperatures of these Ni_xB_y compounds increase. The longitudinal velocity and transverse velocity are correlated to the bulk modulus, shear modulus, and the density of the compound. The compounds with large bulk modulus and the low density will have large sound velocity. Hence, NiB has the highest sound velocity among these Ni_xB_y compounds because it has the highest shear modulus and bulk modulus. This conclusion can also be used to explain the sound velocity of other compounds.

4. Conclusions

In this study, the results showed that mechanical properties, electronic properties, and Debye temperature of Ni_xB_y compounds with different atomic ratios are different. This can provide theoretical guidance for material design in the laser cladding layer and help to account for changes in performance under different Ni and B ratios. The conclusions are drawn as follows:

(1)

The calculated lattice parameters are consistent with the predecessor calculation data. The formation energy of all Ni_xB_y compounds is negative, which indicates that all Ni_xB_y compounds have stable structures.

(2)

Ni_xB_y compounds have mechanical stability. NiB has the largest bulk modulus, shear modulus and Young’s modulus and the smallest Poisson’s ratio, which imply that the hardness of NiB is higher than other Ni_xB_y compounds. Ni_xB_y compounds exhibit anisotropic characteristics, and Ni₄B₃ had the greatest anisotropy. The mechanical properties of the Ni_xB_y compounds increase with the increase of the B atomic ratio.

(3)

Ni_xB_y compounds exhibit p-d hybridization and they exhibit the metal bond and the covalent bond.

(4)

NiB has largest Debye temperature (7681.8 K), which indicates that NiB has the highest thermodynamic stability. Debye temperature of the Ni_xB_y compounds increase with the increase of the B atomic ratio.