Mechanical Properties , Electronic Structures , and Debye Temperature of Ni x B y Compounds Obtained by the First Principles Calculations

Mechanical properties, electronic properties, and Debye temperatures of NixBy (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 NixBy 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 NixBy compounds studied were composed of both metal bond and covalent bond, and the mechanical properties and Debye temperature of the NixBy compounds increased with the increase of the B atomic ratio.


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 x B y compounds.Researchers have studied some Ni x B y compounds in different systems.Shein et al. [15] calculated the lattice constant, magnetic properties, and formation energy of Ni 3 B, which indicated that the strong stability of Ni 3 B was due to the strong Ni-B hybridization.Zhou et al. [16] researched the stability, electronic, and structural properties of X 2 B (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 x B 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 x B y compounds.In this study, the mechanical properties, electronic properties, and Debye temperature of Ni x B y compounds were calculated by using the first-principles.The obtained results could provide guidance for the properties of Ni-based laser cladding layers.

Methods
In this paper, mechanical properties, electronic properties, and Debye temperature of Ni x B 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 3p 6 3d 8 4s 2 and 2s 2 2p 1 , respectively.The crystals structures of these Ni

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 error between our results and predecessors calculation data was less than 3.0%, which proved that 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]: where ∆ r H m (Ni 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.

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 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 x B y compounds calculated using Equation ( 1) is shown in

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]: 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]: 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 Ni 3 B, Ni 4 B 3 , and NiB) Tetragonal phase (for Ni 2 B) Table 2 shows the elastic constants of Ni x B y compounds.Examining the data against Equations ( 2)-( 3), it can be found that these Ni x B y compounds meet the criteria of mechanical stability.Therefore, Ni 3 B, Ni 2 B, Ni 4 B 3 , 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]: 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 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 Ni 3 B, Ni 2 B, Ni 4 B 3 , and NiB are 234.8GPa, 256.6 GPa, 262.1 GPa, and 260.7 GPa, respectively.With the increase of B atomic ratio, bulk modulus of Ni x B 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 (122.8GPa) 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 x B 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 x B 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 x B y compounds gradually decreases as the B atomic ratio increases.

Anisotropy
Anisotropy can help to understand mechanical properties of compounds [34].Elastic anisotropy of Ni x B 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]: 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 x B y compounds can assume orthorhombic and tetragonal crystal structures.The shear anisotropic factors A 1 , A 2 , and A 3 are defined as follows [38]: The results of A U , A B , A G , A 1 , A 2 , and A 3 are shown in Table 4.It can be seen from Table 4 that the values of A 1 , A 2 , and A 3 are not equal, which indicates that these Ni x B y compounds are shear anisotropic.The A B values are very small, which indicate that these Ni x B 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]  To characterize the Young's modulus anisotropy of these Ni x B 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] 1/E = l where S ij are the elastic compliance constants, l 1 , l 2 and l 3 are the directional cosines in spherical coordinates with respect to θ and ϕ (l 1 = sin θcos ϕ, l 2 = sin θsin ϕ, l 3 = cos ϕ).The more irregular the sphere, the greater is the anisotropy [42].It can be seen from Figure 2 that the shape of Ni 4 B 3 is the most irregular and Ni 4 B 3 is the most regular, which indicate that the anisotropy of Ni 4 B 3 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 x B y compounds have great difference in different axles.The planar contour of Ni 4 B 3 is the most irregular and Ni 2 B is the most regular, which indicate that the anisotropy of Ni 4 B 3 is the greatest and the anisotropy of Ni 2 B is the least.This is also consistent with the result of the anisotropy index presented above.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 NixBy 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.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 NixBy 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.

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 x B y compounds was calculated using the following equations [44]: where G and B are shear modulus and bulk modulus, respectively.The hardness of the Ni x B y compounds is shown in

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 (HV) of the NixBy compounds was calculated using the following equations [44]: . ( ) where G and B are shear modulus and bulk modulus, respectively.The hardness of the NixBy compounds is shown in Figure 4.The hardness values of Ni3B, Ni2B, Ni4B3, 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 NixBy compound decreases.

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 NixBy compounds.Two main peaks are observed in the TDOS of NixBy 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 NixBy compounds exhibit p-d hybridization and the bond between the Ni and B atom is in the form of a covalent bond.NixBy 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 NixBy compounds are composed of both metal bond and covalent bond.

Electronic Structures
Electronic structure and characteristics of chemical bonds can be characterized by density state (DOS).Figure 5 shows total density of states (TDOS) and partial electronic density of states (PDOS) of these Ni x B y compounds.Two main peaks are observed in the TDOS of Ni x B 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 x B y compounds exhibit p-d hybridization and the bond between the Ni and B atom is in the form of a covalent bond.Ni x B 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 x B 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 x B y compounds, the results are listed in Table 5.The Mulliken method is used to calculate the overlap population and the charge.Average bond strength (L(AB)) and mean overlap population (n AB ) can be calculated by the following equations [16]: where N i is the total number of i bond and the L i is the bond length of i type.The population analysis can provide more insightful information on chemical bonds of these NixBy compounds, the results are listed in Table 5.The Mulliken method is used to calculate the overlap population and the charge.Average bond strength ( (AB) L ) and mean overlap population ( AB n ) can be calculated by the following equations [16]: where Ni is the total number of i bond and the Li is the bond length of i type.
As can be seen from Table 5, for all of the NixBy compounds, B atoms and Ni atoms carry the negative charge and the positive charge, respectively.The charge value of Ni atom varies from 0.

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 NixBy compounds [45].Debye temperature and sound velocity can be calculated by following equations [46][47][48]: where h, kB, n, ρ, NA, 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.m v , 1 v , and s v 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.The order of Debye temperatures for NixBy compounds are NiB > Ni4B3 > Ni2B > Ni3B.The Debye temperature of NiB is the highest (681.8K), 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 NixBy 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 NixBy compounds because it has the highest shear

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 x B y compounds [45].Debye temperature and sound velocity can be calculated by following equations [46][47][48]: 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 Ni 3 B, Ni 2 B, Ni 4 B 3 , and NiB are calculated and listed in Table 6.The order of Debye temperatures for Ni x B y compounds are NiB > Ni 4 B 3 > Ni 2 B > Ni 3 B. The Debye temperature of NiB is the highest (681.8K), 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 x B 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 x B 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.

Conclusions
In this study, the results showed that mechanical properties, electronic properties, and Debye temperature of Ni x B 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 x B y compounds is negative, which indicates that all Ni x B y compounds have stable structures.

Figure 1 .
Figure 1.The crystal structure of Ni x B y compounds (red balls represent B atoms, blue balls represent Ni atoms) (a) Ni 3 B; (b) Ni 2 B; (c) Ni 4 B 3 ; (d) NiB.
. The A U values of Ni 3 B, Ni 2 B, Ni 4 B 3 , and NiB are 0.167, 0.099, 0.196, and 0.174, respectively.Therefore, the order of the anisotropy for these Ni x B y compound is Ni 4 B 3 > NiB > Ni 3 B > Ni 2 B. With the increase of the B atomic ratio, the anisotropy is strengthened.

Figure 4 .
Figure 4.The hardness of Ni x B y compounds.
19 (Ni3B) to 0.54 (NiB).With the increase of B atomic ratio, the charge values of Ni in the NixBy compounds increase.There are two electron transfer paths in the NixBy 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

Figure 5 .
Figure 5.Total density of states (TDOS) and partial density of states (PDOS) for Ni x B y compounds.Dashed lines represent the Fermi level: (a) Ni 3 B; (b) Ni 2 B; (c) Ni 4 B 3 ; (d) NiB.As can be seen from Table5, for all of the Ni x B 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 (Ni 3 B) to 0.54 (NiB).With the increase of B atomic ratio, the charge values of Ni in the Ni x B y compounds increase.There are two electron transfer paths in the Ni x B 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 x B y compounds are composed of the metal bond and the covalent bond.

( 2 )
Ni x B 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 x B y Ni x B y compounds exhibit anisotropic characteristics, and Ni 4 B 3 had the greatest anisotropy.The mechanical properties of the Ni x B y compounds increase with the increase of the B atomic ratio.(3) Ni x B y compounds exhibit p-d hybridization and they exhibit the metal bond and the covalent bond.(4) NiB has largest Debye temperature (7681.8K), which indicates that NiB has the highest thermodynamic stability.Debye temperature of the Ni x B y compounds increase with the increase of the B atomic ratio.

Table 1 .
The formation enthalpies of Ni 3 B, Ni 2 B, Ni 4 B 3 , and NiB are −32.819kJ/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 x B y compounds are thermodynamically stable.Ni 4 B 3 has more thermodynamic stability than other compounds because its formation enthalpy is the lowest.

Table 1 .
The optimized parameters of Ni x B y compounds.

Table 2 .
The calculated elastic constants (in GPa) of Ni x B y compounds.

Table 4 .
The calculated universal anisotropic index (A U ), bulk anisotropy (A B and A G ), and shear anisotropic factors (A 1 , A 2 , A 3 ) of Ni x B y compounds.

Table 5 .
Milliken population analysis results of Ni x B y , the unit of bond length is Å.

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

Table 6 .
Theoretically calculated thermal properties of the Ni x B y compounds, including v s , v 1 , v m , and Θ D.