Sun Min, Wang Chong-Yu, Liu Ji-Ping. Anisotropic elastic properties and ideal uniaxial compressive strength of TiB2 from first principles calculations. Chinese Physics B, 2018, 27(7): 077103
Permissions
Anisotropic elastic properties and ideal uniaxial compressive strength of TiB2 from first principles calculations
Sun Min1, Wang Chong-Yu2, ‡, Liu Ji-Ping1, †
School of Material Science and Engineering, Beijing Institute of Technology, Beijing 100081, China
Department of Physics, Tsinghua University, Beijing 100084, China
The structural, anisotropic elastic properties and the ideal compressive and tensile strengths of titanium diboride (TiB2) were investigated using first-principles calculations based on density functional theory. The stress–strain relationships of TiB2 under , , and ⟨0001⟩ compressive loads were calculated. Our results showed that the ideal uniaxial compressive strengths are GPa, GPa, and GPa, at strains −0.16, −0.32, and −0.24, respectively. The variational trend is just the opposite to that of the ideal tensile strength with GPa, GPa, and , at strains 0.14, 0.28, and 0.22, respectively. Furthermore, it was found that TiB2 is much stronger under compression than in tension. The ratios of the ideal compressive to tensile strengths are 5.56, 2.55, and 4.01 for crystallographic directions , , and ⟨0001⟩, respectively. The present results are in excellent agreement with the most recent experimental data and should be helpful to the understanding of the compressive property of TiB2.
TiB2 is well known as a ceramic material with many unique physical properties, such as high hardness, elastic modulus, strength to weight ratio, wear resistance, good thermal and electrical conductivities.[1] The range of potential applications of TiB2 is extensive, including nose tip and sharp leading edges of hypersonic vehicles, hot structure components for scramjet engines, impact resistant armor, cutting tools, wear resistant parts, grain refiner, as well as reinforcements in various composite materials such as special steel. In addition, the high electrical conductivity enables machining of titanium borides using the electrical discharge machining.[2,3] Thus, in the last two decades, TiB2 has received increasing attention in its experimental and theoretical researches.
The physical and mechanical properties are important to the practical performance. Especially, profound insight of the ideal strength of TiB2 is of great importance not only in deepening the understanding of intrinsic properties of TiB2 but also in realistic structural applications. The ideal strength of a material is the minimum stress at which a perfect crystal begins to deform plastically, whereas the yield point is the point where nonlinear deformation begins.[4] The ideal strength of materials, which is dominated by the behavior of valence electrons and ions, represents an upper limit of the strength that real materials can have under a given load, and thus, has attracted great interest in material science.[5]
Based on the first principles approaches and the developments in super-computers, the ideal strength can now be investigated at the atomic level. The strength is commonly considered in terms of compressive strength, tensile strength, and shear strength with corresponding applied loads.[6,7] The atomistic deformation mechanisms and stress–strain relationships of tensile and shear deformations have been extensively examined for TiB2. Zhou et al.[8] investigated the structural, stability, chemical bonding, hardness, and mechanical properties of 3d, 4d, 5d transition metal diboride TMB2 using first principles calculations and pointed out that the number of valence electrons of the transition metals has strong effects on the above properties. The ideal shear and tensile strengths of ZrB2, HfB2, and TiB2 were studied by Zhang et al.[9] using the first-principles calculations. Chen et al.[10] estimated the temperature-dependent ideal tensile strength of the AlB2 like diborides. The anisotropic elastic and thermal expansions of the titanium borides were calculated by Sun et al.[3] using first-principles methods. Afaf Saai et al.[11] used a sequential multi-scale approach to evaluate the degradation of TiB2 as a function of its microstructure.
Experimental works about the compressive strength of TiB2 were also reported. Munro[12] measured the physical, mechanical, and thermal properties of polycrystalline TiB2, and determined a consistent set of trend values of these properties on the dependence of density and grain size. At room temperature, Munro estimated that the dependence of the compressive strength on the density is approximately linear, ranging from 1.1 GPa at 3.8 g/cm to 1.8 GPa at 4.5 g/cm. The Hugoniot-measurements were performed on the TiB2 polycrystal by Mashimo et al.[13] to investigate the elastoplastic transition using the inclined-mirror method combined with the powder gun and two-stage gas gun. The Hugoniot-elastic limit stress was measured to be approximately 12 GPa. These studies have deepened the understanding of the single crystals of TiB2.
However, the calculation of the ideal compressive strength of TiB2 is still an open question, and there is scant confirmed reports on this issue. In fact, the ideal compressive strength, like the ideal tensile and shear strength, is one of the primary intrinsic mechanical properties of a solid material and is fundamental for the development of practical applications.[7]
In the light of this background, in the present work, by performing the first-principles calculations based on density functional theory, the ideal uniaxial compressive strength of TiB2 single crystal is studied. The organization of the paper is as follows. Section 2 introduces the methodology for calculating the theoretical strength under a uniaxial load. Section 3 presents and discusses the main results of this work. Finally, the conclusions are presented in Section 4. The results fill the void in the literature regarding the ideal compressive strength of TiB2.
2. Calculational details
The first principles calculations were performed within the generalized-gradient approximation of Perdew–Burke–Ernzerhoff[14] for the exchange–correlation functional as implemented in the Vienna abinitio simulation package code.[15] The cut-off energy of atomic wave functions was set to be 415 eV in all calculations after the convergence test. A special k-point sampling method was utilized for the integration by setting 11 × 11 × 10 for TiB2 with Monkhorst–Pack scheme[16] in the first irreducible Brillouin zone. The break condition for the electronic self-consistency was set to be 10−5 eV. The ionic relaxation was stopped when the forces on all of the atoms were less than 0.02 eV/Å.
In order to investigate the anisotropic mechanical properties, the elastic stiffness and other mechanical properties were estimated. The crystal structure of TiB2 considered in the current paper has the hexagonal symmetry with Ti atoms forming a metallic hexagonal structure and B atoms situated interstitially between the Ti-layers as is shown in Fig. 1. The elastic constants were estimated from first-principles strain-stress relationships as implemented by Milman et al.[17] Specifically, according to Hooke’s law, the elastic constants can be defined by means of a Taylor expansion of the total energy E(V, {εi}) with respect to a small strain δ for the system[18]
where Cij are the elastic constants (i,j,k = 1,...,6), V0 is the unstrained volume of the unit cell, τi are related to the stress on the lattice (i,j = 1,...,6), and ei are the components of the strain tensor ε,
in conventional elastic theory notation and I represents the identity matrix. TiB2 crystallizes in AlB2-type structure with the space group P6/mmm (No. 191) and therefore the number of independent elastic constants is reduced to 5, i.e., C11, C12, C13, C33, and C44,
The elastic constants can be obtained by applying an appropriate set of distortions with the distortion parameter δ as shown in Table 1.
Fig. 1. Schematic illustrations of (a) the hexagonal crystal structure, (b) the primitive cell, and (c) the orthogonal unit cell of TiB2. The blue and red atoms are Ti and B atoms, respectively.
Table 1.
Table 1.
Table 1.
The five applied strain configurations used to calculate the five independent elastic constants of the hexagonal TiB2 and the corresponding strain energy density–strain (ΔE/V0 ∼ δ) relation.
.
Strain configuration
Strain energy density–strain relation
e = (δ, δ, 0,0,0,0)
ΔE/V0 = (C11 + C12) δ2
e = (0,0,0,0,0,δ)
e = (0,0,δ,0,0,0)
e = (0,0,0,δ,δ,0)
ΔE/V0 = C44δ2
e = (δ,δ,δ,0,0,0)
ΔE/V0 = (C11 + C12 + 2C13 + C33/2)δ2
Table 1.
The five applied strain configurations used to calculate the five independent elastic constants of the hexagonal TiB2 and the corresponding strain energy density–strain (ΔE/V0 ∼ δ) relation.
.
For the purpose of analyzing the stress response to strain in the main crystal directions, an orthogonal unit cell having six atoms was used to calculate the ideal strength of TiB2[9] as shown in Fig. 1(b). The axes orientations of the orthogonal unit cell are along the (a axis), (b axis), and ⟨0001⟩ (c axis) directions of the primitive cell. The calculation approaches of tensile strength and compressive strength are similar to those of previous studies.[19,20] The stress–strain curves were calculated by incrementally deforming the unit cell in the selected strain direction. The atomic basis vectors orthogonal to the applied strain were simultaneously relaxed until the other stress components disappeared. Meanwhile, all the internal freedoms of the atom were relaxed until the atoms reached their equilibrium at each step.
3. Results and discussion
3.1. Equilibrium geometry of TiB2
TiB2 crystallizes in the hexagonal AlB2 structure designated as C32 with the space group P6/mmm. There are three atoms in the unit cell, one Ti atom at the origin and two B atoms at (1/3, 2/3, 1/2) and (2/3, 1/3, 1/2), respectively. The optimized cell constants and the previously calculated and experimental values[2,21,22] are listed in Table 2. The lattice constants a and c are in excellent agreement with the available experimental and calculated results. The average deviation is less than 0.3%, which demonstrates the reliability of the present calculations.
Table 2.
Table 2.
Table 2.
Calculated equilibrium lattice constants of TiB2 compared with available experimental and theoretical data.
Calculated equilibrium lattice constants of TiB2 compared with available experimental and theoretical data.
.
3.2. Elastic constants, polycrystalline moduli, and hardness
A hexagonal TiB2 crystal has six different elastic constants (c11, c12, c13, c33, c44, and c66), but only five of them are independent since c66 = (c11 − c12)/2. The compliance constants {S} = {C−1} were also calculated by solving the inverse of the elastic constant matrix by using MATLAB. The calculated elastic and compliance constants of TiB2 are shown in Table 3. Notably, the anisotropic parameters in elasticity, c33/c11 and c44/c66 are smaller than unity, which indicates that the atomic bonding along the a-axis is stronger than that along the c-axis. This is consistent with the picture that the crystal structure of TiB2 is of the layered-type. Intra the B layer, the B atoms form a 2D graphite-like geometric structure by sharing the valence electrons through sp2 hybridization. Each B atom establishes three σ-type sp2–sp2 bonds with its adjacent B atoms. The bonding in the Ti plane is mainly metallic. While the nature of the bonding inter the Ti and B sub-layers is the mixture of ionic–covalent interactions.[23] Thus, the B–B bond is stronger than the Ti–B bond, which could be the main reason for the anisotropic elasticity of the TiB2 structure. The intrinsic mechanical stability of TiB2 can be justified by the Born–Huang criterion. For a hexagonal structure, the criterion can be expressed as[24]
The elastic constants in Table 3 completely satisfy the above stability conditions.
Table 3.
Table 3.
Table 3.
Calculated elastic constants and compliance constants of TiB2 from the primitive cells. Availabel experimental and theoretical data are listed for comparison.
Calculated elastic constants and compliance constants of TiB2 from the primitive cells. Availabel experimental and theoretical data are listed for comparison.
.
Based on the obtained elastic and compliance constants, the bulk modulus B and shear modulus G can be calculated according to the Voigt–Reuss–Hill method.[27] The Voigt approximation assumes a uniform strain throughout a crystalline material and has[28]
The Reuss approximation is based on the assumption of a uniform stress applied on the crystalline material and has[29]
where
The Voigt and Reuss approximations represent the upper and lower bounds of the real modulus. Using energy considerations, Hill recommended that a practical estimate of the bulk and shear moduli are the arithmetic means of the two extremes[30]
Furthermore, the Young’s modulus E and Poisson ratio ν are obtained in the light of the following equations:
It is well acknowledged that hardness can be utilized as an indication of machinability of ceramics. Chen et al.[31] has given a theoretical definition for the Vicker’s hardness, which is correlated to shear modulus G and can be estimated by the following formula:
The calculated bulk modulus, shear modulus, Young’s modulus, Poisson ratio, and hardness are listed in Table 4. The calculated results in Table 4 show a good agreement with the reported theoretical and experimental data.[3,12,25,32,33] Bulk modulus B reflects the resistance to volume change, shear modulus G reflects the resistance to shape change, and the Young’s modulus E describes the material’s strain response to uniaxial stress in the direction of the stress.[8] The relatively high values of B, G, and E of TiB2 make it possess many superior mechanical properties. According to Pugh’s empirical rule,[34] the ductile/brittle behaviors of a material are closely related to the ratio of G/B. It is considered that the ductile behavior is associated with G/B < 0.5 and G/B > 0.5 means brittleness. For TiB2, the calculated results evidently indicate that G/B is larger than 0.5, implying its brittle nature. The small value of Possion’s ratio 0.13 indicates the strong covalence in TiB2. The calculated hardness is 38.2 GPa, which is a bit larger than the reported experimental datum 35 GPa.[33]
Table 4.
Table 4.
Table 4.
The bulk modulus (GPa), shear modulus (GPa), Young’s modulus (GPa), Poisson’s ratio and Vicker’s hardness (GPa) for TiB2 by means of Voigt–Reuss–Hill approximation.
The bulk modulus (GPa), shear modulus (GPa), Young’s modulus (GPa), Poisson’s ratio and Vicker’s hardness (GPa) for TiB2 by means of Voigt–Reuss–Hill approximation.
.
3.3. Theoretical tensile and compressive strength of TiB2
Figures 2(a)–2(f) show the relationships of energy and stress with the applied strain for TiB2 that is pulled in , , and ⟨0001⟩ directions respectively with full relaxations along the perpendicular axes.
Fig. 2. (a), (c), (e) Energy and (b), (d), (f) stress as a function of deformed strain for TiB2 under uniaxial tension and compression along the , , and ⟨0001⟩ directions.
It is seen from Figs. 2(a), 2(c), and 2(e) that the total energy profiles have a parabolic, convex character. With increasing elongations, the curves reach their inflexion points (, where E is the total energy and ε is the corresponding deformed strain) and become concave. The inflexion points for [], [], and [0001] uniaxial loadings occur at ε = 0.16, 0.26, 0.36 for tension and ε = −0.22, −0.16, −0.30 for compression.
For uniaxial compressive deformation, the stress τ is given by ,[19] where V(ε) is the volume at strain ε. The ideal strength of crystal refers to the first maximum or minimum point of the stress–strain curve reached during the tensile or compressive deformation, respectively. The corresponding stress and strain at this point are presented as the first maximum (minimum) stress (σFM) and the first maximum (minimum) strain (εFM), respectively.[6] The lines in Figs. 2(b), 2(d), and 2(f) formed by the red triangles represent the first derivatives of the stress–strain curve, , which could measure the sensitivity to change of the stress with respect to a change in the strain. The calculation results indicate that the behaviors of the stress–strain curves of TiB2 differ little at small strains (−0.1, −0.1) along any of the three primary directions. Monotonically increasing proportionality up to almost 25% the total strain domain can be found and indicates that the present deformation is an elastic deformation. However, the anisotropy of the stress–strain relation of TiB2 begins to manifest at larger strains, both in tension and in compression.
The ideal tensile strengths of the three principle directions of TiB2 show an increasing tendency GPa; < σ⟨0001⟩ = 47.03 GPa < GPa, at strains 0.14, 0.28, and 0.22, respectively. The ideal compressive strengths of the three principle directions of TiB2 show an increasing tendency GPa < GPa < GPa, at strains −0.16, −0.32, and −0.24, respectively. The trends for tensile and compressive strengths are just the opposite. Along the direction, the compressive strength is about 72% lower than that of the strongest direction, while its tensile strength is almost 21% higher than that of the weakest direction. This manifests a strong mechanical anisotropy in the (0001) plane.
From Figs. 2(b), 2(d), and 2(f), it can be seen that the stress–strain curves are noticeably compression–tension anisomerous for all of the three directions. Like most structural ceramics, TiB2 is considerably stronger under compression than in tension. The asymmetric ratios of the ideal compressive to tensile strengths are 5.56, 2.55, and 4.01 for the crystallographic directions , , and ⟨0001⟩, respectively. It can be expected that this compression tension asymmetry also exists in other AlB2-type transition metal diborides, such as ZrB2 and HfB2.
3.4. Electronic structure analysis
Figure 3 gives the total and partial densities of states (TDOS and PDOS) of unstrained TiB2 and TiB2 at εFM 0.14, −0.24 for tension and compression along the direction , 0.22 and −0.16 for tension and compression along the direction , and 0.28, −0.32 for tension and compression along the direction ⟨0001⟩, where the vertical line indicates the Fermi level EF. Three features are obvious in the density of states of TiB2. First, the peak at low energies (∼ −15 to −7 eV) can mainly be attributed to localized B-2s electrons. Second, around −1 eV to −5 eV, the bonding states of Ti-d and B-2p electrons can be found. It indicates a covalent interaction between them resulting from the strong hybridization. The strong hybridization of the Ti-3d and B-2p can lower the energy of the bonding states and increase that of the anti-bonding states, which produces the pseudogap near the Fermi level.[35,36] The existence of the pseudogap helps effectively resist the elastic and plastic deformations and results in high elastic moduli and ideal strengths.[37,38] Third, above the pseudogap there are antibonding states dominated by Ti.
Fig. 3. The total and site projected densities of states (TDOS and PDOS) of TiB2 at strains (a) 0.00, (b) 0.14, (c) −0.24 along the direction , (d) 0.22, (e) −0.16 along the direction , and (f) 0.28, (g) −0.32 along the direction ⟨0001⟩. The vertical line indicates the Fermi level EF.
Going from the unstrained system to TiB2 under tension and compression, the bonding peaks of Ti-3d and B-2p are moved to higher energy. In Figs. 3(g) and 3(f), EF is no longer located at the bottom of the pseudogap, and is shifted to the region implying occupation of anti-bonding states instead. The above two findings are consistent with the predicted reduced structural stability of TiB2.
4. Conclusions
We have investigated the anisotropic elastic properties, ideal compressive and tensile strength, and critical strain of TiB2 by first-principles calculations. The lattice constant c is larger than a due to the anisotropic chemical bonding intra- and inter-the boron layers. The obtained elastic constants could clearly indicate the anisotropic elasticity of the TiB2 structure. Because the covalent B–B bond is much stronger than the ionic-covalent Ti–B bond. Based on the obtained elastic and compliance constants, the bulk modulus and shear modulus were calculated according to the Voigt–Reuss–Hill method. The Young’s modulus, Poisson’ ratio, and hardness were also estimated and discussed. Our data fit the other theoretical and experimental results quite well. The inflexion points of the energy–strain curves for [, [], and [0001] uniaxial loadings occur at ε = 0.16, 0.26, 0.36 for tension and ε = −0.22, −0.16, −0.30 for compression. The ideal compressive and tensile strengths of TiB2 show opposite changing tendencies for the three primary directions. The ideal tensile strengths vary from GPa, σ⟨0001⟩ = 47.03 GPa to GPa, at strains 0.14, 0.28, and 0.22, respectively. While the ideal compressive strengths vary from GPa, |σ⟨0001⟩| = 188.75 GPa to GPa, at strains −0.16, −0.32, and −0.24, respectively. The theoretical investigation shows that the stress–strain curves of TiB2 are compression–tension asymmetric, and TiB2 is much stronger under compression than in tension. The ratios of the ideal compressive to tensile strengths are 5.56, 2.55, and 4.01 for the directions , , and ⟨0001⟩. The existence of the pseudogap in the DOS of TiB2 is the result of strong hybridization between Ti-3d and B-2p and indicates a covalent interaction between them. As for TiB2 under tension and compression, the main peaks of Ti-d and B-2p are shifted towards high energy and EF moves to the region of anti-bonding states, which manifest the reduced structural stability of TiB2.
