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Crystals 2018, 8(8), 307; doi:10.3390/cryst8080307

Article
The Mechanical Properties and Elastic Anisotropies of Cubic Ni3Al from First Principles Calculations
1
Guangdong Provincial Key Laboratory of Advanced Welding Technology, Guangdong Welding Institute (China-Ukraine E.O. Paton Institute of Welding), Guangzhou 541630, China
2
Key Laboratory of Guangxi Manufacturing System and Advanced Manufacturing Technology, Guilin University of Electronic Technology, Guilin 541004, China
*
Authors to whom correspondence should be addressed.
Received: 21 June 2018 / Accepted: 22 July 2018 / Published: 25 July 2018

Abstract

:
Ni3Al-based superalloys have excellent mechanical properties which have been widely used in civilian and military fields. In this study, the mechanical properties of the face-centred cubic structure Ni3Al were investigated by a first principles study based on density functional theory (DFT), and the generalized gradient approximation (GGA) was used as the exchange-correlation function. The bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio of Ni3Al polycrystal were calculated by Voigt-Reuss approximation method, which are in good agreement with the existing experimental values. Moreover, directional dependences of bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio of Ni3Al single crystal were explored. In addition, the thermodynamic properties (e.g., Debye temperature) of Ni3Al were investigated based on the calculated elastic constants, indicating an improved accuracy in this study, verified with a small deviation from the previous experimental value.
Keywords:
intermetallic compound; Ni3Al single crystal; first principles; mechanical property; elastic anisotropy

1. Introduction

Ni-Al alloys, especially Ni-Al single crystal alloys, have been widely applied as structural materials and functional materials in civilian and military fields, due to their high strength, high temperature stability, corrosion and oxidation resistances in aggressive environments [1,2,3,4,5]. As the most promising and valued Ni-Al alloys, L12-ordered Ni3Al-based single crystal alloys are well known as superalloys because of their excellent mechanical properties at high temperature [6], which have been extensively used for the hot components of gas turbines [7]. Ni3Al is a kind of intermetallic compound, and its elastic modulus, bulk modulus, shear modulus have been explored by experimental methods. It should be noted that, due to the variation of material preparation, processing and test methods, the mechanical properties of Ni3Al measured by experimental methods are discrete. For example, the bulk modulus of L12-ordered Ni3Al explored by Pearson et al. is 229.2 GPa [8] while the value obtained by Yasuda et al. is 171.0 GPa [9]. In order to break through the limitation derived from experimental methods, in recent years, first principles calculations have been successfully conducted to calculate the elastic properties of alloys and intermetallics [10,11,12,13,14]. Based on first principles, Wen et al. [1] calculated the bulk modulus and Young’s modulus of Ni3Al, as well as the effects of pressure on the mechanical properties. Kim et al. [15] investigated the mechanical parameters of Ni3Al, such as bulk modulus, Young’s modulus, shear modulus, and the influences of several typical doped alloying elements on them. Huang et al. [16] computed various elastic moduli and elastic constants of Ni3Al via first principles and pointed out these values coincide well with other theoretical and experimental results. In addition, Wen et al. [2] investigated the behaviour of Ni3Al by calculating the stress-strain curve and the corresponding mechanical stability. Ni3Al-based alloys have been largely applied as single crystal materials [17,18,19], while most of the previous studies based on experiments and first principles calculations neglect the anisotropic behaviour of Ni3Al single crystal, which has great influences on the abnormal crystal grains growth, microstructure transformation/formation, and the microcrack development [20,21]. So, it is of great significance to study the anisotropic mechanical properties for the reliability evaluation of Ni3Al-based alloys.
Thus far, anisotropic mechanical properties of Ni3Al single crystal alloys is not yet clear due to difficulties of accurately measuring the anisotropic mechanical properties. In the present study, the mechanical properties and elastic anisotropies, such as the anisotropy of elastic modulus, bulk modulus, shear modulus and Poisson’s ratio, were investigated by first principles calculations.

2. Computational Methods and Details

The L12-ordered Ni3Al compound has a face-centred cubic structure with a space group of Pm-3m with a = b = c = 3.572 Å and α = β = γ = 90°, wherein the atomic locations of Ni and Al atoms in an elementary cell are 1a (0.5, 0.5, 0) and 1a (0, 0, 0), respectively, as depicted in Figure 1 [22]. In this study, the ab initio density functional theory calculation was performed using the Cambridge Sequential Total Energy Package (CASTEP) program [23]. Meanwhile, the generalized gradient approximation (GGA) [24] of the revised Perdew-Burke-Ernzerhof formalism [25] and the local density approximation (LDA) [26] proposed by Ceperley and Alder were operated to calculate the exchange-correlation potential, respectively. The Vanderbilt ultra-soft pseudopotentials [27] and Broyden Fletcher Goldfarb Shanno algorithm [28,29] were used to optimise the crystal models. The cutoff energy and k point were set to be 600 eV and 10 × 10 × 10, respectively. The convergence tolerance of energy was set at 5.0 × 10−6 eV/atom. Meanwhile, the self-consistent field (SCF) convergence threshold was 5.0 × 10−7 eV/atom with a maximum atomic displacement of 5.0 × 10−4 Å. The maximum ionic Hellmann-Feynman force and maximum stress were less than 0.01 eV/Å and 0.02 GPa, respectively.

3. Results and Discussion

3.1. Lattice Constants

The optimised lattice constants and previous experimental results are listed in Table 1 for comparison. It is conspicuous that the GGA results are in agreement with the experimental values: the maximum difference of lattice constants is merely around 0.14% and that of volumes is 0.397%, demonstrating the effectiveness of the proposed simulation model. Generally, the previous and present calculated data obtained from GGA are similar, and the difference is less than 1%. However, when using LDA, the calculated lattice volume is about 7% smaller than the experimental data, which is induced by the difference between GGA and LDA. LDA assumes that the local exchange-correlation energy is same in a uniform electron gas with the same density, resulting in the underestimation of the exchange-correlation in regions of low electron density. Accordingly, LDA induces smaller lattice constants, larger cohesive energies and bulk modulus. GGA introduces a dependence of exchange-correlation function on the local gradient of the electron density and typically improves the underestimation of LDA in predicting lattice constants [30]. The present work and many other works [31,32,33] show that, compared to LDA, GGA can get a higher accuracy in predicting lattice constants and elastic properties, therefore GGA was used in the following calculations. In our research, elastic constants of Ni3Al were obtained by linear fitting using four strains of ±0.001 and ±0.003 under nine strain conditions [34].

3.2. Elastic Properties

The elastic constants are essential parameters that can correlate the microscopic properties of materials with macroscopic mechanical behaviours and then provide the information of crystal stability and stiffness. The relationship between stress and strain is directly determined by Hooke’s law σij = Cijklεij, where Cijkl represents the elastic constants. If Hooke’s law is applied to the lattice dynamics, combining the lattice symmetry, thus the stress-strain matrix of Ni3Al single crystal can be written in the following form.
( σ 1 σ 2 σ 3 τ 1 τ 2 τ 3 ) = ( C 11 C 12 C 13 0 0 0 C 22 C 23 0 0 0 C 33 0 0 0 C 44 0 0 S y m . C 55 0 C 66 ) ( ε 1 ε 2 ε 3 γ 1 γ 2 γ 3 )
where σi, τi, εi, γi are the normal stress, shearing stress, the corresponding normal strain and shearing strain, respectively. The elastic flexibility matrix Sij can be written as the inverse matrix of the elastic stiffness matrix Cij, i.e., [Sij] = [Cij]−1. As for cubic crystal, because of lattice symmetry, there are three independent variables in Cij and Sij, respectively: C11, C12, C44 and S11, S12, S44. In this study, the elastic constants calculated based on the cubic Ni3Al crystal structure are tabulated in Table 2, and the calculated values of S11, S12 and S44 in this paper are 0.009, −0.004 and 0.008, respectively.
The calculated outcomes of first principles (C11 = 240.1 GPa, C12 = 160.0 GPa, C44 = 123.8 GPa) are consistent with previous calculations and experimental values (C11 = 224.3 GPa, C12 = 148.6 GPa, C44 = 125.8 GPa). The elastic constants calculated for the cubic crystal should satisfy the following mechanical stability criteria [38]:
C 11 C 12 > 0 , C 11 > 0 , C 44 > 0 , C 11 + 2 C 12 > 0
By substituting the three calculated elastic constants into the above equation, we found that the present results match the above criteria obviously, indicating that Ni3Al single crystal is an intrinsically stable system. Moreover, the bulk modulus, shear modulus and elastic modulus were calculated using the Voigt-Reuss approximation [39]. The upper limit and lower limit of the actual effective modulus correspond to Voigt bound and Reuss bound obtained by the average polycrystalline modulus based on two assumptions: uniform strain and uniform stress throughout a polycrystal, respectively [30]. For cubic lattices, Voigt bulk modulus (BV) and shear modulus (GV) are
B V = 1 3 ( C 11 + 2 C 12 )
G V = 1 5 ( C 11 C 12 + 3 C 44 )
and the Reuss bulk modulus (BR) and Reuss shear modulus (GR) are defined as
B R = 1 3 S 11 + 6 S 12
G R = 15 4 S 11 4 S 12 + 3 S 44
Voigt and Reuss equations represent upper and lower limits of the true polycrystalline constants, and the practical estimate of the bulk and shear moduli can be treated as the arithmetic mean of the two extremes [40]. The Hill’s average for the shear modulus (G) and bulk modulus (B) is given by
B = 1 2 ( B R + B V )
G = 1 2 ( G R + G V )
while Young’s modulus (E) and Poisson’s ratio (v) are given by
E = 9 B G 3 B + G
ν = 3 B E 6 B
the bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio calculated in this study are listed in Table 3. The results calculated in this paper are very close to previously calculated values and experimental values measured by Prikhodko et al. [41], which can effectively verify the reliability of the calculation results. In addition, either the ductility or the brittleness of metallic material can affect the mechanical properties directly, and determine the failure mode. Pugh et al. [42] introduced the ratio of bulk modulus to shear modulus B/G as a reference for the judgments of the ductility of a material. If the value of B/G is large, it means that the material has a good ductility, otherwise it will be characterised by prominent brittleness. For a normal material, if its B/G value exceeds 1.75, indicating it is a ductile material, otherwise it is brittle [30]. According to the data given in Table 3, the B/G value of Ni3Al alloy is 2.368, which is obviously higher than the critical value 1.75, demonstrating that Ni3Al is a ductile material which agrees with the experimental result reported in a previous study [43].
Further, the Vicker’s hardness Hv is another important mechanical property of materials, which can be predicted by Equation (10) [44].
Hv = 0.92(B/G)1.3137 G0.708
It is reported that materials with Vickers hardness larger than 40 GPa can be classified as superhard materials [45]. For the calculated value of Hv of Ni3Al is 54.004 GPa, it is supposed that N3Al and N3Al-based materials have an excellent ability to resist being scratched or dented.

3.3. Elastic Anisotropy

The elastic anisotropy can be described by the universal anisotropic index AU [46] and the percent anisotropy indexes of compression and shear (AB and AG) [47,48], respectively. The calculation formulas are expressed as follows
A U = 5 G V G R + B V B R 6
A B = B V B R B V + B R
A G = G V G R G V + G R
for the three indexes, the value of zero indicates elastic isotropy while the variation from zero means anisotropic elastic properties. After calculation, it is found that the AU value of Ni3Al is 1.670, indicating that Ni3Al exhibits anisotropic behaviour. The value 0.000 for AB indicates the isotropic compression of Ni3Al, while the AG value of 0.145 shows that Ni3Al exhibits shear anisotropy. As a measure of the anisotropy degree for the bondings between atoms in different crystal planes, the shear anisotropic factors are used. The shear anisotropic factor A1 for {100} planes is written as [48]
A 1 = 4 C 44 C 22 + C 33 2 C 13
shear anisotropic factors A2 and A3 for the {010} and {001} planes are given as Equations (15) and (16), respectively,
A 2 = 4 C 55 C 33 + C 11 2 C 23
A 3 = 4 C 66 C 11 + C 22 2 C 12
For the case of isotropic crystals, the value of shear anisotropic factor is 1, and the anisotropy is reflected in the deviation from 1 [30]. In this calculation, the three values are all 3.093 (see Table 4), indicating that the Ni3Al crystals exhibit shear anisotropy in the {100}, {010} and {001} planes evidently. In addition, the anisotropic indexes of the bulk modulus along the a axis and c axis with respect to the b axis can be written as Equations (17) and (18) [48], respectively.
ABa = α
ABc = β
where
α = ( C 11 C 12 ) ( C 33 C 13 ) ( C 23 C 13 ) ( C 11 C 13 ) ( C 33 C 13 ) ( C 22 C 12 ) ( C 13 C 23 ) ( C 12 C 23 )
β = ( C 22 C 12 ) ( C 11 C 13 ) ( C 11 C 12 ) ( C 23 C 12 ) ( C 22 C 12 ) ( C 33 C 13 ) ( C 12 C 23 ) ( C 13 C 23 )
For these parameters, the value of 1 indicates elastic isotropy, and similarly any deviation from 1 corresponds to a degree of elastic anisotropy. From the calculation result of the elastic constant, it can be observed that the bulk modulus anisotropy index ABa = ABc = 1. Therefore, the calculations show once again that Ni3Al exhibit isotropic bulk modulus.
To further investigate the anisotropic features of Ni3Al crystal, the three-dimensional surface representation of the elastic anisotropy of the crystal was performed. The direction dependences of the Bulk modulus and Young’s modulus of cubic crystal system are shown in the following Equations (21) and (22), respectively [49]
1 B = ( S 11 + 2 S 12 ) ( l 1 2 + l 2 2 + l 3 2 )
1 E = S 11 2 ( S 11 S 12 S 44 2 ) ( l 1 2 l 2 2 + l 2 2 l 3 2 + l 1 2 l 3 2 )
where 11, 12, and 13 denote the direction cosines with respect to the a, b, and c directions of the lattice. As mentioned before, for Ni3Al, the bulk modulus B has no anisotropy and its spatial three-dimensional (3D) surface representation is spherical, as shown in Figure 2a. For Young’s modulus, its 3D surface representation is shown in Figure 2b, which exhibits obvious anisotropy.
In order to better understand the anisotropic characteristics, the Young’s modulus of the Ni3Al single crystal in the normal direction of the three low-index crystal planes {100}, {110} and {111} were calculated. The formula can be written as follows [50]
1 E h k l = S 11 2 S 0 ( h k ) 2 + ( h l ) 2 + ( l k ) 2 ( h 2 + k 2 + l 2 ) 2
where
S 0 = S 11 S 12 1 2 S 44
The corresponding calculation results are listed in Table 5. The maximum and minimum values of Young’s moduli are 304.241 GPa and 112.094 GPa along the normal direction of {111} and {100} planes, respectively, which are consistent with the results shown in Figure 2b.
In addition, the three-dimensional surface of the shear modulus of Ni3Al can be expressed by Equation (25) [51]:
1 G = ( S 44 + 4 S 0 J )
where
J = sin 2 θ cos 2 θ + 0.125 sin 4 θ ( 1 cos 4 φ )
wherein θ and ϕ are Euler angles, as shown in Figure 3. Figure 2c shows the shear moduli of Ni3Al along different orientations. Clearly, shear modulus anisotropy can be deduced, with the maximum and minimum values are 123.831 GPa along <001> directions and 51.696 GPa along <111> directions, respectively. Generally, the direction dependence of shear modulus is opposite to that of Young’s modulus.
Further, anisotropies of Poisson’s ratios were also investigated by calculating the values along three lower-index planes. As for cubic crystal, the Poisson’s ratio ν along arbitrary directions in (hkl) planes can be expressed as [50].
ν ( h k l , θ ) = { S 12 + S 0 h 2 + k 2 + l 2 [ ( h 2 l h 2 + k 2 h 2 + k 2 + l 2 cos θ h k h 2 + k 2 sin θ ) 2 + ( k 2 l h 2 + k 2 h 2 + k 2 + l 2 × cos θ + h k h 2 + k 2 sin θ ) 2 + ( l h 2 + k 2 h 2 + k 2 + l 2 cos θ ) 2 ] } / [ S 11 + 2 S 0 ( h k ) 2 + ( h l ) 2 + ( l k ) 2 ( h 2 + k 2 + l 2 ) 2 ]
Variations of Poisson’s ratios in two low index planes, (100) and (111) are shown in Figure 4a,b, where Poisson’s ratios are 0.400 and 0.228 correspondently, with no direction dependences. However, ν in (110) plane exhibits conspicuous anisotropic behaviour, see Figure 4c, where the maximum value 0.76 appears along [001] and [ 00 1 ¯ ] directions, while the minimum value is −0.14 along the orientations of [ 1 ¯ 10 ] and [ 1 1 ¯ 0 ]. Poisson’s ratios along two orthogonal symmetric directions in three low index planes of Ni3Al are listed in Table 6.

3.4. Thermodynamic Properties

As a fundamental parameter, the Debye temperature (θD) correlates with many physical properties of solids, such as specific heat, elastic constant and melting temperature [48]. One of the standard methods to calculate the Debye temperature can be estimated from the averaged sound velocity (vm) by Equation (28) [52]
θ D = k B ( 3 n N A ρ 4 π M ) 1 / 3 v m
where ћ is Planck’s constant, kB is Boltzmann’s constant, NA is Avogadro’s number, ρ is the density, M is the molecular weight and n is the number of atoms in a molecule. The average wave velocity vm is approximately given by [52]
v m = [ 1 3 ( 1 v l 3 + 2 v t 3 ) ] 1 / 3
v l = ( 3 B + 4 G 3 ρ ) 1 / 2
v t = ( G ρ ) 1 / 2
among them, vl and vt are the longitudinal and transverse elastic wave velocities of the polycrystalline material that are determined by bulk modulus (B) and shear modulus (G). Based on the calculated elastic properties, the results are listed in Table 7 accordingly. It can be pointed out that the calculated Debye temperature is 475.734 K, which is similar to the experimental value 460 K [53], approving the accuracy of the calculation results in this study.

4. Conclusions

First principles density functional calculations were applied to extensively explore the mechanical properties and elastic anisotropies of cubic Ni3Al single crystal. The calculated lattice constants are found to be in good agreement with experimental data, revealing the effectiveness of the proposed theoretical models. The calculated bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio of polycrystalline Ni3Al are 186.724 GPa, 207.378 GPa, 78.857 GPa and 0.315 respectively, which are consistent with experimental values. The calculated B/G ratio implies that Ni3Al single crystal is a ductile material. Anisotropies of mechanical properties were studied by computing varied anisotropic indices and directional dependences of bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio of monocrystalline Ni3Al. There is no anisotropy for bulk modulus of Ni3Al, while the Young’s modulus exhibits evident anisotropy, with the largest value 304.241 GPa along the normals of {111} planes and the minimum value 112.094 GPa along the {100} planes nomals. In addition, the direction dependence of shear modulus is opposite to that of Young’s modulus, and the shear modulus takes the maximum value of 123.831 GPa in the <100> directions while the minimum value 51.696 GPa is assigned along the <111> directions. Poisson’s ratio shows isotropy on the (100) and (111) planes respectively. In (110) plane, however, Poisson’s ratio depends strongly on the direction, with the minimum value of only 0.140, while the maximum value is 0.760. Furthermore, the calculated Debye temperature based on the calculated elastic constants is 475.734 K, which is close to the experimental value 460K; thus the accuracy of the calculation results is further approved.

Author Contributions

H.Q. and F.L. conceived and designed the research; H.Q., X.L., F.L. and Q.L. performed the first principles calculation; X.L. wrote the manuscript; Y.Y. and Z.D. reviewed and edited the manuscript. All authors read and approved the final manuscript.

Funding

This research was funded by the GDAS’ Project of Science and Technology Development (No. 2018GDASCX-1005), Key Program for International Cooperation of Science and Technology (No. 2015DFR50310), National Natural Science Foundation of China (No. 51505095) and Science and Technology Plan Project Public Welfare Fund and Ability Construction Project of Guangdong province (No. 2017A070701026).

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Crystal structure of cubic Ni3Al.
Figure 1. Crystal structure of cubic Ni3Al.
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Figure 2. Directional dependence of bulk modulus (a), Young’s modulus (b) and shear modulus (c).
Figure 2. Directional dependence of bulk modulus (a), Young’s modulus (b) and shear modulus (c).
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Figure 3. Coordinates system and Eulerian angles.
Figure 3. Coordinates system and Eulerian angles.
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Figure 4. Curves of direction dependences of Poisson’s ratios for typical low index planes (100) (a), (111) (b) and (110) (c).
Figure 4. Curves of direction dependences of Poisson’s ratios for typical low index planes (100) (a), (111) (b) and (110) (c).
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Table 1. Calculated and experimental lattice parameters of Ni3Al.
Table 1. Calculated and experimental lattice parameters of Ni3Al.
Lattice Parametersa(Å)b(Å)c(Å)V3)ΔV/Ve, %
Present work (GGA)3.5773.5773.57745.757+0.397
Present work (LDA)3.4863.4863.48642.377−7.019
GGA [16]3.5693.5693.56945.461−0.252
GGA [35]3.5613.5613.56145.156+0.922
LDA [35]3.4863.4863.48642.363−7.050
Experiment [22]3.5723.5723.57245.576--
Table 2. Calculated and experimental elastic constants for Ni3Al single crystal.
Table 2. Calculated and experimental elastic constants for Ni3Al single crystal.
Elastic ConstantC11C12C44
Present work240.104160.034123.831
GGA [6]225.3157.6121.1
GGA [36]232.7154.5123.0
GGA [35]230.31162.51124.79
Experiment [37]224.3148.6125.8
Table 3. Calculated elastic properties of Ni3Al.
Table 3. Calculated elastic properties of Ni3Al.
BV, GPaBR, GPaGV, GPaGR, GPaB, GPaG, GPaE, GPavB/GHv
Present Work186.724186.72490.31367.401186.72478.857207.3780.3152.36854.004
GGA [6]--------180.272.9--------
GGA [36]--------180.677.8204.0------
GGA [35]--------184.4973.05196.650.320----
Exp. [41]--------173.977.8203.10.3052.235--
Table 4. The calculated anisotropic index of Ni3Al.
Table 4. The calculated anisotropic index of Ni3Al.
Anisotropic IndexAUABAGA1A2A3ABaABc
Value1.6700.0000.1453.0933.0933.0931.0001.000
Table 5. Values of Young’s moduli along normal directions of planes {100}, {110} and {111}.
Table 5. Values of Young’s moduli along normal directions of planes {100}, {110} and {111}.
Planes{100}{110}{111}
Ehkl, GPa112.094212.973304.241
Table 6. Poisson’s ratios along two orthogonal symmetric directions in three low index planes of Ni3Al.
Table 6. Poisson’s ratios along two orthogonal symmetric directions in three low index planes of Ni3Al.
Plane(100)(110)(111)
Direction[ 010 ][ 00 1 ¯ ][ 1 ¯ 10 ][ 00 1 ¯ ][ 1 ¯ 10 ][ 11 2 ¯ ]
Poisson’s ratio0.4000.400−0.1400.7600.2280.228
Table 7. The calculated thermodynamic properties of Ni3Al at zero pressure.
Table 7. The calculated thermodynamic properties of Ni3Al at zero pressure.
Propertyρ, g/cm3vt, m/svl, m/svm, m/sθD, K
Value7.3713270.8096292.5593660.396475.734

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