- Email: [email protected]
First-principles Study of NiAl Alloyed with Rare Earth Element Ce
J. Mater. Sci. Technol., 2011, 27(8), 719-724.
First-principles Study of NiAl Alloyed with Rare Earth Element Ce You Wang† , Junqi He, Mufu Yan, Chonggui Li, Liang Wang and Ye Zhou Department of Materials Science, School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China [Manuscript received February 24, 2010, in revised form April 27, 2010]
The structural, elastic, and electronic properties of NiAl alloyed with rare earth element (REE) Ce have been investigated by using density functional theory (DFT). It is found that Ce has a strong Al site preference and causes lattice distortion of NiAl. The calculation of elastic constants shows that Ce increased both the hardness and the ductility of NiAl, which could be explained by the formation of new ionic bonds between Al (and Ni) and Ce and the enhancement of covalent bonds in Ni8 Al7 Ce. Our results are in good agreement with the available experimental data and other theoretical results. KEY WORDS: Point defects; Electronic states; First-principles
1. Introduction The intermetallic compound NiAl is one of the most promising engineering materials with attractive properties including good high-temperature strength, resistance to oxidation and corrosion, relatively low material density and high melting point. However, the application of NiAl is restricted as aerospace materials for its brittle fracture and low tensile ductility at room temperature (RM)[1] . Many efforts have been made to improve the ductility of NiAl[2,3] since Aoki and Lzumi[4] discovered that Ni3 Al added boron exhibits high ductility at room temperature. Among these efforts, alloying is one of the most efficient methods to improve the quality of NiAl at RM. Alloying elements which form 3d and 4d transition series have been studied extensively by both experimental and theoretical methods[5–7] . While rare earth elements (REEs) as alloying elements are more attractive and have had great effect on improvement of materials. For example, Guo et al.[8] have shown that the 0.05 wt% Ce-doped NiAl-Cr alloy attained † Corresponding author. Prof., Ph.D.; Tel.: +86 451 82921251; E-mail address: [email protected] (Y. Wang).
the highest compressive ductility of 14.8% at RT, while the REEs-free alloy only possessed 9% compressive ductility. The experimental results in literature [9] showed that adding 0.025 at.% Ce to NiAl (Ni/Al ratio=1.14) has increased the compressive plastic strain to more than 4, and increased the strength simultaneously. Wang et al.[10] have proved that the hardness of NiAl coating with 2 wt% CeO2 is 764.9 HV100g , while the hardness for NiAl coating without CeO2 is only 530 HV100g . This work also proved that the Young s elastic modulus and the η ratio (the ratio of the recoverable deformation energy to the total deformation energy) of NiAl coating with 2 wt% CeO2 are 179.7 GPa and 33%, respectively, while Young s elastic modulus and the η ratio of NiAl coating without CeO2 are 150 GPa and 28%, respectively. These results indicated that alloying element Ce plays an important role in improving the hardness and ductility of NiAl. However, the research on alloying effect of Ce in NiAl is still limited. Moreover, the studies of Ce addition on the atomic level are not sufficient, and the effect of Ce on the ductility, especially the micromechanism of the interaction between NiAl and the alloyed element Ce is not clear. In this paper, first-principles approach is used to calculate the structural, elastic,
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
720
and electronic properties of NiAl alloyed with Ce in order to illuminate the intrinsic effect of Ce. Though the amount of Ce in this study is 6.25 at.% and is larger than the experimental requirement (less than 3 at.%), it can still provide fundamental understanding of alloying effects of Ce on atomic and even on electronic levels as a complement of experimental results. 2. Computational Method Density functional theory (DFT) with ultrasoft pseudopotentials[11] is employed in the CASTEP code (Cambridge Sequential Total Energy Package), which utilizes plane-wave pseudopotential to perform firstprinciples quantum mechanics calculations. Generalized gradient approximation (GGA) as described by Perdew et al.[12] is adopted as exchange-correlation functionals for all elements in our models. This program is employed to investigate the lattice parameters, elastic characteristics as well as electronic structures of the NiAl 2×2×2 supercell with addition of REEs. The unit cell of NiAl with the space group PM-3M and the lattice constant of 0.29 nm is established, which is composed of one Ni atom at (1/2, 1/2, 1/2) and one Al atom at (0, 0, 0), respectively. The values of kinetic energy cutoff Ecut and the k points number are increased until the calculated energy converges within the required tolerance, where Ecut determines number of plane waves and k points does the sampling of the irreducible wedge of the Brillouin zone. Ecut is set to 350 eV and the k point is set to 4×4×4 with a regular Monkhorst-Pack scheme. In the calculation of self-consistent field (SCF), the Pulay scheme of density mixing[13] is adopted. The calculation of the elastic constants and the electronic structure are followed by cell optimization with a convergence tolerance of energy of 5.0×10−6 eV·atom−1 , a maximum displacement of 5.0×10−5 nm and a maximum force of 0.1 eV·nm−1 . A supercell model of NiAl with REE Ce, consists of 16 atoms, as shown in Fig. 1. 3. Results and Discussion 3.1 Site preference and lattice parameter of Ce in NiAl To understand the alloying effects of Ce requires the knowledge of site distributions of ternary elements M . Formation enthalpy is used to choose the more stable site that REE M occupied (Ni site or Al site) in NiAl. The site preference energy is described as Esite = E1 − E2
(1)
where E1 and E2 are the formation enthalpies of Ni7 Al8 M and Ni8 Al7 M , respectively. The usual definition of formation enthalpy is the total energy difference between compound and constituent elements in the solid state[14] :
Fig. 1 16-atom supercell used in this work. The cubic supercell is built up from 8 unit cells of NiAl. The rare earth atom Ce is placed at the centre of the supercell, substituting for Al, which is supercell Ni8 Al7 Ce; we also use a similar supercell where the rare earth alloying atom is placed at a Ni site, which is supercell Ni7 Al8 Ce
E1 = ENi7 Al8 M − 7ENi − 8EAl − EM
(2)
E2 = ENi8 Al7 M − 8ENi − 7EAl − EM
(3)
where ENi , EAl and EM are the total energies of pure Ni, Al and M , respectively. So we can deduce the form of Esite from the above two formation energies: Esite = E1 − E2 = ENi7 Al8 M − ENi8 Al7 M + ENi − EAl (4) If Esite <0, the substitution for Ni is more favorable, and if Esite >0, M is on the Al site. Facecentered cubic (fcc) Al and Ni are calculated, and the total energies of pure Ni and Al are −1348.9034 and −52.6996 eV, respectively. The calculated total energy of ENi7 Al8 Ce and ENi8 Al7 Ce are −10997.9529 and −12299.9075 eV, respectively, and the Esite is 5.7508 eV. So, Ce tends to substitute for Al site. The formation enthalpy of Ni8 Al8 , Ni8 Al7 Ce and Ni7 Al8 Ce are also calculated by Eqs. (2) and (3) (the total energies of pure Ce is −1056.5083 eV), so the formation enthalpy of the three compounds are −84.8337, −83.0893 and −77.312 eV, respectively. Though the formation enthalpy of the latter two compounds is a little larger than Ni8 Al8 , they are supposed to exist stably because of their negative formation enthalpy. Therefore, the models Ni8 Al7 Ce is used for the following calculation. The relaxed lattice parameter of a structure is the one which causes the energy to be minimized. The lattice parameters of Ni8 Al8 and Ni8 Al7 Ce are shown in Table 1. The equilibrium lattice constant of Ni8 Al8 is set to 0.58 nm, according to the experimental results[15,16] . In Table 1, the lattice parameter of Ni8 Al8 increased 3.28% after alloying with Ce.
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
721
Table 1 Equilibrium lattice constants and its change in Ni8 Al8 and Ni8 Al7 Ce [δ=(a-5.8)/5.8] Model Ni8 Al8 Ni8 Al7 Ce
a/nm 0.580 0.599
δ/% 0 3.28
3.2. Elastic properties It is known that the elastic moduli of materials can be used to assess certain mechanical properties[17,18] , such as ductility, hardness and strength. Those macroscopic parameters are related to the microscopic elastic constants. However, to measure the elastic constants of materials with defects is difficult and demands testing facilities with high accuracy. Here, theoretical calculation based on first principles is on prior consideration. The independent elastic constants in the cubic phase are C11 , C12 and C44 , according to Hooke s law. C11 represents the ratio of stress along <100> direction to strain along <100> direction; C12 represents the ratio of stress along <100> direction to strain along <010> direction; C44 represents the ratio of stress along <110> direction to strain along <110> direction. From C11 , C12 and C44 , we could receive bulk modulus B0 , shear modulus G, Young’s modulus E and the Poisson s ratio ν [19] : B0 = (C11 + 2C12 )/3
(5)
G = (3C44 + C11 − C12 )/5
(6)
E = 9B0 G/(3B0 + G)
(7)
ν = (1 − E/3B0 )/2
(8)
The elastic constants C11 , C12 , C44 and the bulk modulus B0 are shown in Table 2. There is a good agreement between our results and the experimental or theoretical ones[20,21] . The mechanical stability criteria in cubic crystal are: C11 − C12 >0; C44 >0; B0 >0[22,23] . So, the calculated elastic constants show that Ni8 Al7 Ce is stable. In Fig. 2, the negative Cauchy pressure −(C12 −C44 ), G/B0 , C12 and 1/v are calculated and showed together with other elastic constants in order to show the alloying effect of Ce on the properties of Ni8 Al8 . Figure 2 (a) are the histograms of Young s modulus E, shear modulus G and elastic constants C11 . It is generally believed that the hardness of materials can be related to their elastic modulus, such as
Fig. 2 Histograms of (a) Young s modulus E, shear modulus G and the elastic constants C11 , (b) the ratios of shear modulus to bulk modulus G/B0 , negative Cauchy pressure parameters –(C12 −C44 ) and (c) the reciprocal of Poisson s ratio 1/v bulk modulus B0 , elastic constants C44 and C12 of Ni8 Al8 and Ni8 Al7 Ce
the Young s modulus E, and the shear modulus G[24] . Although the relationships between hardness and the modulus are not identical for different materials, the general trend is, the larger the modulus, the harder the material. Therefore, the calculated Young s modulus and shear modulus can be used as a general guidance for selection of materials. From Fig. 2(a), it can be seen that E, G and C11 follow the similar trend, and in C11 , the difference between Ni8 Al8 and Ni8 Al7 Ce is larger than that in E and G. So alloying with Ce enhances the hardness of NiAl on the whole.
Table 2 Elastic constants C11 , C12 , C44 and the bulk modulus B0 of all supercells Model C11 /GPa C12 /GPa C44 /GPa C11 − C12 Ni8 Al8 Cal. 161.11 149.71 116.78 11.4 199 137 116 62 Ni8 Al8 Exp 204.6 135.4 116.8 69.2 Ni8 Al8 Exp 173.48 149.37 117.41 24.21 Ni8 Al8 295.56 127.85 74.43 167.71 Ni8 Al7 Ce Note: Cal/Exp: results from other calculations/experiments
B0 /GPa 153.18 158.7 158.5 157.41 183.75
722
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
Fig. 3 DOS of NiAl supercell (a) and Ni8 Al7 Ce (b)
Figure 2(b) shows the ratios of shear modulus to bulk modulus G/B0 , negative Cauchy pressure −(C12 −C44 ) and the reciprocal of Poisson s ratio 1/v. In order to predict the brittle and ductile behavior of solids, Pugh[25] has proposed a simple relationship that links empirically the plastic properties of metals with their elastic modulus by G/B0 . If G/B0 <0.5. The material behaves in a ductile manner, otherwise, behaves in a brittle manner. In Fig. 2(b), both the values of Ni8 Al8 and Ni8 Al7 Ce are smaller than 0.5. The smaller the G/B0 value, the better the ductility of NiAl. Cauchy pressure as an evaluation norm for ductile and brittle, generally, for ductile material is positive; and brittle material is negative. Figure 2(b) shows negative Cauchy pressure for easy comparison, smaller negative Cauchy pressure represents more ductile materials. Poisson s ratio reflects the stability of a crystal against shear. This ratio can formally take values between –1 and 0.5, and the smaller Poisson s ratio leads to more brittle behavior of materials. In other words, the smaller the reciprocal of Poisson s ratio 1/v, the more ductile the materials. Therefore, in Fig. 2(b), comparing Ni8 Al8 with Ni8 Al7 Ce, we can get the information that the rare earth alloying element Ce improves the ductility of NiAl. Figure 2(c) presents the bulk modulus B0 , elastic constants C44 and C12 of Ni8 Al8 and Ni8 Al7 Ce. The bulk modulus B0 represents the resistance to volume change and is related to over all atomic binding properties in a material. The bulk modulus B0 indi-
cates that the bond strength among atoms increased after adding Ce into NiAl, which is coincident with the electronic results in the following section 3.4. The shear modulus G is an average of single crystal elastic constants according to the Voigt approximation G=(C11 −C12 )/5+3C44 /5 for a unit cell with cubic symmetry. As mentioned above, shear modulus G is a good hardness predictor for a wide variety of covalent materials. C12 and C44 can be used to evaluate the hardness in the direction of <010> and <110>. Comparing C12 , C44 with G, we can get the conclusion that Ce can increase the hardness of NiAl on the whole, however, the hardness along the direction of <010> and <110> are decreased. 3.3 Density of states (DOS) and charge density contours The DOS as well as the charge density difference of Ni8 Al8 and Ni8 Al7 Ce are compared and shown in Figs. 3 and 4. The dominant feature of the Ni8 Al8 DOS is the existence of the sharp peaks contributed by d orbitals of Ni, which hybridizes weakly with the other orbitals. Placing the Ce atom at the Al site in Ni8 Al7 Ce adds two sharp peaks around –18 eV and –35 eV, which is mainly the contribution of s and p orbitals of Ce. This increases the height of the peak of the Ni-d state. There is a pseudogap, which presents stronger covalent bonding in Ni8 Al7 Ce compared to Ni8 Al8 , probably due to the partly filled 4f orbitals of Ce. Therefore alloyed with Ce, the system is relatively more stable.
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
723
Table 3 Charge populations of unit bond length before and after alloying with Ce Bond Population Length/nm P /(charge/nm) Al 1–Ni 1 0.14 0.251147 0.557 Al 2–Ni 1 0.14 0.251147 0.557 Al 4–Ni 1 0.14 0.251147 0.557 Al 1–Al 2 0.22 0.290000 0.759 Without Ce Al 2–Al 4 0.22 0.290000 0.759 Ni 1–Ni 2 −0.03 0.290000 −0.103 Al 4–Al 8 (Ce) 0.22 0.290000 0.759 Ni 1–Al 8 (Ce) 0.14 0.251147 0.557 Al 1–Ni 1 0.15 0.245999 0.610 Al 2–Ni 1 0.25 0.255252 0.979 Al 4–Ni 1 0.29 0.264181 1.098 Al 1–Al 2 0.20 0.299536 0.668 With Ce Al 2–Al 4 0.25 0.299539 0.835 Ni 1–Ni 2 0.13 0.284053 0.458 Al 4–Ce −1.03 0.299540 −3.439 Ni 1–Ce −1.16 0.272817 −4.252 Notes: Al 4, Al 6 and Al 7 are the first nearest neighbor of Ce; Al 2, Al 3 and Al 5 are the second nearest neighbor of Ce; Al 1 is the third nearest neighbor of Ce Table 4 Nanohardness, elastic modulus and elastic recoverability of NiAl with and without alloying elements Samples NiAl NiAl1CeO2 NiAl2CeO2
Ce/at.% 0 0.25 1.15
Nanohardness/GPa 9.471 9.777 10.023
Elastic modulus/GPa 193.78 194.64 194.90
Elastic recoverability/% 20.79 20.80 21.45
Fig. 4 Charge density difference of the center atom of (001) plane before (a) and after (b) placing Ce in Ni8 Al8 . Atoms at the site of Ni 2, Ni 4, Ni 6 and Ni 8 are on the upper layer of the (001) plane, and Ce is to replacing Al 8 which is under the site of Al 9
Figure 4 shows the charge density difference of the center atom of (001) plane along <001> direction. Figure 4 (a) and (b) are the charge density difference contours of Ni8 Al8 before and after being alloyed with Ce. The blue area represents the depletion (decrease) of electronic charge while the red area represents the accumulation (increase) of electronic charge. Contours start from ±0.2, and increase successively by a factor of 0.025. Comparing with the two figures, we can see that, after replacing Al with Ce, the bonding
charge accumulation at the Ce site is along the Ce–Ni directions (Ni atoms are the nearest neighbors of Ce), and the bonding charge depletion is along the nearest neighbor Al (Ce–Al directions). 3.4 Population analysis The bond populations indicate the overlap degree of the electron cloud of two bonding atoms and can be used to access covalent or ionic nature of a chemical bond. For the bond populations, the lowest and
724
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
highest values imply that the chemical bonds exhibit strong ionicity and covalency, respectively. In Table 3, the bond population between Ni and Al are increased, and Al 4–Ni 1 shows the highest value of bond population of unit bond length. The values of Al 2–Al 4, Ni 1–Ni 2 are also increased, which means the covalent effect in Ni8 Al8 is increased and the ionic effect is relatively decreased after alloying with Ce. The population values of Ni 1 (FNN—first nearest neighbor of Ce)–Ce and Al 4 (FNN)–Ce are relatively low, so the bonds Ce–Ni (FNN) and Ce–Al (FNN) show great ionicity, though they are instability; the ionicity tendency is evident, and the bond of Ce–Ni (FNN) is more stronger than Ce–Al (FNN), which consists with the result of Fig. 4(b). 4. Additional Experimental Results To further verify the former calculation results, we do the nanoindentation test on NiAl, which is doped by CeO2 . In this investigation, NiAl powders were mixed with two different amounts of CeO2 (shown in Table 4), and were fabricated by spark-plasma sintering (SPS) at 1250◦ C for a holding time of 5 min. Nanoindentation test was performed on the polished surface of three specimens: NiAl, NiAl1CeO2 (NiAl alloyed with 1.15 at.% Ce), NiAl2CeO2 (the amount of Ce is 0.25 at.%). The nanoindentation tester (TriboIndenter, Hysitron, USA) was equipped with a sharp Berkovich pyramid indenter. In the tests, maximal indentation force was 120 mN, and loading time and unloading time were 8 and 80 s, respectively. This study aims to analyze the different nanomechanical properties (i.e. elastic modulus, nanohardness and elastic recoverability) of NiAl with and without Ce. According to literature [26], the elastic recoverability was evaluated by the ratio of recoverable deformation energy to the total deformation energy, throughout loading and unloading cycles. Table 4 shows the nanohardness, elastic modulus and elastic recoverability of samples. From Table 4 we can get the information that with increasing the amount of Ce, the nanohardness the elastic modulus and the elastic recoverability all increased; NiAl with 1.15 at.% Ce shows the highest increase. In other words, Ce increases both the hardness and the ductility of NiAl. These results are coincident with our calculation results above. 5. Conclusions Ab initio density functional theory calculations have been performed to study the effects of rare earth alloying element Ce on the crystal structural, elastic, and electronic properties of NiAl. In general, the following conclusions are drawn: (1) Ce has a strong site preference of Al site and increases the lattice parameters of Ni8 Al8 . (2) Ce increases the hardness and ductility of NiAl simultaneously.
(3) Alloying with Ce enhances covalent bonds in Ni8 Al7 Ce. (4) The bonds between Ce and its first nearest neighbor Ni and Al show great ionicity.
Acknowledgement This work was financially supported by the National Natural Science Foundation of China under grant No. 50871035. REFERENCES [1 ] C.T. Liu, J. Stringer and J.N. Mundy: Intermetallics, 1997, 5, 579. [2 ] A. Albiter, M. Salazar, E. Bedolla, R.A.L. Drewc and R. Perez: Mater. Sci. Eng. A, 2003, 347, 154. [3 ] C.Y. Chung, C.Y. Xie and T.Y. Hsu: Scripta Mater., 1997, 37, 99. [4 ] K. Aoki and O. Lzumi: J. Jpn. Inst. Met., 1979, 43, 1190. [5 ] A.I. Kovalev, R.A. Barskaya and D.L. Wainstein: Surf. Sci., 2003, 532-535, 35. [6 ] P. Lazar and R. Podloucky: Phys. Rev. B, 2006, 73, 1. [7 ] C.L. Fu, C.T. Liu and X.L. Wang: Intermetallics, 2004, 12, 911. [8 ] J.T. Guo, K.W. Huai, Q. Gao, W.L. Ren and G.S. Li: Intermetallics, 2007, 15, 727. [9 ] W.X. Chen, J.R. Hines and Y. Wang: Adv. Eng. Mater., 2004, 6, 876. [10] Y. Wang, Z. Wang, Y. Yang and W. Chen: Intermetallics, 2008, 16, 682. [11] D. Vanderbilt: Phys. Rev. B, 1990, 41, 7892. [12] J.P. Perdew, J.A. Chevary and S.H. Vosko: Phys. Rev. B, 1992, 46, 6671. [13] B. Hammer, L.B. Hansen and J.K. Nørskov: Phys. Rev. B, 1999, 59, 7413. [14] W. Wolf, R. Podloucky, P. Rogl and H. Erschbaumer: Intermetallics, 1996, 4, 201. [15] J.P. Perdew, K. Burke and M. Ernzerhof: Phys. Rev. Lett., 1996, 77, 3865. [16] K.Y. Chen, L.R. Zhao, J. Rodgers, J.S. Tse and R. John: J. Phys. D: Appl. Phys., 2003, 36, 2725. [17] D.G. Clerc and H.M. Ledbetter: J. Phys. Chem. Solids, 1998, 59, 1071. [18] S.H. Jhi, J. Ihm, S.G. Louie and M.L. Cohen: Nature, 1999, 399, 132. [19] K.Y. Chen, L.R. Zhao and J.S. Tse: J. Appl. Phys., 2003, 93, 2414. [20] T. Davenport, L. Zhou and J. Trivisonno: Phys. Rev. B, 1999, 59, 3421. [21] D. Farkas, B. Mutasa, C. Vailhe and K. Ternes: Model. Simul. Mater. Sci. Eng., 1995, 3, 201. [22] J.F. Nye: Physical Properties of Crystals, Oxford University Press, Oxford, 1985. [23] B.B. Karki, G.J. Ackland and J. Crain: J. Phys: Condens. Mater., 1997, 9, 8579. [24] I.N. Frantsevich, F.F. Voronov and S.A. Bokuta: Elastic Constants and Elastic Moduli of Metals and Insulators, ed I.N Frantsevich, Naukova Dumka, Kiev, 1983, 60. [25] S.F. Pugh: Philos. Mag., 1954, 45, 823. [26] R. Liu, D.Y. Li, Y.S. Xie, R. Llewellyn and H.M. Hawthorne: Scripta Mater., 1999, 41, 691.
First-principles Study of NiAl Alloyed with Rare Earth Element Ce You Wang† , Junqi He, Mufu Yan, Chonggui Li, Liang Wang and Ye Zhou Department of Materials Science, School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, China [Manuscript received February 24, 2010, in revised form April 27, 2010]
The structural, elastic, and electronic properties of NiAl alloyed with rare earth element (REE) Ce have been investigated by using density functional theory (DFT). It is found that Ce has a strong Al site preference and causes lattice distortion of NiAl. The calculation of elastic constants shows that Ce increased both the hardness and the ductility of NiAl, which could be explained by the formation of new ionic bonds between Al (and Ni) and Ce and the enhancement of covalent bonds in Ni8 Al7 Ce. Our results are in good agreement with the available experimental data and other theoretical results. KEY WORDS: Point defects; Electronic states; First-principles
1. Introduction The intermetallic compound NiAl is one of the most promising engineering materials with attractive properties including good high-temperature strength, resistance to oxidation and corrosion, relatively low material density and high melting point. However, the application of NiAl is restricted as aerospace materials for its brittle fracture and low tensile ductility at room temperature (RM)[1] . Many efforts have been made to improve the ductility of NiAl[2,3] since Aoki and Lzumi[4] discovered that Ni3 Al added boron exhibits high ductility at room temperature. Among these efforts, alloying is one of the most efficient methods to improve the quality of NiAl at RM. Alloying elements which form 3d and 4d transition series have been studied extensively by both experimental and theoretical methods[5–7] . While rare earth elements (REEs) as alloying elements are more attractive and have had great effect on improvement of materials. For example, Guo et al.[8] have shown that the 0.05 wt% Ce-doped NiAl-Cr alloy attained † Corresponding author. Prof., Ph.D.; Tel.: +86 451 82921251; E-mail address: [email protected] (Y. Wang).
the highest compressive ductility of 14.8% at RT, while the REEs-free alloy only possessed 9% compressive ductility. The experimental results in literature [9] showed that adding 0.025 at.% Ce to NiAl (Ni/Al ratio=1.14) has increased the compressive plastic strain to more than 4, and increased the strength simultaneously. Wang et al.[10] have proved that the hardness of NiAl coating with 2 wt% CeO2 is 764.9 HV100g , while the hardness for NiAl coating without CeO2 is only 530 HV100g . This work also proved that the Young s elastic modulus and the η ratio (the ratio of the recoverable deformation energy to the total deformation energy) of NiAl coating with 2 wt% CeO2 are 179.7 GPa and 33%, respectively, while Young s elastic modulus and the η ratio of NiAl coating without CeO2 are 150 GPa and 28%, respectively. These results indicated that alloying element Ce plays an important role in improving the hardness and ductility of NiAl. However, the research on alloying effect of Ce in NiAl is still limited. Moreover, the studies of Ce addition on the atomic level are not sufficient, and the effect of Ce on the ductility, especially the micromechanism of the interaction between NiAl and the alloyed element Ce is not clear. In this paper, first-principles approach is used to calculate the structural, elastic,
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
720
and electronic properties of NiAl alloyed with Ce in order to illuminate the intrinsic effect of Ce. Though the amount of Ce in this study is 6.25 at.% and is larger than the experimental requirement (less than 3 at.%), it can still provide fundamental understanding of alloying effects of Ce on atomic and even on electronic levels as a complement of experimental results. 2. Computational Method Density functional theory (DFT) with ultrasoft pseudopotentials[11] is employed in the CASTEP code (Cambridge Sequential Total Energy Package), which utilizes plane-wave pseudopotential to perform firstprinciples quantum mechanics calculations. Generalized gradient approximation (GGA) as described by Perdew et al.[12] is adopted as exchange-correlation functionals for all elements in our models. This program is employed to investigate the lattice parameters, elastic characteristics as well as electronic structures of the NiAl 2×2×2 supercell with addition of REEs. The unit cell of NiAl with the space group PM-3M and the lattice constant of 0.29 nm is established, which is composed of one Ni atom at (1/2, 1/2, 1/2) and one Al atom at (0, 0, 0), respectively. The values of kinetic energy cutoff Ecut and the k points number are increased until the calculated energy converges within the required tolerance, where Ecut determines number of plane waves and k points does the sampling of the irreducible wedge of the Brillouin zone. Ecut is set to 350 eV and the k point is set to 4×4×4 with a regular Monkhorst-Pack scheme. In the calculation of self-consistent field (SCF), the Pulay scheme of density mixing[13] is adopted. The calculation of the elastic constants and the electronic structure are followed by cell optimization with a convergence tolerance of energy of 5.0×10−6 eV·atom−1 , a maximum displacement of 5.0×10−5 nm and a maximum force of 0.1 eV·nm−1 . A supercell model of NiAl with REE Ce, consists of 16 atoms, as shown in Fig. 1. 3. Results and Discussion 3.1 Site preference and lattice parameter of Ce in NiAl To understand the alloying effects of Ce requires the knowledge of site distributions of ternary elements M . Formation enthalpy is used to choose the more stable site that REE M occupied (Ni site or Al site) in NiAl. The site preference energy is described as Esite = E1 − E2
(1)
where E1 and E2 are the formation enthalpies of Ni7 Al8 M and Ni8 Al7 M , respectively. The usual definition of formation enthalpy is the total energy difference between compound and constituent elements in the solid state[14] :
Fig. 1 16-atom supercell used in this work. The cubic supercell is built up from 8 unit cells of NiAl. The rare earth atom Ce is placed at the centre of the supercell, substituting for Al, which is supercell Ni8 Al7 Ce; we also use a similar supercell where the rare earth alloying atom is placed at a Ni site, which is supercell Ni7 Al8 Ce
E1 = ENi7 Al8 M − 7ENi − 8EAl − EM
(2)
E2 = ENi8 Al7 M − 8ENi − 7EAl − EM
(3)
where ENi , EAl and EM are the total energies of pure Ni, Al and M , respectively. So we can deduce the form of Esite from the above two formation energies: Esite = E1 − E2 = ENi7 Al8 M − ENi8 Al7 M + ENi − EAl (4) If Esite <0, the substitution for Ni is more favorable, and if Esite >0, M is on the Al site. Facecentered cubic (fcc) Al and Ni are calculated, and the total energies of pure Ni and Al are −1348.9034 and −52.6996 eV, respectively. The calculated total energy of ENi7 Al8 Ce and ENi8 Al7 Ce are −10997.9529 and −12299.9075 eV, respectively, and the Esite is 5.7508 eV. So, Ce tends to substitute for Al site. The formation enthalpy of Ni8 Al8 , Ni8 Al7 Ce and Ni7 Al8 Ce are also calculated by Eqs. (2) and (3) (the total energies of pure Ce is −1056.5083 eV), so the formation enthalpy of the three compounds are −84.8337, −83.0893 and −77.312 eV, respectively. Though the formation enthalpy of the latter two compounds is a little larger than Ni8 Al8 , they are supposed to exist stably because of their negative formation enthalpy. Therefore, the models Ni8 Al7 Ce is used for the following calculation. The relaxed lattice parameter of a structure is the one which causes the energy to be minimized. The lattice parameters of Ni8 Al8 and Ni8 Al7 Ce are shown in Table 1. The equilibrium lattice constant of Ni8 Al8 is set to 0.58 nm, according to the experimental results[15,16] . In Table 1, the lattice parameter of Ni8 Al8 increased 3.28% after alloying with Ce.
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
721
Table 1 Equilibrium lattice constants and its change in Ni8 Al8 and Ni8 Al7 Ce [δ=(a-5.8)/5.8] Model Ni8 Al8 Ni8 Al7 Ce
a/nm 0.580 0.599
δ/% 0 3.28
3.2. Elastic properties It is known that the elastic moduli of materials can be used to assess certain mechanical properties[17,18] , such as ductility, hardness and strength. Those macroscopic parameters are related to the microscopic elastic constants. However, to measure the elastic constants of materials with defects is difficult and demands testing facilities with high accuracy. Here, theoretical calculation based on first principles is on prior consideration. The independent elastic constants in the cubic phase are C11 , C12 and C44 , according to Hooke s law. C11 represents the ratio of stress along <100> direction to strain along <100> direction; C12 represents the ratio of stress along <100> direction to strain along <010> direction; C44 represents the ratio of stress along <110> direction to strain along <110> direction. From C11 , C12 and C44 , we could receive bulk modulus B0 , shear modulus G, Young’s modulus E and the Poisson s ratio ν [19] : B0 = (C11 + 2C12 )/3
(5)
G = (3C44 + C11 − C12 )/5
(6)
E = 9B0 G/(3B0 + G)
(7)
ν = (1 − E/3B0 )/2
(8)
The elastic constants C11 , C12 , C44 and the bulk modulus B0 are shown in Table 2. There is a good agreement between our results and the experimental or theoretical ones[20,21] . The mechanical stability criteria in cubic crystal are: C11 − C12 >0; C44 >0; B0 >0[22,23] . So, the calculated elastic constants show that Ni8 Al7 Ce is stable. In Fig. 2, the negative Cauchy pressure −(C12 −C44 ), G/B0 , C12 and 1/v are calculated and showed together with other elastic constants in order to show the alloying effect of Ce on the properties of Ni8 Al8 . Figure 2 (a) are the histograms of Young s modulus E, shear modulus G and elastic constants C11 . It is generally believed that the hardness of materials can be related to their elastic modulus, such as
Fig. 2 Histograms of (a) Young s modulus E, shear modulus G and the elastic constants C11 , (b) the ratios of shear modulus to bulk modulus G/B0 , negative Cauchy pressure parameters –(C12 −C44 ) and (c) the reciprocal of Poisson s ratio 1/v bulk modulus B0 , elastic constants C44 and C12 of Ni8 Al8 and Ni8 Al7 Ce
the Young s modulus E, and the shear modulus G[24] . Although the relationships between hardness and the modulus are not identical for different materials, the general trend is, the larger the modulus, the harder the material. Therefore, the calculated Young s modulus and shear modulus can be used as a general guidance for selection of materials. From Fig. 2(a), it can be seen that E, G and C11 follow the similar trend, and in C11 , the difference between Ni8 Al8 and Ni8 Al7 Ce is larger than that in E and G. So alloying with Ce enhances the hardness of NiAl on the whole.
Table 2 Elastic constants C11 , C12 , C44 and the bulk modulus B0 of all supercells Model C11 /GPa C12 /GPa C44 /GPa C11 − C12 Ni8 Al8 Cal. 161.11 149.71 116.78 11.4 199 137 116 62 Ni8 Al8 Exp 204.6 135.4 116.8 69.2 Ni8 Al8 Exp 173.48 149.37 117.41 24.21 Ni8 Al8 295.56 127.85 74.43 167.71 Ni8 Al7 Ce Note: Cal/Exp: results from other calculations/experiments
B0 /GPa 153.18 158.7 158.5 157.41 183.75
722
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
Fig. 3 DOS of NiAl supercell (a) and Ni8 Al7 Ce (b)
Figure 2(b) shows the ratios of shear modulus to bulk modulus G/B0 , negative Cauchy pressure −(C12 −C44 ) and the reciprocal of Poisson s ratio 1/v. In order to predict the brittle and ductile behavior of solids, Pugh[25] has proposed a simple relationship that links empirically the plastic properties of metals with their elastic modulus by G/B0 . If G/B0 <0.5. The material behaves in a ductile manner, otherwise, behaves in a brittle manner. In Fig. 2(b), both the values of Ni8 Al8 and Ni8 Al7 Ce are smaller than 0.5. The smaller the G/B0 value, the better the ductility of NiAl. Cauchy pressure as an evaluation norm for ductile and brittle, generally, for ductile material is positive; and brittle material is negative. Figure 2(b) shows negative Cauchy pressure for easy comparison, smaller negative Cauchy pressure represents more ductile materials. Poisson s ratio reflects the stability of a crystal against shear. This ratio can formally take values between –1 and 0.5, and the smaller Poisson s ratio leads to more brittle behavior of materials. In other words, the smaller the reciprocal of Poisson s ratio 1/v, the more ductile the materials. Therefore, in Fig. 2(b), comparing Ni8 Al8 with Ni8 Al7 Ce, we can get the information that the rare earth alloying element Ce improves the ductility of NiAl. Figure 2(c) presents the bulk modulus B0 , elastic constants C44 and C12 of Ni8 Al8 and Ni8 Al7 Ce. The bulk modulus B0 represents the resistance to volume change and is related to over all atomic binding properties in a material. The bulk modulus B0 indi-
cates that the bond strength among atoms increased after adding Ce into NiAl, which is coincident with the electronic results in the following section 3.4. The shear modulus G is an average of single crystal elastic constants according to the Voigt approximation G=(C11 −C12 )/5+3C44 /5 for a unit cell with cubic symmetry. As mentioned above, shear modulus G is a good hardness predictor for a wide variety of covalent materials. C12 and C44 can be used to evaluate the hardness in the direction of <010> and <110>. Comparing C12 , C44 with G, we can get the conclusion that Ce can increase the hardness of NiAl on the whole, however, the hardness along the direction of <010> and <110> are decreased. 3.3 Density of states (DOS) and charge density contours The DOS as well as the charge density difference of Ni8 Al8 and Ni8 Al7 Ce are compared and shown in Figs. 3 and 4. The dominant feature of the Ni8 Al8 DOS is the existence of the sharp peaks contributed by d orbitals of Ni, which hybridizes weakly with the other orbitals. Placing the Ce atom at the Al site in Ni8 Al7 Ce adds two sharp peaks around –18 eV and –35 eV, which is mainly the contribution of s and p orbitals of Ce. This increases the height of the peak of the Ni-d state. There is a pseudogap, which presents stronger covalent bonding in Ni8 Al7 Ce compared to Ni8 Al8 , probably due to the partly filled 4f orbitals of Ce. Therefore alloyed with Ce, the system is relatively more stable.
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
723
Table 3 Charge populations of unit bond length before and after alloying with Ce Bond Population Length/nm P /(charge/nm) Al 1–Ni 1 0.14 0.251147 0.557 Al 2–Ni 1 0.14 0.251147 0.557 Al 4–Ni 1 0.14 0.251147 0.557 Al 1–Al 2 0.22 0.290000 0.759 Without Ce Al 2–Al 4 0.22 0.290000 0.759 Ni 1–Ni 2 −0.03 0.290000 −0.103 Al 4–Al 8 (Ce) 0.22 0.290000 0.759 Ni 1–Al 8 (Ce) 0.14 0.251147 0.557 Al 1–Ni 1 0.15 0.245999 0.610 Al 2–Ni 1 0.25 0.255252 0.979 Al 4–Ni 1 0.29 0.264181 1.098 Al 1–Al 2 0.20 0.299536 0.668 With Ce Al 2–Al 4 0.25 0.299539 0.835 Ni 1–Ni 2 0.13 0.284053 0.458 Al 4–Ce −1.03 0.299540 −3.439 Ni 1–Ce −1.16 0.272817 −4.252 Notes: Al 4, Al 6 and Al 7 are the first nearest neighbor of Ce; Al 2, Al 3 and Al 5 are the second nearest neighbor of Ce; Al 1 is the third nearest neighbor of Ce Table 4 Nanohardness, elastic modulus and elastic recoverability of NiAl with and without alloying elements Samples NiAl NiAl1CeO2 NiAl2CeO2
Ce/at.% 0 0.25 1.15
Nanohardness/GPa 9.471 9.777 10.023
Elastic modulus/GPa 193.78 194.64 194.90
Elastic recoverability/% 20.79 20.80 21.45
Fig. 4 Charge density difference of the center atom of (001) plane before (a) and after (b) placing Ce in Ni8 Al8 . Atoms at the site of Ni 2, Ni 4, Ni 6 and Ni 8 are on the upper layer of the (001) plane, and Ce is to replacing Al 8 which is under the site of Al 9
Figure 4 shows the charge density difference of the center atom of (001) plane along <001> direction. Figure 4 (a) and (b) are the charge density difference contours of Ni8 Al8 before and after being alloyed with Ce. The blue area represents the depletion (decrease) of electronic charge while the red area represents the accumulation (increase) of electronic charge. Contours start from ±0.2, and increase successively by a factor of 0.025. Comparing with the two figures, we can see that, after replacing Al with Ce, the bonding
charge accumulation at the Ce site is along the Ce–Ni directions (Ni atoms are the nearest neighbors of Ce), and the bonding charge depletion is along the nearest neighbor Al (Ce–Al directions). 3.4 Population analysis The bond populations indicate the overlap degree of the electron cloud of two bonding atoms and can be used to access covalent or ionic nature of a chemical bond. For the bond populations, the lowest and
724
Y. Wang et al.: J. Mater. Sci. Technol., 2011, 27(8), 719–724
highest values imply that the chemical bonds exhibit strong ionicity and covalency, respectively. In Table 3, the bond population between Ni and Al are increased, and Al 4–Ni 1 shows the highest value of bond population of unit bond length. The values of Al 2–Al 4, Ni 1–Ni 2 are also increased, which means the covalent effect in Ni8 Al8 is increased and the ionic effect is relatively decreased after alloying with Ce. The population values of Ni 1 (FNN—first nearest neighbor of Ce)–Ce and Al 4 (FNN)–Ce are relatively low, so the bonds Ce–Ni (FNN) and Ce–Al (FNN) show great ionicity, though they are instability; the ionicity tendency is evident, and the bond of Ce–Ni (FNN) is more stronger than Ce–Al (FNN), which consists with the result of Fig. 4(b). 4. Additional Experimental Results To further verify the former calculation results, we do the nanoindentation test on NiAl, which is doped by CeO2 . In this investigation, NiAl powders were mixed with two different amounts of CeO2 (shown in Table 4), and were fabricated by spark-plasma sintering (SPS) at 1250◦ C for a holding time of 5 min. Nanoindentation test was performed on the polished surface of three specimens: NiAl, NiAl1CeO2 (NiAl alloyed with 1.15 at.% Ce), NiAl2CeO2 (the amount of Ce is 0.25 at.%). The nanoindentation tester (TriboIndenter, Hysitron, USA) was equipped with a sharp Berkovich pyramid indenter. In the tests, maximal indentation force was 120 mN, and loading time and unloading time were 8 and 80 s, respectively. This study aims to analyze the different nanomechanical properties (i.e. elastic modulus, nanohardness and elastic recoverability) of NiAl with and without Ce. According to literature [26], the elastic recoverability was evaluated by the ratio of recoverable deformation energy to the total deformation energy, throughout loading and unloading cycles. Table 4 shows the nanohardness, elastic modulus and elastic recoverability of samples. From Table 4 we can get the information that with increasing the amount of Ce, the nanohardness the elastic modulus and the elastic recoverability all increased; NiAl with 1.15 at.% Ce shows the highest increase. In other words, Ce increases both the hardness and the ductility of NiAl. These results are coincident with our calculation results above. 5. Conclusions Ab initio density functional theory calculations have been performed to study the effects of rare earth alloying element Ce on the crystal structural, elastic, and electronic properties of NiAl. In general, the following conclusions are drawn: (1) Ce has a strong site preference of Al site and increases the lattice parameters of Ni8 Al8 . (2) Ce increases the hardness and ductility of NiAl simultaneously.
(3) Alloying with Ce enhances covalent bonds in Ni8 Al7 Ce. (4) The bonds between Ce and its first nearest neighbor Ni and Al show great ionicity.
Acknowledgement This work was financially supported by the National Natural Science Foundation of China under grant No. 50871035. REFERENCES [1 ] C.T. Liu, J. Stringer and J.N. Mundy: Intermetallics, 1997, 5, 579. [2 ] A. Albiter, M. Salazar, E. Bedolla, R.A.L. Drewc and R. Perez: Mater. Sci. Eng. A, 2003, 347, 154. [3 ] C.Y. Chung, C.Y. Xie and T.Y. Hsu: Scripta Mater., 1997, 37, 99. [4 ] K. Aoki and O. Lzumi: J. Jpn. Inst. Met., 1979, 43, 1190. [5 ] A.I. Kovalev, R.A. Barskaya and D.L. Wainstein: Surf. Sci., 2003, 532-535, 35. [6 ] P. Lazar and R. Podloucky: Phys. Rev. B, 2006, 73, 1. [7 ] C.L. Fu, C.T. Liu and X.L. Wang: Intermetallics, 2004, 12, 911. [8 ] J.T. Guo, K.W. Huai, Q. Gao, W.L. Ren and G.S. Li: Intermetallics, 2007, 15, 727. [9 ] W.X. Chen, J.R. Hines and Y. Wang: Adv. Eng. Mater., 2004, 6, 876. [10] Y. Wang, Z. Wang, Y. Yang and W. Chen: Intermetallics, 2008, 16, 682. [11] D. Vanderbilt: Phys. Rev. B, 1990, 41, 7892. [12] J.P. Perdew, J.A. Chevary and S.H. Vosko: Phys. Rev. B, 1992, 46, 6671. [13] B. Hammer, L.B. Hansen and J.K. Nørskov: Phys. Rev. B, 1999, 59, 7413. [14] W. Wolf, R. Podloucky, P. Rogl and H. Erschbaumer: Intermetallics, 1996, 4, 201. [15] J.P. Perdew, K. Burke and M. Ernzerhof: Phys. Rev. Lett., 1996, 77, 3865. [16] K.Y. Chen, L.R. Zhao, J. Rodgers, J.S. Tse and R. John: J. Phys. D: Appl. Phys., 2003, 36, 2725. [17] D.G. Clerc and H.M. Ledbetter: J. Phys. Chem. Solids, 1998, 59, 1071. [18] S.H. Jhi, J. Ihm, S.G. Louie and M.L. Cohen: Nature, 1999, 399, 132. [19] K.Y. Chen, L.R. Zhao and J.S. Tse: J. Appl. Phys., 2003, 93, 2414. [20] T. Davenport, L. Zhou and J. Trivisonno: Phys. Rev. B, 1999, 59, 3421. [21] D. Farkas, B. Mutasa, C. Vailhe and K. Ternes: Model. Simul. Mater. Sci. Eng., 1995, 3, 201. [22] J.F. Nye: Physical Properties of Crystals, Oxford University Press, Oxford, 1985. [23] B.B. Karki, G.J. Ackland and J. Crain: J. Phys: Condens. Mater., 1997, 9, 8579. [24] I.N. Frantsevich, F.F. Voronov and S.A. Bokuta: Elastic Constants and Elastic Moduli of Metals and Insulators, ed I.N Frantsevich, Naukova Dumka, Kiev, 1983, 60. [25] S.F. Pugh: Philos. Mag., 1954, 45, 823. [26] R. Liu, D.Y. Li, Y.S. Xie, R. Llewellyn and H.M. Hawthorne: Scripta Mater., 1999, 41, 691.