Materials and Design 89 (2016) 676–683

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Materials and Design

journal homepage: www.elsevier.com/locate/jmad

Relationship between Si concentration and mechanical properties of

Nb–Si compounds: A first-principles study
Yong Pan ⁎, Yuanhua Lin ⁎, Qi Xue, Chengqiang Ren, Hu Wang
School of Materials Science and Engineering, Southwest Petroleum University, Chengdu 610500, China

a r t i c l e i n f o a b s t r a c t

Article history: The effect of chemical bonding on the mechanical properties of high-temperature materials remains a big chal-
Received 3 August 2015 lenge. In this paper, the relationship between Si concentration and mechanical properties for Nb–Si compounds
Received in revised form 4 October 2015 was investigated by first-principles calculations. One previously unreported crystal structure of Nb2Si with Ta2Si-
Accepted 6 October 2015
type structure was predicted. The elastic properties, hardness and chemical bonding of Nb–Si compounds were
Available online 8 October 2015
calculated in detail. The calculated formation enthalpy shows that Nb5Si3 is more stable than that of other Nb–
Keywords:
Si compounds at the ground state. The obtained shear modulus, Young's modulus and hardness of Nb–Si com-
Transition metal silicides pounds increase with increasing Si concentration. NbSi2 exhibits the strongest shear deformation resistance
Elastic behavior and the biggest hardness among these Nb–Si compounds. It thus can be concluded that the variation of elastic
Hardness properties depends not only on the Si concentration but also on the chemical bonding. With increasing Si concen-
First-principles calculation tration, Nb–Nb metallic bond is disappeared and Si–Si covalent bond is formed.
© 2015 Elsevier Ltd. All rights reserved.

1. Introduction bonds and exhibit excellent mechanical properties [5–7]. Based on the
design principle, Nb–Si based superalloys have received considerable
Despite extensive studies carried out in the past decades, there are attention for high-temperature application such as structural compo-
still many unresolved issues associated with the high-temperature ma- nents in aircraft engines and rockets [8–12].
terials, especially the nature of the mechanical properties. Among those For Nb–Si compounds, early studies mainly focused on the structure
mechanical properties, the shear deformation resistance and elastic and mechanical properties of Nb5Si3 and Nb3Si [13,14]. Recently,
stiffness play a crucial role in high-temperature environment because Kashyap et al. [15] have attempted to study the microstructural and me-
the shear modulus directly controls the dislocation motion of a solid chanical properties of suction cast Nb–Si binary alloys. The microstruc-
and the Young's modulus indicates the degree of linear compression. Es- ture and mechanical properties of oxidation resistant suction cast Nb–
sentially, these elastic properties mainly depend on the cohesive force Si–Al alloy have been reported [16]. Our previous work has shown
per volume, which is entirely determined by the localized electron den- that vacancy results in brittle-to-ductile transition for Nb5Si3 [17]. On
sity and chemical bonding. As we known, the bond strengths of covalent the other hand, alloying additions effectively improve the mechanical
bond and ionic bond are much stronger than that of metallic bond. Thus, properties of Nb–Si alloys [18,19]. With Ti addition, the room-
a large number of covalent bonds in a solid effectively improve the de- temperature fracture toughness of Nb–Si binary alloys increases [20].
formation resistance and enhance the elastic properties. However, the The addition of B in Nb–Si–Ti ternary alloy changes the morphological
metallic bond plays an important role in all high-temperature materials feature and restrains the crack propagation [21]. Actually, the mechan-
[1–4]. This is why these materials have the low elastic properties. There- ical properties of Nb–Si based superalloys mainly determined by the Si
fore, the search for high-temperature material with the covalent bond is concentration because the Si–Si covalent bond obviously influences
a topic of much interest. the resistance to deformation. Unfortunately, the effect of Si concentra-
Till now, C and Si are known as typical covalent elements. Obviously, tion on the mechanical properties of Nb–Si compounds and relevant
the covalent bonds resist the deformation and prevent the propagation mechanism are still unknown.
of dislocation. To explore the high-temperature materials, alloying addi- In the present paper, the formation enthalpy, elastic constants, elas-
tions such as C and Si are considered to be the most promising candi- tic properties, hardness, electronic structure and chemical bonding of
dates because they are expect to have a larger number of covalent Nb–Si compounds were systematically calculated by using a first-
principles approach. The structure and mechanical properties of Nb–Si
⁎ Corresponding authors. compounds as a function of Si concentration were investigated in detail.
E-mail addresses: y_pan@ipm.com.cn (Y. Pan), yhlin28@163.com (Y. Lin). We first found that Nb–Si compounds with high concentration of Si

http://dx.doi.org/10.1016/j.matdes.2015.10.028
0264-1275/© 2015 Elsevier Ltd. All rights reserved.
 Y. Pan et al. / Materials and Design 89 (2016) 676–683 677

exhibit the strong shear deformation resistance and high elastic stiffness Table 1
in comparison with the low concentration of Si. Although the resistance Calculated elastic constants, Cij (GPa) of Nb–Si compounds.

to volume deformation of NbSi2 is weaker than that of Nb5Si3, NbSi2 has Phase Method C11 C12 C13 C33 C44 C66
the strongest shear modulus and the maximum Young's modulus Nb3Si Cal 321 112 72
among these Nb–Si compounds. Theoa 290 151 78
Nb2Si Cal 279 141 143 269 117 108
2. Theoretical methods Nb5Si3 Cal 378 96 117 326 128 119
Theoa 362 104 118 312 121 110
NbSi2 Cal 355 67 74 440 133
According to the Nb–Si binary phase diagram, the structures of
a
Ref [25].
Nb3Si, Nb5Si3 and NbSi2 are represented by the experimental deter-
mined cubic (space group Pm-3n, No. 223) [22], tetragonal (space
group I4/mcm, No. 140) [23] and hexagonal (space group P6222, No.
180) [24], respectively. It is worth noticing that Nb5Si3 has three differ- generalized gradient approximation (GGA) with Perdew–Burke–
ent structures such as α-, β- and γ-phases [25,26]. Considering the Ernzerhof (PBE) [28]. The ultrasoft pseudopotential [29] was used to de-
structural stable at the ground state, we select the α-phase. For other scribe the electronic exchange and correlation. The atomic configura-
Nb–Si compound such as Nb2Si, there is not experimental structural pa- tions of Nb and Si atoms were 4p64d45s1 and 3s23p2, respectively. To
rameter up to now. Here, the structure of Ta2Si-type (space group I4/ ensure the total energy at the ground state to be converged, a plane-
mcm, No. 140) for Nb2Si is considered. The structural models of four dif- wave basis set for electron wave function with cutoff energy of 400 eV
ferent Nb–Si compounds are shown in Fig. 1. was used. Integrations in the Brillouin zone were performed by using
Structure, total energy, elastic properties and electronic structure of special k-points generated with 12 × 12 × 12, 14 × 14 × 14,
Nb–Si compounds were calculated by using the density functional the- 11 × 11 × 14, and 15 × 15 × 10 for Nb3Si, Nb2Si, Nb5Si3 and NbSi2, re-
ory (DFT), as implemented in the CASTEP code [27]. To estimate the cal- spectively. The optimized atomic geometry was achieved through the
culated results, the exchange correlation functional was treated by the minimizing Hellmann-Feynman forces acting on each atom until the

Fig. 1. Crystal structures of Nb–Si compounds, (a) Nb3Si, (b) Nb2Si, (c) Nb5Si3, (d) NbSi2. The blue and gray spheres represent the Nb and Si atoms, respectively.
678 Y. Pan et al. / Materials and Design 89 (2016) 676–683

Fig. 2. Calculated bulk modulus and Poisson's ratio of Nb–Si compounds as a function of Si concentration.

maximum forces on the ions were smaller than 1 × 10− 5 eV/Å. The For hexagonal structure, it has five different elastic constants: C11,
atoms of all structures were obtained by fully relaxing unit cells. C12, C13, C33 and C44. Thus, the requirement of mechanical stability in a
hexagonal system leads to the following restrictions on these elastic
3. Results and discussions constants [6]:

The mechanical properties play a crucial role in the high- C44 N0; C11 NjC12 j; ðC11 þ 2C12 ÞC33 N2C213 : ð3Þ
temperature materials. Generally speaking, the mechanical properties
of a solid are evaluated by the elastic properties such as bulk modulus Table 1 lists the calculated elastic constants of Nb–Si compounds as a
(B), shear modulus (G), Young's modulus (E) and Poisson's ratio (δ). function of Si concentration. It is clear that Nb–Si compounds are me-
In particular, the elastic properties are obtained by the elastic constants chanical stability at the ground state because these obtained elastic con-
(Cij), which are defined as the stress tensor vs small strain. In addition, stants meet the Born stability criteria [31]. On the other hand, the
the mechanical stable of a material is also measured by the elastic con- calculated elastic constants of Nb3Si and Nb5Si3 are in good agreement
stants. To estimate the relationship between the mechanical properties with the previous theoretical results [32]. For Nb3Si, it is found that
of Nb–Si compounds and Si concentration, the elastic constants of those the calculated elastic constant C11 is bigger than that of other theoretical
compounds are first calculated and discussed, here. For Nb–Si com- result by 10.7%, in contrast to the obtained elastic constants C12 and C44
pounds, there are three different crystal structures such as cubic are lower than the previous theoretical values by 25.8% and 7.7%, re-
(Nb3Si), tetragonal (Nb2Si and Nb5Si3) and hexagonal (NbSi2) spectively. For Nb5Si3, the calculated elastic constants C11, C33, C44 and
structures. C66 are bigger by about 4.4%, 4.5%, 5.8% and 8.2%, respectively, than
For cubic structure, there are three independent elastic constants: that of other theoretical results. On the contrary, the elastic constants
C11, C12 and C44. Therefore, the mechanical stability of a cubic structure C12 and C13 are less than 7.7% and 0.8% for theoretical data.
can be tested by following equations [30]: To our knowledge, the elastic constants C11 and C33 represent the de-
gree of deformation resistance along the a-axis and c-axis, respectively.
C11 N0; C44 N0; C11 NjC12 j; ðC11 þ 2C12 ÞN0: ð1Þ For Nb2Si and Nb5Si3, the calculated elastic constant C11 is larger than
that of elastic constant C33 by about 3.7% and 16.0%, respectively, indi-
For tetragonal system, there are six different elastic constants: C11, cating that these Nb–Si compounds have strong deformation resistance
C12, C13, C33, C44 and C66. The mechanical stability of this structure can along the a-axis in comparison to the c-axis. However, the calculated
be obtained by: elastic constant C11 of NbSi2 is lower than that of elastic constant C33
by about 23.9%. In other words, the deformation resistance of NbSi2
along the c-axis is stronger than the a-axis. As mentioned above, it is
C11 N0; C33 N0; C44 N0; C66 N0; C11 −C12 N0; C11 suggested that this discrepancy is derived from the structural feature
þ C33 −2C13 N0; 2ðC11 þ C12 Þ þ C33 þ 4C13 N0: ð2Þ and atomic arrangement (see Fig. 1). As listed in Table 1, the calculated

Fig. 3. Calculated elastic modulus of Nb–Si compounds as a function of Si concentration. (a) Shear modulus, (b) Young's modulus.
 Y. Pan et al. / Materials and Design 89 (2016) 676–683 679

Fig. 6. Calculated formation enthalpy of Nb–Si compounds as a function of Si concentra-

Fig. 4. Calculated B/G ratio of Nb–Si compounds as a function of Si concentration.
tion. The solid line denotes the convex hull.

elastic constant C33 of NbSi2 (440 GPa) is larger than the corresponding modulus and Poisson's ratio are given by:
elastic constant of other Nb–Si compounds, meaning that NbSi2 exhibits
the strongest deformation resistance along the c-axis. 9BG
E¼ ð4Þ
On the other hand, the elastic constant C44 represents the degree of 3B þ G
shear distortion in the (100) plane and elastic constant C66 indicates the
shear resistance in the b110N direction. From Table 1, the variations of 3B−2G
δ¼ ð5Þ
elastic constants C33, C44 and C66 depend on the Si concentration. The 6B þ 2G
elastic constants of Nb–Si compounds increase with increasing Si con-
centration. In particular, the obtained elastic constants C44 and C66 of where B and G represent the bulk modulus and shear modulus,
NbSi2 are 133 GPa and 144 GPa, which are larger than that of other respectively.
Nb–Si compounds. Obviously, NbSi2 shows the higher deformation re- Fig. 2 displays the calculated bulk modulus and Poisson's ratio of Nb–
sistance along the c-axis and the stronger shear deformation resistance Si compounds as a function of Si concentration. It can be seen that the
in comparison with other Nb–Si compounds. This result may be origi- obtained bulk modulus of Nb3Si and Nb5Si3 is 181 GPa and 193 GPa, re-
nated from the strong Si–Si covalent bond in NbSi2 because the Si–Si co- spectively, which are in excellent agreement with the previous theoret-
valent bond along the c-axis obviously improves the shear deformation ical results [32]. Note that the calculated bulk modulus of Nb–Si
resistance. increases with increasing Si concentration when Si b 37.5 at.%
To explore the effect of Si concentration on the mechanical proper- (Nb5Si3). On the contrary, the bulk modulus of Nb–Si compounds de-
ties of Nb–Si compounds, following, the correlation between the elastic creases with increasing Si concentration when Si N 37.5 at.%. The convex
properties and Si concentration is investigated in detail. Considering the hull indicates that Nb5Si3 has the strongest volume deformation resis-
structural symmetry, the bulk modulus and shear modulus are calculat- tance among these Nb–Si compounds. In addition, the calculated bulk
ed according to the Voigt–Reuss–Hill (VRH) approximation [33]. In ad- modulus of NbSi2 is 175 GPa, which is lower than that of other Nb–Si
dition, the Young's modulus is defined as the ratio of linear stress vs compounds. Namely, the bulk modulus of Nb–Si compounds is not de-
linear strain, indicating the degree of elastic stiffness [34]. The value of termined by Si concentration. As we known, the bulk modulus of a
the Poisson's ratio is indicative of the degree of the covalent bond. solid measures the size stability. Therefore, it can be concluded that
Therefore, the larger value of Poisson's ratio implies the ionic bond the variation of bulk modulus is related to the cohesive strength be-
and metallic bond. In Voigt–Reuss–Hill approximation, the Young's tween atoms.
From Fig. 2(b), it is observed that the obtained Poisson's ratio of Nb–
Si compounds decreases with increasing Si concentration. The calculat-
ed Poisson's ratio of Nb3Si (0.299) is close to that of metal and alloy.
However, the obtained Poisson's ratio of NbSi2 is only about of 0.177,

Table 2
Calculated lattice parameters, a-, b- and c-axis (nm), c/a ratio and density, ρ (g/cm3) of
Nb–Si compounds.

Phase Method Structure a c c/a ρ

Nb Cal Cubic 0.331 8.52

Nb3Si Cal Cubic 0.511 7.65
Expa 0.516
Nb2Si Cal Tetragonal 0.619 5.134 0.830 7.23
Nb5Si3 Cal Tetragonal 0.659 11.914 1.807 7.036
Theob 0.660 11.878
NbSi2 Cal Hexagonal 0.481 6.606 1.375 5.62
Expc 0.479 6.590
Si Cal Cubic 0.547 2.29
a
Ref [22].
b
Ref [41].
c
Fig. 5. Calculated hardness of Nb–Si compounds as a function of Si concentration. Ref [42].
680 Y. Pan et al. / Materials and Design 89 (2016) 676–683

Fig. 7. Total and partial density of states (DOS) of Nb–Si compounds as a function of Si concentration, (a) Nb3Si, (b) Nb2Si, (c) Nb5Si3, (d) NbSi2, respectively. The black vertical dashed
indicates the Fermi level (EF).

which is lower than that of metal and alloy. That is to say, the variation For high-temperature materials, the brittle or ductile behavior
of Poisson's ratio indicates that the Si concentration increases the cova- should be considered in order to accelerate their practical application.
lency between atoms. According to the Pugh rule [35], the brittle or ductile of a solid is esti-
However, the trends of shear modulus and Young's modulus are dif- mated by B/G ratio. The material exhibits brittle behavior when B/
ferent from that of bulk modulus. Fig. 3 shows the calculated shear mod- G b 1.75, and vice versa, a material demonstrates ductile when B/
ulus and Young's modulus of Nb–Si compounds as a function of Si G N 1.75. Fig. 4 depicts the calculated B/G ratio of Nb–Si compounds as
concentration. It is obvious that the shear modulus and Young's modu- a function of Si concentration. It is found that the calculated B/G ratio
lus of Nb–Si compounds increase with increasing Si concentration. As of Nb–Si compounds decreases with increasing Si concentration. Nb3Si
seen in Fig. 3, the obtained shear modulus and Young's modulus of and Nb2Si exhibit ductile behavior in contrast to Nb5Si3 and NbSi2
NbSi2 are 144 GPa and 339 GPa, respectively, which are larger than show brittle behavior, which are consistent with the experimental re-
the corresponding elastic properties of other Nb–Si compounds. These sults [36,37]. Obviously, the variation of B/G ratio is also related to the
results imply that NbSi2 exhibits the strongest shear deformation resis- Si concentration and chemical bonding. It should be mentioned that
tance and the highest elastic stiffness among these Nb–Si compounds. the calculated B/G ratio of NbSi2 (1.22) is lower than that of other Nb–
To the best of our knowledge, the shear modulus measures the shear Si compounds, implying that NbSi2 exhibits the strongest brittle behav-
strength of a solid. There is no doubt that the value of shear modulus ior among these Nb–Si compounds.
plays an important role in the mechanical properties because it is direct- On the other hand, the hardness is an important mechanical param-
ly related to the dislocation motion. At atomic level, the shear modulus eter because the hardness determines the resistance of surfaces to wear.
is the energy needed to shear pairs of atom, which depends on the local- In other words, the mechanical properties of a material are also directly
ized electronic density. In particular, the type of chemical bonding and estimated by the hardness. In this paper, the hardness of Nb–Si com-
orientation of bond determine the degree of shear deformation. With pounds is obtained by the semi-empirical hard model [38]:
increasing Si concentration, a large number of Si–Si covalent bonds are
formed in Nb–Si compounds. On the other hand, the increased concen-  0:585
2
Hv ¼ 2  k  G −3 ð6Þ
tration of Si decreases the number of Nb–Nb metallic bond and Nb–Si
bond. This result is demonstrated by the structural feature. From
Fig. 1, the Si concentration results in charge transfer from Nb–Si to Si– where k is the B/G ratio and G is the shear modulus, respectively.
Si. According to the structural feature of NbSi2, the Nb–Si–Nb sandwich Fig. 5 shows the calculated hardness of Nb–Si compounds as a func-
structure at each layer forms a two-dimensional (2D) bonding state. tion of Si concentration. It can be seen that the calculated theoretical
Therefore, Si–Si covalent bonds along the b-axis improve the elastic de- hardness of Nb5Si3 (19.1 GPa) is slightly larger than the previous theo-
formation resistance. It explains why NbSi2 shows the strong elastic retical result (19.0 GPa) [39]. It is worth noticing that the trend of hard-
properties. ness of Nb–Si compounds is consistent with the shear modulus and
 Y. Pan et al. / Materials and Design 89 (2016) 676–683 681

Fig. 8. The difference charge density of contour plots of Nb–Si compounds. (a) Nb3Si along the (001) plane, (b) Nb2Si along the (001) plane, (c) Nb5Si3 along the (010) plane, (d) NbSi2
along the (100) plane, respectively.

Young's modulus. The calculated hardness of NbSi2 is 27.5 GPa, which is excellent agreement with the previous theoretical result [40]. The con-
much larger than that of other Nb–Si compounds. Essentially, the hard- vex hull implies that Nb5Si3 is more stable than that of other Nb–Si com-
ness of a solid is related to the chemical bonding. Therefore, the increas- pounds. This is well explained why Nb5Si3 has received considerable
ing of Si–Si covalent bond improves the deformation resistance and attention in comparison with other Nb–Si compounds. Importantly, it
enhances its hardness. is predicted that Nb2Si possibly exists because the calculated formation
For high-temperature materials, the structural stability also plays an enthalpy of Nb2Si is lower than that of Nb3Si.
important role in high-temperature environment in order to ensure To understand the stable structure of Nb–Si compounds, the struc-
working processes and improve their lifetime. To estimate the structural tural information of Nb–Si compounds is calculated and discussed. The
stability, the formation enthalpies of Nb–Si compounds as a function of calculated lattice parameters and density of Nb–Si compounds are listed
Si concentration are calculated and discussed. The equation of formation in Table 2. For Nb3Si, the calculated lattice parameter (a = 0.511 nm) is
enthalpy (ΔH) is defined as: in good agreement with the experimental value (a = 0.516 nm) [22]. In
this structure (see Fig. 1(a)), each Si atom is surrounded by 8 Nb atoms,
1 and the bond length of Nb–Si is 0.286 nm. It should be pointed out that
ΔHðNbm Sin Þ ¼ ðE ðNbm Sin Þ−mENb −nESi Þ ð7Þ
m þ n total Nb–Nb metallic bond (0.255 nm) exists in Nb3Si compound. Therefore,
the structural stability and mechanical properties of Nb3Si are deter-
where Etotal(NbmSin), ENb and ESi are the total energy of NbmSin com- mined by both Nb–Si and Nb–Nb bonds. However, the Nb–Nb metallic
pounds, Nb and Si atoms at the ground state. m and n are the number bond weakens the mechanical properties of Nb3Si.
of Nb and Si atoms in a system, respectively. For Nb5Si3, the calculated lattice parameters (a = 0.659 nm and c =
Fig. 6 represents the formation enthalpy of Nb–Si compounds as a 1.191 nm) are in accordance with the previous theoretical results [41].
function of Si concentration. To our knowledge, the formation enthalpy In this structure (see Fig. 1(c)), the alternative stacking of the Nb layer
of a material strongly depends on the atomic potential. The negative for- and Si layer can be viewed along the crystallographic c-axis. The layered
mation enthalpy indicates the thermodynamic stable at the ground structure is constructed by the graphite-like Nb and Si layers with a
state. The general trend is, the lower the formation enthalpy, the more clear sliding between the two adjacent layers. The calculated bond
stable the structure. It can be seen from Fig. 6 that those compounds length of Nb–Si bond between the Nb layer and Si layer is 0.261 nm.
are thermodynamic stable at the ground state. Furthermore, Nb5Si3 However, the alternative Nb and Si atoms in a sub-boundary layer
has the lowest formation enthalpy as 0.700 eV/atom, which is in form the Nb–Si–Nb sandwich structure. The calculated bond length of
682 Y. Pan et al. / Materials and Design 89 (2016) 676–683

Nb–Si bond is 0.259 nm. It is obvious that the bond strength in Nb–Si– 1. The calculated lattice parameters of Nb3Si, Nb5Si3 and NbSi2 are in
Nb sandwich structure is stronger than that of layer–layer. Thus, the good agreement with the experimental data and theoretical results.
structural stability and mechanical properties of this alloy are deter- The convex hull indicates that Nb5Si3 is more stable than that of
mined by the bond strength between Nb and Si layers. other Nb–Nb compounds.
For NbSi2 (see Fig. 1(d)), the calculated lattice parameters (a = 2. The Nb–Nb compounds are mechanically stable at the ground state.
0.481 nm and c = 0.661 nm) are in excellent agreement with the exper- The obtained elastic constants C33, C44 and C66 increase with increas-
imental data [42]. In this structure, the stacking sequence of Nb layer ing Si concentration.
and Si layer is ABABABAB along the crystallographic a-axis. Note that 3. The deformation resistance of Nb2Si and Nb5Si3 along the a-axis is
Si layer separates the two different sub-boundary layers. Therefore, stronger than that of the c-axis, in contrast to the deformation resis-
the mechanical properties of Nb–Si layer and Si–Si layer are mainly at- tance of NbSi2 along the c-axis is stronger than that of the a-axis.
tributed to Nb–Si (0.262 nm) and Si–Si (0.257 nm) bonds, respectively. 4. The calculated elastic properties show that the bulk modulus of
As mentioned above, it is concluded that the mechanical properties Nb5Si3 is larger than that of other Nb–Nb compounds. However,
of Nb–Nb compounds are related to the Si concentration. NbSi2 is ex- the shear modulus, Young's modulus and hardness of Nb–Nb com-
pected to have excellent mechanical properties. However, the nature pounds mainly depend on the Si concentration.
of mechanical properties is very complex involving the elastic and plas- 5. The calculated shear modulus, Young's modulus and hardness of
tic deformations, which are contributed to the intrinsic factors such as NbSi2 are 144 GPa, 339 GPa and 27.5 GPa, respectively, which are
chemical bonding, cohesive energy and crystal structure etc. To gain in- larger than that of other Nb–Nb compounds. Nb5Si3 and NbSi2 exhib-
sight into the nature of mechanical properties, the electronic structure it brittle behavior, in contrast to Nb3Si and Nb2Si show ductile
and chemical bonding of Nb–Nb compounds are calculated and ana- behavior.
lyzed, here. Fig. 7 shows the total and partial density of states (DOS) 6. The increasing of elastic properties and hardness are originated from
of Nb3Si, Nb2Si, Nb5Si3 and NbSi2. It is observed that these DOS profiles the variation of chemical boding. With increasing Si concentration,
of Nb–Nb compounds are mainly contributed by Si-3s state, Si-3p state, the Nb–Nb metallic bond is disappeared and Si–Si covalent bond is
Nb-4d state and Nb-4p state, meaning that the strong hybridization be- formed. For Nb–Nb compounds, the strong Nb–Nb and Si–Si bonds
tween Nb and Si atoms, forming the Nb–Nb bond. obviously improve the deformation resistance, and enhance the elas-
For Nb–Nb compounds, the PDOS profiles of Si atom have obvious tic properties and hardness.
discrepancy. With increasing Si concentration, the PDOS profile of Si
atom shows the charge transfer from Si-3s state to Si-3p state. It is Acknowledgments
worth noticing that Si concentration results in the main peak shifted
from valence band to Fermi level (EF). In going from Nb3Si to NbSi2, This work is supported by National Natural Science Foundation of
the Si-3s state stretches into the Si-3p state near EF, meaning that the China (Grant No. 51274170) and Natural Science Foundation of Yunnan
Si–Si covalent bond is formed in NbSi2. There is the main reason why province (Grant No: 2009CD134).
the elastic properties and hardness of NbSi2 are larger than that of
other Nb–Nb compounds. References
Obviously, the high concentration of Si changes the localized hybrid-
ization between Nb and Si atoms, and chemical bonding. To reveal the [1] I. Papadinitriou, C. Utton, P. Tsakiropoulos, Ab initio investigation of the intermetal-
lics in the Nb-Sn binary system, Acta Mater. 86 (2015) 23–33.
nature of chemical bonding, the calculated contours of electronic local- [2] Y. Yan, H. Ding, Y. Kang, J. Song, Microstructure evolution and mechanical properties
ized function (ELF) of Nb–Nb compounds are shown in Fig. 8, and the of Nb–Nb based alloy processed by electromagnetic cold crucible directional solidi-
critical feature are labeled. It can be seen that the directional Nb–Nb me- fication, Mater. Des. 55 (2014) 450–455.
[3] J.J. Han, W.Y. Wang, X.J. Liu, C.P. Wang, X.D. Hui, Z.K. Liu, Effect of solute atoms on
tallic bond, Nb–Nb bond and Si–Si covalent bond are observed in Nb–Nb glass-forming ability for Fe-Y-B alloy: an ab initio molecular dynamics study, Acta
compounds. However, the chemical bonding of Nb–Nb compounds has Mater. 77 (2014) 96–110.
obvious discrepancy. For Nb3Si (see Fig. 8(a)) and Nb2Si (see Fig. 8(b)), [4] X. Zhou, Z. Xu, R. Mu, L. He, G. Huang, X. Cao, Thermal barrier coatings with a
double-layer bond coat on Ni3Al based single-crystal superalloy, Journal Of Alloys
they have Nb–Nb metallic bond and Nb–Nb bond. There is no doubt that
and Compounds 591 (2014) 41–51.
Nb–Nb metallic bond weakens the deformation resistance, and reduces [5] A.I.Z. Farahat, A.S. Hamdy, Influence of hot forging and alloying with Al on the elec-
the elastic properties and hardness. It must be mentioned that the bond trochemical behavior and mechanical properties of austenitic stainless steel, Mater.
Des. 57 (2014) 538–545.
length Nb–Nb metallic bond becomes longer and the bond length of
[6] Y. Liu, H. Huang, Y. Pan, G. Zhao, Z. Liang, First-principles study on the phase transi-
Nb–Nb bond becomes short with increasing Si concentration. That is tion, elastic properties and electronic structure of Pt3Al alloys under high pressure,
to say, the increasing of Si atom results in charge transfer from Nb–Nb Journal Of Alloys and Compounds 597 (2014) 200–204.
to Nb–Nb, and enhances the elastic properties. [7] Y.C. Lin, J. Deng, Y.Q. Jiang, D.X. Wen, G. Liu, Hot tensile deformation behaviors and
fracture characteristics of a typical Ni-based superalloy, Mater. Des. 55 (2014)
With further increasing Si concentration such as Nb5Si3 (see 949–957.
Fig. 8(c)) and NbSi2 (see Fig. 8(d)), the Si–Si covalent bond is observed [8] B. Xiong, C. Cai, Z. Wang, Microstructures and room temperature fracture toughness
and the Nb–Nb metallic bond is disappeared in these compounds. For of Nb/Nb5Si3 composites alloyed with W, Mo and W, Mo and W-Mo fabricated by
spark plasma sintering, J. Alloys Compd. 604 (2014) 211–216.
Nb5Si3, it has three types of Nb–Nb bonds (0.259 nm, 0.261 nm and [9] W. Wang, C. Zhou, Hot corrosion behaviour of Nbss/Nb5Si3 in situ composites in the
0.265 nm), which form the network Nb–Nb bonds. Therefore, the Nb– mixture of Na2SO4 and NaCl melts, Corros. Sci. 74 (2013) 345–352.
Nb bond plays an important role in Nb5Si3. For NbSi2, the directional [10] S. Shi, L. Zhu, L. Jia, H. Zhang, Z. Sun, Ab-initio study of alloying effects on structure
stability and mechanical properties of a-Nb5Si3, Comput. Mater. Sci. 108 (2015)
Si–Si covalent bond (0.257 nm) and Nb–Nb bonds (0.262 nm and 121–127.
0.274 nm) are observed. In particular, the Si–Si covalent bond and [11] W. Wei, H. Wang, C. Zou, Z. Zhu, Z. Wei, Microstructure and oxidation behavior of
Nb–Nb bond improve the deformation resistance, and enhance the elas- Nb-based multi-phase alloys, Mater. Des. 46 (2013) 1–7.
[12] S. Lu, H. Zheng, L. Deng, J. Yao, Effect of silicon on the fracture toughness and oxida-
tic properties and hardness.
tion behavior of hot pressed NbCr2 alloys, Mater. Des. 51 (2013) 432–437.
[13] S. Kashyap, K. Chattopadhyay, Synthesis and phase evolution in Nb/Si multilayers
obtained by sequential laser ablation, Thin Solid Films 531 (2013) 312–319.
[14] C.S. Tiwary, S. Kashyap, K. Chattopadhyay, Effect of Mg addition on
4. Conclusions
microstructural,mechanical and environmental properties of Nb–Si eutectic com-
posite, Mater. Sci. Eng. A 560 (2013) 200–207.
In summary, we employed first-principles approach to investigate [15] S. Kashyap, C.S. Tiwary, K. Chattopadhyay, Microstructural and mechanical behavior
the structural stability, elastic constants, elastic properties, B/G ratio, study of suction cast Nb–Si binary alloys, Mater. Sci. Eng. A 583 (2013) 188–198.
[16] S. Kashyap, C.S. Tiwary, K. Chattopadhyay, Microstructure and mechanical proper-
hardness and chemical bonding of Nb–Nb compounds as a function of ties of oxidation resistant suction cast Nb–Si–Al alloy, Mater. Sci. Eng. A 559
Si concentration. The results are given by: (2013) 74–85.
 Y. Pan et al. / Materials and Design 89 (2016) 676–683 683

[17] Y. Pan, Y. Lin, H. Wang, C. Zhang, Vacancy induced brittle-to-ductile transition of [31] A. Sezgin, S. Mehmet, First-principles calculations of MnB2, TcB2, and ReB2 within
Nb5Si3 alloy from first-principles, Mater. Des. 86 (2015) 259–265. the ReB2-type structure, Phys. Rev. B 80 (2009) 134107.
[18] I. Grammenos, P. Tsakiropoulos, Study of the role of Al, Cr and Ti additions in the mi- [32] I. Papadimitriou, C. Utton, A. Scott, P. Tsakiropoulos, Ab initio study of the interme-
crostructure of Nb–18Si–5Hf base alloys, Intermetallics 18 (2010) 242–253. tallics in Nb–Si binary system, Intermetallics 54 (2014) 125–132.
[19] N. Vellios, P. Tsakiropoulos, The role of Sn and Ti additions in the microstructure of [33] M. Ellner, Bond energy in palladium and platinum rich alloys with the A4 transition
Nb–18Si base alloys, Intermetallics 15 (2007) 1518–1528. metals, Journal of Alloys and Compounds 366 (2004) 222–227.
[20] Z. Li, L.M. Peng, Microstructural and mechanical characterization of Nb-based in situ [34] I.R. Shein, A.L. Ivanovskii, Electronic band structure, Fermi surface, and elastic prop-
composites from Nb–Si–Ti ternary system, Acta Mater. 55 (2007) 6573–6585. erties of polymorphs of the 5.2 K iron-free superconductor SrPt2As2 from first-
[21] Huang Q, Kang Y, Song J, Qu S, Han Y, Guo X. Effects of Ni, Co, B, and Ge on the mi- principles calculations, Phys. Rev. B 83 (2011) 104501.
crostructures and mechanical properties of Nb–Ti–Si ternary alloys. Met. Mater. Int. [35] S.F. Pugh, Relations between the elastic moduli and the plastic properties of poly-
2014;20:475–81. crystalline pure metals, Philos. Mag. 45 (1954) 823–843.
[22] W.K. Wang, H. Iwasaki, C. Suryanarayana, T. Masumoto, N. Toyota, T. Fukase, et al., [36] N. Sekido, S. Miura, Y.Y. Mitarai, Y. Kimura, Y. Mishima, Dislocation character and
Crystallization characteristics of an amorphous Nb81 Si19 alloy under high pressure operative slip systems in a-Nb5Si3 tested at 1673 K, Intermetallics 18 (2010)
and formation of the A15 phase, Journal of Materials Science 17 (1982) 1523–1532. 841–845.
[23] S.V. Meschel, O.J. Kleppa, Standard enthalpies of formation of some 4d transition [37] S. Miura, Y. Murasato, Y. Sekito, Y. Tsutsumi, K. Ohkubo, Y. Kimura, et al., Effect of
metal silicides by high temperature direct synthesis calorimetry, Journal of Alloys microstructure on the high-temperature deformation behavior of Nb–Si alloys,
and Compounds 274 (1998) 193–200. Mater. Sci. Eng. A 510–511 (2009) 317–321.
[24] G. Cheng, H. Qian, L. He, H. Ye, Characterization of a new Nb–Nblicide (Nb11Si4) in [38] X.Q. Chen, H.Y. Niu, D.Z. Li, Y.Y. Li, Modeling hardness of polycrystalline materials
Nb–Si binary systems, Philosophical Magazine A 90 (2010) 2557–2568. and bulk metallic glasses, Intermetallics 19 (2011) 1275–1281.
[25] S. Kashyap, C.S. Tiwary, K. Chattopadhyay, Effect of gallium on microstructure and [39] W. Xu, J. Han, C. Wang, Y. Zhou, Y. Wang, Y. Kang, et al., Temperature-dependent
mechanical properties of Nb–Si eutectic alloy, Intermetallics 19 (2011) 1943–1952. mechanical properties of alpha−/beta-Nb5Si3 phases from first-principles calcula-
[26] C.S. Tiwary, S. Kashyap, K. Chattopadhyay, Effect of indium addition on microstruc- tions, Intermetallics 46 (2014) 72–79.
tural, mechanical and oxidation properties of suction cast Nb–Nb eutectic alloy, [40] Y. Chen, T. Hammerschmidt, D.G. Petifor, J.X. Shang, Y. Zhang, Influence of vibration-
Mater. Sci. Technol. 29 (2013) 702–709. al entropy on structural stability of Nb–Nb and Mo-Si systems at elevated tempera-
[27] M.D. Segall, P.J.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, et al., First- tures, Acta Mater. 57 (2009) 2657–2664.
principles simulation: ideas, llustrations and the CASTEP code, Journal Of Physics- [41] Y. Chen, J.X. Shang, Y. Zhang, Bonding characteristics and site occupancies of alloying
condensed Matter 14 (2002) 2717–2744. elements in different Nb5Si3 phases from first-principles, Phys. Rev. B 76 (2007)
[28] J.P. Perdew, Y. Wang, Accurate and simple analytic representation of the electron- 184204.
gas correlation energy, Phys. Rev. B 45 (1992) 13244–13249. [42] F. Chu, M. Lei, S.A. Maloy, J.J. Petrovic, T.E. Mitchell, Elastic properties of C40 transi-
[29] D. Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue for- tion metal disilicides, Acta Mater. 44 (1996) 3035–3048.
malism, Phys. Rev. B 41 (1990) 7892–7895.
[30] W.W. Xu, J.J. Han, Y. Wang, C.P. Wang, X.J. Liu, Z.K. Liu, First-principles investigation
of electronic, mechanical and thermodynamic properties of L12 ordered Co3(M,
W) (M = Al, Ge, Ga) phases, Acta Mater. 61 (2013) 5437–5448.