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Al2O3 interface with yttrium dopant under tension
Engineering Fracture Mechanics xxx (2015) xxx–xxx
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An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension Zeying Bao, Xiancong Guo, Fulin Shang ⇑ State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, PR China
a r t i c l e
i n f o
Article history: Received 28 November 2014 Received in revised form 30 March 2015 Accepted 20 May 2015 Available online xxxx Keywords: Adhesion Fracture First-principle method Ni/Al2O3 interface Yttrium dopant
a b s t r a c t This study performs a first principle study of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant, and aims to understand the interfacial fracture behavior of a thermal barrier coating at atomic level. Uniaxial tensile simulations by density functional theory approach are conducted for a stable configuration of this interface, and its theoretical tensile strength and work of separation are extracted. By observing the variations of the atomic bonds and valence charge densities near interface region, we conclude that the Y doped Ni/Al2O3 interface breaks along the Y–Al and Y–O atomic bonds, tends to fracture in a ductile way. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Thermal barrier coatings (TBCs) are often used to improve the efficiency of gas turbine engines by protection against thermal and corrosive degradation of superalloy turbine blades in the harsh combustion environment. Premature failure has been observed to occur often along the interface between the bond coat and the thermally grown oxide (TGO) by microcrack nucleation, propagation and coalescence, leading eventually to spalling and delaminating of the TBC [1–8]. To prevent such failure in TBCs, it is important to improve the adhesion strength between the bond coat and the TGO. While the exact micro-structure and compositions of the real interface between bond coat metal (typically made of a Ni-based alloy) and TGO (predominantly a-alumina) could be very complicated, this study focused on modeling of Ni/Al2O3 interface. The reason of this choice has been explained elsewhere [6,7,9]. In current gas turbine engine applications, yttria-stabilized zirconia (YSZ) is widely used as a ceramic top coat, owing to its high thermal expansion coefficient and low thermal conductivity. For the YSZ, the reactive-element dopants, such as Y, play a very important and unique role, in terms of improving the adhesion strength between the bond coat and the TGO by absorbing the oxygen from Ni-based alloy. In addition, the Y dopant from bond coat (e.g. NiCrAlY) may affect the structure and properties of the interface. Recent studies [3,7] demonstrated that addition of the reactive elements of Hf, Y and Zr into the bond coat could lower the weight gain of oxides, inhibit the void formation, enhance the mental-oxide adhesion, and reduce the S diffusion rate to the interface by forming sulfides in the bond coat, thus improving the strength of interfacial adhesion. Jarvis and Carter [10] found that doping Y to the Ni/Al2O3 interface can improve the cohesive energy of about 70–80%. The first principle calculation of Ozfidan et al. [11] indicated that the effect of Y to the adhesion strength of NiAl/Al2O3(0 0 0 1) was parallel to Hf and Zr. So far, the effect of Y on adhesion of the Ni(1 1 1)/a-Al2O3(0 0 0 1) interface under ⇑ Corresponding author. Tel.: +86 29 8266 5490; fax: +86 29 8266 5397. E-mail address: shangfl@mail.xjtu.edu.cn (F. Shang). http://dx.doi.org/10.1016/j.engfracmech.2015.05.040 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Nomenclature a0 , c 0 lattice constants A interface area B0 bulk modulus of the crystal E0 ðV 0 Þ binding energy of the crystal EAl2 O3 total energy of Al2O3 slab ENi=Y total energy of Ni/Y slab ENi=Y=Al2 O3 total energy of Ni/Y/Al2O3 slab l0 initial length of the model cell along the tensile loading direction l length of the model cell pulled after a load incremental r tensile stress increment V0 cell volume at equilibrium state of the crystal W ad work of adhesion
external loadings has not reported yet. Before doing this kind of study, Ni/Al2O3 interface has been thoroughly examined by the authors group, and its mechanical strength and fracture characteristics have been checked under uniaxial tension, shear, and also mixed mode loading conditions [9,12,13]. To have a deeper understanding of the mechanical properties and failure behavior of the Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with Y dopant, first principle method is very helpful in checking the fundamental aspects of deformation and fracture processes at atomic and electronic levels. In this work, we perform density functional theory (DFT) calculation to address the basics by investigating the Ni/Al2O3 interface with Y dopant under uniaxial tension, and discuss the nature of fracture such as failure mode, bonds to break, and fracture parameters. 2. Method and model 2.1. First principle tensile simulation First principle calculations are performed using the DFT [14,15] code VASP (The Vienna ab initio Simulation Package) [16,17], together with the generalized gradient approximation (GGA) and with projector augmented wave (PAW) potentials [18,19]. Monkhorst–Pack (MP) scheme is used for the k-point sampling of the Brillouin zone of the model cell. To ensure the convergence of results in each relaxation step, a plane-wave cutoff energy of 500 eV and an energy convergence criterion of 105 eV for self-consistency are adopted throughout all the calculations. Relaxation is done until the Hellman–Feynman forces are less than 0.01 eV/Å using a conjugate gradient (CG) algorithm. The magnetic contributions are not considered. For the relaxed configurations of the model cell, the so-called uniaxial tension is applied [6,9]. Namely, the simulation cell is stretched gradually in a small increment and uniformly along the direction of tensile loading, and the atomic positions are changed linearly. Then, the lattice relaxation is carried out while keeping lateral lattice vectors fixed. The lateral lattice vectors of the cell are relaxed along with the ionic positions so that the stresses in the direction orthogonal to the loading direction vanish (less than 0.01 GPa). During this relaxation, the symmetry and periodic arrangement of the model cell are preserved. This process is iterated until the final rupture of the cell. As one can see, this kind of first principle simulation (also called ab initio tensile test in the literature [6]) mimics the actual experimental tensile test of a real specimen, and intends to observe the deformation behavior of the material cell, in some sense. 2.2. Simulation model Preliminary calculations for the bulk and surface properties of Ni and a-Al2O3 have been performed (see Ref. [9] for details). In order to assess the accuracy of the computational methods used for the present interface systems, the bulk property of yttrium is also a necessity to verify. Fig. 1 shows the crystalline structure of unit cell of bulk yttrium, with a closed-packed hexagonal structure, and having two Y atoms in each unit cell, a0 and c0 represent the lattice constants. Test calculations are performed for the bulk property of yttrium. The total energy of yttrium is computed and fitted to Murnaghan equation of state [20]
" # B00 B0 V V0 V0 0 þ EðVÞ ¼ 0 0 B 1 1 þ E0 ðV 0 Þ V V B0 ðB0 1Þ 0
ð1Þ
where B0 is the bulk modulus, V 0 is the cell volume at equilibrium state, E0 ðV 0 Þ is the binding energy of the crystal. From this equation of state, the equilibrium lattice constants for yttrium are determined to be a0 = 3.635 Å and c0 = 5.714 Å, and its bulk modulus is B0 = 40.5 GPa, which are very close to the available experimental and other first principle results [21,22], Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 1. Crystalline structure of unit cell of bulk yttrium.
i.e., 3.647 Å, 5.731 Å, 41 GPa, respectively. In other words, the computational method within the framework of PAW and GGA is suitable for the present case. Next, we construct a physically plausible model with an addition of yttrium into the Ni(1 1 1)/a-Al2O3(0 0 0 1) interface. Based on our previous studies [9,12,13], the most stable Ni/Al2O3 interface model is considered, namely Al-terminated O-site Ni(1 1 1)/a-Al2O3(0 0 0 1) model (hereinafter, Al–O model for short) as seen in Fig. 2. The atoms of Ni, Al, O and Y are represented by blue, pink, red and green spheres, respectively. To better illustrate the interface structural feature, the atomic arrangements involving three periodic atomic layers along transverse direction are displayed in the figure. Three axis directions a, b, c in Fig. 2a and b represent ½1 2 1 0, ½1 1 0k½1 0 1 0, and [111]||k½0 0 0 1, respectively. The physically possible positions that are occupied by reactive element Y atoms might be Al substitution site, Ni substitution site, or an interstitial site. The interface models consist of five Ni atomic layers, five O atomic layers, and ten Al atomic layers. And the numbers of atoms for the Al substitution site model and the Ni substitution site model are identical, i.e. 55. For the interstitial site model, the simulation model totally involves 56 atoms. This study considered a substitution of Y at the first or second Ni atomic layer (i.e. Ni1 or Ni2 hereinafter), excluding any further possible substitution positions, as a trade-off between computational accuracy and cost. The work of adhesion, W ad , is often used to assess the stability of an interface in the literature, see e.g. Ref. [23]. It is understood that the higher the work of adhesion, more stable the interfacial structure. W ad is defined as the energy required (per unit area) to reversibly separate an interface into two free surfaces. These dissipative processes are responsible for the fact that energy needed in an actual cleavage experiment is always considerably greater than the ideal work of adhesion. Therefore, our predictions may be considered as lower bounds for the work of adhesion obtained by any cleavage experiment. Formally, W ad is defined in terms of either the surface and interface energies relative to respective bulk materials or by the difference in total energy between the interface slab and its isolated component slabs (substrate and coating) [23]:
W ad ¼
ENi=Y þ EAl2 O3 ENi=Y=Al2 O3 A
ð2Þ
where ENi=Y , EAl2 O3 , and ENi=Y=Al2 O3 are the total energies of the corresponding slab, respectively, A is the interface area. In order to approximately cancel the effect of lattice mismatch, we calculated the energy of the isolated coating using the same lattice vectors as that for the interface. In the present simulation, convergence tests demonstrated that 1–2 meV/Å per atom degree of convergence with respect to k-point sampling was attained upon using 5 5 1 points for the interface systems. By fully relaxing the entire slab, the bonding energy of the Al–O Ni(1 1 1)/a-Al2O3(0 0 0 1) interfaces with yttrium dopant at different sites are obtained, see Table 1. It is observed that, when Y is placed into the interstitial site, the interface is most stable, with an interfacial energy of 3.87 J/m2. That is, a significant enhancement of adhesion is seen by doping Y to the interstitial site of its clean interface, approximately 2.65 times higher in terms of bonding energy. Fig. 3 is the schematic diagram of the most stable interface model with Y dopant at the interstitial site, stretched normal to the interface. l0 is the initial length of the model along the tensile loading direction, and l is its length pulled after a tensile stress increment r. 3. Results and discussion 3.1. Interfacial strength and fracture parameter Fig. 4 shows the stress–strain curves of the clean Ni/Al2O3 interface and the Ni/Al2O3 interface with yttrium dopant. For the clean Ni/Al2O3 interface, normal stress reaches its maximum value of 11.1 GPa at strain of 8%, while for the doped interface, the stress reaches the maximum value of 18.75 GPa at strain of 10%. That is to say, the theoretical tensile strength of the Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 2. The schematic diagrams of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant. (a) Front view; (b) top view; (c) Al-substitution site; (d) interstitial site; (e) Ni1-substitution site; and (f) Ni2-substitution site.
Al–O Ni/Al2O3 interface can be enhanced (here, 69% higher) by Y dopant at the interstitial sites near interface region. Meanwhile, Y dopant also increases the rupture tensile strain by up to 25%. This implies that adhesion of the Ni/a-Al2O3 interface could be improved efficiently by doping with Y. Further, we look at the stress–strain curves obtained by first principle simulations. Here, the important parameter to characterize the property of the interface is the area under the stress–strain curve. This parameter gives an approximate evaluation of the work required to separate the interface, W sep , which is often referred as the work of separation in the literatures [e.g. 7]. From our calculations, W sep for the clean Ni/Al2O3 interface is 1.44 J/m2, while that for the doped interface is
Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Z. Bao et al. / Engineering Fracture Mechanics xxx (2015) xxx–xxx Table 1 The work of adhesion for the Al-terminated O-site Ni(1 1 1)/a-Al2O3(0 0 0 1) interfaces with yttrium dopant at different sites. Clean Ni/Al2O3
ENi=Y (eV) EAl2 O3 (eV) ENi=Y=Al2 O3 (eV) a0 ðÅÞ Work of adhesion (J/m2)
Ni/Al2O3 with Y dopant Al substitution site
Ni1 substitution site
Ni2 substitution site
Interstitial site
77.291736 222.239385 297.840002 4.756
73.1236 225.56875 342.982271 4.784
77.100155 222.573227 301.540233 4.81
77.485408 222.418245 301.251027 4.78
73.748702 228.972763 307.775659 4.762
1.46
1.13
1.49
1.09
3.87
Fig. 3. Schematic diagram of the uniaxial tension simulation of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant at the interstitial site.
Fig. 4. Stress–strain curves of the clean Ni/Al2O3 interface and the Ni/Al2O3 interface with yttrium dopant.
3.82 J/m2. It is again seen that its work of separation increases about 165% with interstitial Y dopant. It is noticed that Jarvis Carter [10] reported an adhesion of 3.24 J/m2 for the Ni/Al2O3 interface with Y dopant, which is slightly lower than our prediction. The reason could be that, in their calculation, Y atom occupied the Ni substitution site, which is different from the interstitial site in our model. As already explained before, the most stable configuration of the Ni/Al2O3 interface with Y dopant could be that with Y at an interstitial site. 3.2. Fracture characteristic Fig. 4 contains some information on the failure behavior of the interface as well, noticing the shapes of both the stress– strain curves. As have been discussed in Refs. [8,9,23], the Al–O Ni/Al2O3 interface tends to fail in a ductile manner with some plastic dissipation in the metal. For the doped interface, similarly the stress descends gradually after the maximum, instead of a sudden drop. And this softening period during interface cleavage is quite remarkable, compared to that of the clean O–Al Ni/Al2O3 interface. This feature suggests that the failure may occur in a progressive way, instead of a catastrophic one. To better understand the failure process of the Y doped interface, the variations of the atomic bonds near the interface during tension are plotted in Fig. 5. When the stretching goes on, the atomic bond length of Ni1–Y1 does not vary much, while the bond length of Y1–Al1 and Y1–O1 increase gradually until the tensile strain reaches its critical value. In particular, the latter two bonds elongate apparently when beyond the critical strain. During this dramatic elongation period, the interfacial stress drops gradually as well (Fig. 4). It is thus clear that the Y doped interface breaks at Y1–Al1 bond and Y1–O1 bond, when subject to tensile loading. The valence charge density distribution can also reflect the character of atomic bonds near the interface. Fig. 6 shows the evolution of the valence charge density distribution at the Ni/Al2O3 interface with yttrium dopant during the stretching Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 5. Variations of the atomic bonds of Ni/Al2O3 interface with yttrium dopant during the stretching process. The vertical line indicates the critical strain. Pictorial showing the interfacial bonds are shown in the inset panel. The Ni, Al, O and Y atoms are represented by blue, pink, red and green spheres, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
process. The dashed line in Fig. 6a (a top view of the interface model) indicates the observing plane from a top-down view, and the contour plots of the valence charge densities on this plane for different tensile strains are shown below in Fig. 6c–k. For comparison purpose, the charge density distribution for the clean interface is given in Fig. 6b, showing the ionic Ni–O bonds with each Ni atop each O (Ref. [9]). By comparing to that for the Y doped interface (Fig. 6c), one can see that two ionic Y–O bonds and one metallic bond Y–Al are formed near the Y doped interface, which consequently increase the interfacial adhesion. For the increase of the strain as shown in Fig. 6d–f, the charge density between the interfacial Y and O atoms decreases gradually, as well as that between the Y and Al atoms. When the strain goes over the critical value (Fig. 6g–i), this decreasing tendency becomes more pronounced, along with the appearance of a charge depletion region around the interfacial atoms. With the further increase of the strain until approaching to the ultimate rupture, the charge depletion region extends above the interfacial Al and O atoms (Fig. 6j), and finally, the interfacial region fully separates into two surfaces, one with the original Ni slab and the other with Al2O3 slab. By collecting the results in Figs. 4–6, we can conclude that the Y doped Ni/Al2O3 interface breaks along the at a relatively higher level of tensile stress compared to the clean interface, tends to fracture in a ductile way, and separates exactly along the interlayer between the initial Ni slab and Al2O3 slab. 3.3. Discussion As a typical metal/ceramic interface within a TBC system, Ni alloy/Al2O3 is commonly understood to be relatively brittle and not strong enough [8]. Our simulation results also indicate that the interfaces of clean Ni/Al2O3 and that with yttrium dopant have high theoretical tensile strengths, i.e. 11.1 GPa and 18.75 GPa. Though this brittle nature, its interface separation process at the atomic scale is not a scenario of catastrophic breakage. As clearly shown by our results, Figs. 4–6, it is much like a gradual fracturing of the atomic layers between Ni and Al2O3 slabs, with certain breaking of Y–Al and Y–O bonds. In particular, one may notice both the distinct enhancements of the tensile strength and the work of separation parameters of the Ni/Al2O3 interface with Y dopant relative to the ideal clean Ni/Al2O3 interface, and the former might be much closer to a real Ni/Al2O3 interface in a TBC system. As already noted by Ref. [8], small changes in interface property could induce significant changes in its toughness at the macroscopic scale. Therefore, the above enhancements could be expected to contribute an increase of this interface toughness, and in turn that of a TBC system. Of course, the toughness of Ni/Al2O3 interface, as a macroscopic index of material resistance, is much dependent on many other deformation mechanisms such as plastic dissipation in a Ni alloy, and does not directly correspond with either the work of separation or the theoretical strength. Rather, we would say that, by observing at atomic level, the failure process of brittle Ni/Al2O3 interface does not advance as a Griffith cleavage, but may possess a nature of progressive ductile fracturing. To help us understand the fracture of Ni/Al2O3 interface, one needs certainly to go up from bottom, here the atomic and electronic scales. Jiang et al. [24] presented an example to predict the macroscopic toughness of a c-Ni(Al)/a-Al2O3 interface by utilizing the basic knowledge obtained from first principle calculations, and they also gave a quantitative implementation of this computational strategy. One can see from this nice study that, (i) a traction–separation relationship for the interface is computed using first principle results, which serves as a separation criterion for the interface; (ii) The results of first principle tensile simulation, together with that of shear and mixed mode simulations, are necessities to formulate the so-called interfacial potential. This interfacial potential, as recognized in a pioneering work of Sun et al. [25] and also confirmed by our recent study [13], is much related or even quite sensitive to the characteristic physical phenomena such as the effects of tension–shear coupling and interface relaxation. In developing a proper interfacial potential, its analytical expression with Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 6. Variations of the valence charge densities of Ni/Al2O3 interface with yttrium dopant during the stretching process. (a) Top view; (b) clean interface; (c) tensile strain = 0%; (d) 4%; (e) 8%; (f) 9.5%; (g) 10% (critical strain); (h) 10.5%; (i) 12%; (j) 16%; and (k) 20%. Units are e/Å3.
specific form must effectively describe all the basic physics taking place at the atomic level. By making use of this potential, one can do more in understanding the fracture of Ni/Al2O3 interface. For example, it is a straightforward work to derive from this potential a traction–separation law of a cohesive zone model, which is commonly adopted in continuum mechanics
Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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based finite element analysis, e.g. Ref. [26]. More complicated crack problems are also possible to be investigated from this potential, such as a dislocation nucleation from a crack tip, as discussed in Ref. [25]. When going up from an electronic level up to atomic and molecular levels, one would often rely on a physically reliable interatomic potential function, for example, in performing molecular dynamics simulations. In this case, first principle results are valuable in constructing these potential functions between atoms, since first principle calculations provide exact predictions of the chemical, physical and also mechanical properties of the Ni/Al2O3 interface. Normally, when considering the interactions between dissimilar atoms, like here Y–Al and Y–O, the potential functions given in the literature are often inadequate in reproducing its mechanical property. Hence, a task having to face for a mechanics person is to modify or even re-construct such functions, which cannot be an easy work. Then, the mechanical responses of the interface under external loadings should be considered as inputs in the fitting procedure of these functions. In this sense, the first principle result of tensile simulation is of great value. It is the present authors’ opinion that, using the interatomic potential function derived in this way, a reasonable prediction of the interface mechanics from a molecular dynamics simulation could be guaranteed. As mentioned above, first principle tensile simulation results are conducive to the determination of a cohesive zone model with a proper constitutive law. To date, a variety of cohesive zone models are still popular in analyzing the complicated fracture processes, and conventional cohesive finite element method in a continuum mechanics framework is under a continuous development, such as Refs. [27,28]. It is pointed out [29] that, an accurate cohesive zone model should capture the characteristics of atomic-level bond breaking between adjacent atomic layers of, here, the interface region. Thus, an ideal cohesive finite element method would be based on the interface potential that is linked to the atomistic potential obtained from first principle calculations, according to Ref. [27]. This kind of cohesive zone modeling would naturally have some merits in advancing our understanding of fracture. 4. Summary The fracture characteristics of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant have been investigated at atomic and electronic levels through a DFT-based first principle calculation. Tensile simulations are performed to extract the fundamental mechanical properties of this interface, including the theoretical tensile strength and the work of separation. Relative to the ideal clean Ni/Al2O3 interface, both the distinct enhancements of the tensile strength (69% higher) and the work of separation (165% higher) are seen for the Ni/Al2O3 interface with Y dopant. The variations of the atomic bonds and valence charge densities near interface region during tension clarify the failure and fracture characteristics of the interface. We found that the Y doped Ni/Al2O3 interface breaks along the Y–Al and Y–O atomic bonds at a relatively higher level of tensile stress compared to the clean interface, tends to fracture in a ductile way, and separates along the interlayer between the initial Ni slab and Al2O3 slab. We also discussed the approaches to use these first principle results to assess the TBC adhesion, including prediction of macroscopic interface toughness, development of a reliable interatomic potential function, and determination of an accurate cohesive zone model. 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An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension Zeying Bao, Xiancong Guo, Fulin Shang ⇑ State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, PR China
a r t i c l e
i n f o
Article history: Received 28 November 2014 Received in revised form 30 March 2015 Accepted 20 May 2015 Available online xxxx Keywords: Adhesion Fracture First-principle method Ni/Al2O3 interface Yttrium dopant
a b s t r a c t This study performs a first principle study of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant, and aims to understand the interfacial fracture behavior of a thermal barrier coating at atomic level. Uniaxial tensile simulations by density functional theory approach are conducted for a stable configuration of this interface, and its theoretical tensile strength and work of separation are extracted. By observing the variations of the atomic bonds and valence charge densities near interface region, we conclude that the Y doped Ni/Al2O3 interface breaks along the Y–Al and Y–O atomic bonds, tends to fracture in a ductile way. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Thermal barrier coatings (TBCs) are often used to improve the efficiency of gas turbine engines by protection against thermal and corrosive degradation of superalloy turbine blades in the harsh combustion environment. Premature failure has been observed to occur often along the interface between the bond coat and the thermally grown oxide (TGO) by microcrack nucleation, propagation and coalescence, leading eventually to spalling and delaminating of the TBC [1–8]. To prevent such failure in TBCs, it is important to improve the adhesion strength between the bond coat and the TGO. While the exact micro-structure and compositions of the real interface between bond coat metal (typically made of a Ni-based alloy) and TGO (predominantly a-alumina) could be very complicated, this study focused on modeling of Ni/Al2O3 interface. The reason of this choice has been explained elsewhere [6,7,9]. In current gas turbine engine applications, yttria-stabilized zirconia (YSZ) is widely used as a ceramic top coat, owing to its high thermal expansion coefficient and low thermal conductivity. For the YSZ, the reactive-element dopants, such as Y, play a very important and unique role, in terms of improving the adhesion strength between the bond coat and the TGO by absorbing the oxygen from Ni-based alloy. In addition, the Y dopant from bond coat (e.g. NiCrAlY) may affect the structure and properties of the interface. Recent studies [3,7] demonstrated that addition of the reactive elements of Hf, Y and Zr into the bond coat could lower the weight gain of oxides, inhibit the void formation, enhance the mental-oxide adhesion, and reduce the S diffusion rate to the interface by forming sulfides in the bond coat, thus improving the strength of interfacial adhesion. Jarvis and Carter [10] found that doping Y to the Ni/Al2O3 interface can improve the cohesive energy of about 70–80%. The first principle calculation of Ozfidan et al. [11] indicated that the effect of Y to the adhesion strength of NiAl/Al2O3(0 0 0 1) was parallel to Hf and Zr. So far, the effect of Y on adhesion of the Ni(1 1 1)/a-Al2O3(0 0 0 1) interface under ⇑ Corresponding author. Tel.: +86 29 8266 5490; fax: +86 29 8266 5397. E-mail address: shangfl@mail.xjtu.edu.cn (F. Shang). http://dx.doi.org/10.1016/j.engfracmech.2015.05.040 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Nomenclature a0 , c 0 lattice constants A interface area B0 bulk modulus of the crystal E0 ðV 0 Þ binding energy of the crystal EAl2 O3 total energy of Al2O3 slab ENi=Y total energy of Ni/Y slab ENi=Y=Al2 O3 total energy of Ni/Y/Al2O3 slab l0 initial length of the model cell along the tensile loading direction l length of the model cell pulled after a load incremental r tensile stress increment V0 cell volume at equilibrium state of the crystal W ad work of adhesion
external loadings has not reported yet. Before doing this kind of study, Ni/Al2O3 interface has been thoroughly examined by the authors group, and its mechanical strength and fracture characteristics have been checked under uniaxial tension, shear, and also mixed mode loading conditions [9,12,13]. To have a deeper understanding of the mechanical properties and failure behavior of the Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with Y dopant, first principle method is very helpful in checking the fundamental aspects of deformation and fracture processes at atomic and electronic levels. In this work, we perform density functional theory (DFT) calculation to address the basics by investigating the Ni/Al2O3 interface with Y dopant under uniaxial tension, and discuss the nature of fracture such as failure mode, bonds to break, and fracture parameters. 2. Method and model 2.1. First principle tensile simulation First principle calculations are performed using the DFT [14,15] code VASP (The Vienna ab initio Simulation Package) [16,17], together with the generalized gradient approximation (GGA) and with projector augmented wave (PAW) potentials [18,19]. Monkhorst–Pack (MP) scheme is used for the k-point sampling of the Brillouin zone of the model cell. To ensure the convergence of results in each relaxation step, a plane-wave cutoff energy of 500 eV and an energy convergence criterion of 105 eV for self-consistency are adopted throughout all the calculations. Relaxation is done until the Hellman–Feynman forces are less than 0.01 eV/Å using a conjugate gradient (CG) algorithm. The magnetic contributions are not considered. For the relaxed configurations of the model cell, the so-called uniaxial tension is applied [6,9]. Namely, the simulation cell is stretched gradually in a small increment and uniformly along the direction of tensile loading, and the atomic positions are changed linearly. Then, the lattice relaxation is carried out while keeping lateral lattice vectors fixed. The lateral lattice vectors of the cell are relaxed along with the ionic positions so that the stresses in the direction orthogonal to the loading direction vanish (less than 0.01 GPa). During this relaxation, the symmetry and periodic arrangement of the model cell are preserved. This process is iterated until the final rupture of the cell. As one can see, this kind of first principle simulation (also called ab initio tensile test in the literature [6]) mimics the actual experimental tensile test of a real specimen, and intends to observe the deformation behavior of the material cell, in some sense. 2.2. Simulation model Preliminary calculations for the bulk and surface properties of Ni and a-Al2O3 have been performed (see Ref. [9] for details). In order to assess the accuracy of the computational methods used for the present interface systems, the bulk property of yttrium is also a necessity to verify. Fig. 1 shows the crystalline structure of unit cell of bulk yttrium, with a closed-packed hexagonal structure, and having two Y atoms in each unit cell, a0 and c0 represent the lattice constants. Test calculations are performed for the bulk property of yttrium. The total energy of yttrium is computed and fitted to Murnaghan equation of state [20]
" # B00 B0 V V0 V0 0 þ EðVÞ ¼ 0 0 B 1 1 þ E0 ðV 0 Þ V V B0 ðB0 1Þ 0
ð1Þ
where B0 is the bulk modulus, V 0 is the cell volume at equilibrium state, E0 ðV 0 Þ is the binding energy of the crystal. From this equation of state, the equilibrium lattice constants for yttrium are determined to be a0 = 3.635 Å and c0 = 5.714 Å, and its bulk modulus is B0 = 40.5 GPa, which are very close to the available experimental and other first principle results [21,22], Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 1. Crystalline structure of unit cell of bulk yttrium.
i.e., 3.647 Å, 5.731 Å, 41 GPa, respectively. In other words, the computational method within the framework of PAW and GGA is suitable for the present case. Next, we construct a physically plausible model with an addition of yttrium into the Ni(1 1 1)/a-Al2O3(0 0 0 1) interface. Based on our previous studies [9,12,13], the most stable Ni/Al2O3 interface model is considered, namely Al-terminated O-site Ni(1 1 1)/a-Al2O3(0 0 0 1) model (hereinafter, Al–O model for short) as seen in Fig. 2. The atoms of Ni, Al, O and Y are represented by blue, pink, red and green spheres, respectively. To better illustrate the interface structural feature, the atomic arrangements involving three periodic atomic layers along transverse direction are displayed in the figure. Three axis directions a, b, c in Fig. 2a and b represent ½1 2 1 0, ½1 1 0k½1 0 1 0, and [111]||k½0 0 0 1, respectively. The physically possible positions that are occupied by reactive element Y atoms might be Al substitution site, Ni substitution site, or an interstitial site. The interface models consist of five Ni atomic layers, five O atomic layers, and ten Al atomic layers. And the numbers of atoms for the Al substitution site model and the Ni substitution site model are identical, i.e. 55. For the interstitial site model, the simulation model totally involves 56 atoms. This study considered a substitution of Y at the first or second Ni atomic layer (i.e. Ni1 or Ni2 hereinafter), excluding any further possible substitution positions, as a trade-off between computational accuracy and cost. The work of adhesion, W ad , is often used to assess the stability of an interface in the literature, see e.g. Ref. [23]. It is understood that the higher the work of adhesion, more stable the interfacial structure. W ad is defined as the energy required (per unit area) to reversibly separate an interface into two free surfaces. These dissipative processes are responsible for the fact that energy needed in an actual cleavage experiment is always considerably greater than the ideal work of adhesion. Therefore, our predictions may be considered as lower bounds for the work of adhesion obtained by any cleavage experiment. Formally, W ad is defined in terms of either the surface and interface energies relative to respective bulk materials or by the difference in total energy between the interface slab and its isolated component slabs (substrate and coating) [23]:
W ad ¼
ENi=Y þ EAl2 O3 ENi=Y=Al2 O3 A
ð2Þ
where ENi=Y , EAl2 O3 , and ENi=Y=Al2 O3 are the total energies of the corresponding slab, respectively, A is the interface area. In order to approximately cancel the effect of lattice mismatch, we calculated the energy of the isolated coating using the same lattice vectors as that for the interface. In the present simulation, convergence tests demonstrated that 1–2 meV/Å per atom degree of convergence with respect to k-point sampling was attained upon using 5 5 1 points for the interface systems. By fully relaxing the entire slab, the bonding energy of the Al–O Ni(1 1 1)/a-Al2O3(0 0 0 1) interfaces with yttrium dopant at different sites are obtained, see Table 1. It is observed that, when Y is placed into the interstitial site, the interface is most stable, with an interfacial energy of 3.87 J/m2. That is, a significant enhancement of adhesion is seen by doping Y to the interstitial site of its clean interface, approximately 2.65 times higher in terms of bonding energy. Fig. 3 is the schematic diagram of the most stable interface model with Y dopant at the interstitial site, stretched normal to the interface. l0 is the initial length of the model along the tensile loading direction, and l is its length pulled after a tensile stress increment r. 3. Results and discussion 3.1. Interfacial strength and fracture parameter Fig. 4 shows the stress–strain curves of the clean Ni/Al2O3 interface and the Ni/Al2O3 interface with yttrium dopant. For the clean Ni/Al2O3 interface, normal stress reaches its maximum value of 11.1 GPa at strain of 8%, while for the doped interface, the stress reaches the maximum value of 18.75 GPa at strain of 10%. That is to say, the theoretical tensile strength of the Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 2. The schematic diagrams of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant. (a) Front view; (b) top view; (c) Al-substitution site; (d) interstitial site; (e) Ni1-substitution site; and (f) Ni2-substitution site.
Al–O Ni/Al2O3 interface can be enhanced (here, 69% higher) by Y dopant at the interstitial sites near interface region. Meanwhile, Y dopant also increases the rupture tensile strain by up to 25%. This implies that adhesion of the Ni/a-Al2O3 interface could be improved efficiently by doping with Y. Further, we look at the stress–strain curves obtained by first principle simulations. Here, the important parameter to characterize the property of the interface is the area under the stress–strain curve. This parameter gives an approximate evaluation of the work required to separate the interface, W sep , which is often referred as the work of separation in the literatures [e.g. 7]. From our calculations, W sep for the clean Ni/Al2O3 interface is 1.44 J/m2, while that for the doped interface is
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Z. Bao et al. / Engineering Fracture Mechanics xxx (2015) xxx–xxx Table 1 The work of adhesion for the Al-terminated O-site Ni(1 1 1)/a-Al2O3(0 0 0 1) interfaces with yttrium dopant at different sites. Clean Ni/Al2O3
ENi=Y (eV) EAl2 O3 (eV) ENi=Y=Al2 O3 (eV) a0 ðÅÞ Work of adhesion (J/m2)
Ni/Al2O3 with Y dopant Al substitution site
Ni1 substitution site
Ni2 substitution site
Interstitial site
77.291736 222.239385 297.840002 4.756
73.1236 225.56875 342.982271 4.784
77.100155 222.573227 301.540233 4.81
77.485408 222.418245 301.251027 4.78
73.748702 228.972763 307.775659 4.762
1.46
1.13
1.49
1.09
3.87
Fig. 3. Schematic diagram of the uniaxial tension simulation of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant at the interstitial site.
Fig. 4. Stress–strain curves of the clean Ni/Al2O3 interface and the Ni/Al2O3 interface with yttrium dopant.
3.82 J/m2. It is again seen that its work of separation increases about 165% with interstitial Y dopant. It is noticed that Jarvis Carter [10] reported an adhesion of 3.24 J/m2 for the Ni/Al2O3 interface with Y dopant, which is slightly lower than our prediction. The reason could be that, in their calculation, Y atom occupied the Ni substitution site, which is different from the interstitial site in our model. As already explained before, the most stable configuration of the Ni/Al2O3 interface with Y dopant could be that with Y at an interstitial site. 3.2. Fracture characteristic Fig. 4 contains some information on the failure behavior of the interface as well, noticing the shapes of both the stress– strain curves. As have been discussed in Refs. [8,9,23], the Al–O Ni/Al2O3 interface tends to fail in a ductile manner with some plastic dissipation in the metal. For the doped interface, similarly the stress descends gradually after the maximum, instead of a sudden drop. And this softening period during interface cleavage is quite remarkable, compared to that of the clean O–Al Ni/Al2O3 interface. This feature suggests that the failure may occur in a progressive way, instead of a catastrophic one. To better understand the failure process of the Y doped interface, the variations of the atomic bonds near the interface during tension are plotted in Fig. 5. When the stretching goes on, the atomic bond length of Ni1–Y1 does not vary much, while the bond length of Y1–Al1 and Y1–O1 increase gradually until the tensile strain reaches its critical value. In particular, the latter two bonds elongate apparently when beyond the critical strain. During this dramatic elongation period, the interfacial stress drops gradually as well (Fig. 4). It is thus clear that the Y doped interface breaks at Y1–Al1 bond and Y1–O1 bond, when subject to tensile loading. The valence charge density distribution can also reflect the character of atomic bonds near the interface. Fig. 6 shows the evolution of the valence charge density distribution at the Ni/Al2O3 interface with yttrium dopant during the stretching Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 5. Variations of the atomic bonds of Ni/Al2O3 interface with yttrium dopant during the stretching process. The vertical line indicates the critical strain. Pictorial showing the interfacial bonds are shown in the inset panel. The Ni, Al, O and Y atoms are represented by blue, pink, red and green spheres, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
process. The dashed line in Fig. 6a (a top view of the interface model) indicates the observing plane from a top-down view, and the contour plots of the valence charge densities on this plane for different tensile strains are shown below in Fig. 6c–k. For comparison purpose, the charge density distribution for the clean interface is given in Fig. 6b, showing the ionic Ni–O bonds with each Ni atop each O (Ref. [9]). By comparing to that for the Y doped interface (Fig. 6c), one can see that two ionic Y–O bonds and one metallic bond Y–Al are formed near the Y doped interface, which consequently increase the interfacial adhesion. For the increase of the strain as shown in Fig. 6d–f, the charge density between the interfacial Y and O atoms decreases gradually, as well as that between the Y and Al atoms. When the strain goes over the critical value (Fig. 6g–i), this decreasing tendency becomes more pronounced, along with the appearance of a charge depletion region around the interfacial atoms. With the further increase of the strain until approaching to the ultimate rupture, the charge depletion region extends above the interfacial Al and O atoms (Fig. 6j), and finally, the interfacial region fully separates into two surfaces, one with the original Ni slab and the other with Al2O3 slab. By collecting the results in Figs. 4–6, we can conclude that the Y doped Ni/Al2O3 interface breaks along the at a relatively higher level of tensile stress compared to the clean interface, tends to fracture in a ductile way, and separates exactly along the interlayer between the initial Ni slab and Al2O3 slab. 3.3. Discussion As a typical metal/ceramic interface within a TBC system, Ni alloy/Al2O3 is commonly understood to be relatively brittle and not strong enough [8]. Our simulation results also indicate that the interfaces of clean Ni/Al2O3 and that with yttrium dopant have high theoretical tensile strengths, i.e. 11.1 GPa and 18.75 GPa. Though this brittle nature, its interface separation process at the atomic scale is not a scenario of catastrophic breakage. As clearly shown by our results, Figs. 4–6, it is much like a gradual fracturing of the atomic layers between Ni and Al2O3 slabs, with certain breaking of Y–Al and Y–O bonds. In particular, one may notice both the distinct enhancements of the tensile strength and the work of separation parameters of the Ni/Al2O3 interface with Y dopant relative to the ideal clean Ni/Al2O3 interface, and the former might be much closer to a real Ni/Al2O3 interface in a TBC system. As already noted by Ref. [8], small changes in interface property could induce significant changes in its toughness at the macroscopic scale. Therefore, the above enhancements could be expected to contribute an increase of this interface toughness, and in turn that of a TBC system. Of course, the toughness of Ni/Al2O3 interface, as a macroscopic index of material resistance, is much dependent on many other deformation mechanisms such as plastic dissipation in a Ni alloy, and does not directly correspond with either the work of separation or the theoretical strength. Rather, we would say that, by observing at atomic level, the failure process of brittle Ni/Al2O3 interface does not advance as a Griffith cleavage, but may possess a nature of progressive ductile fracturing. To help us understand the fracture of Ni/Al2O3 interface, one needs certainly to go up from bottom, here the atomic and electronic scales. Jiang et al. [24] presented an example to predict the macroscopic toughness of a c-Ni(Al)/a-Al2O3 interface by utilizing the basic knowledge obtained from first principle calculations, and they also gave a quantitative implementation of this computational strategy. One can see from this nice study that, (i) a traction–separation relationship for the interface is computed using first principle results, which serves as a separation criterion for the interface; (ii) The results of first principle tensile simulation, together with that of shear and mixed mode simulations, are necessities to formulate the so-called interfacial potential. This interfacial potential, as recognized in a pioneering work of Sun et al. [25] and also confirmed by our recent study [13], is much related or even quite sensitive to the characteristic physical phenomena such as the effects of tension–shear coupling and interface relaxation. In developing a proper interfacial potential, its analytical expression with Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040
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Fig. 6. Variations of the valence charge densities of Ni/Al2O3 interface with yttrium dopant during the stretching process. (a) Top view; (b) clean interface; (c) tensile strain = 0%; (d) 4%; (e) 8%; (f) 9.5%; (g) 10% (critical strain); (h) 10.5%; (i) 12%; (j) 16%; and (k) 20%. Units are e/Å3.
specific form must effectively describe all the basic physics taking place at the atomic level. By making use of this potential, one can do more in understanding the fracture of Ni/Al2O3 interface. For example, it is a straightforward work to derive from this potential a traction–separation law of a cohesive zone model, which is commonly adopted in continuum mechanics
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based finite element analysis, e.g. Ref. [26]. More complicated crack problems are also possible to be investigated from this potential, such as a dislocation nucleation from a crack tip, as discussed in Ref. [25]. When going up from an electronic level up to atomic and molecular levels, one would often rely on a physically reliable interatomic potential function, for example, in performing molecular dynamics simulations. In this case, first principle results are valuable in constructing these potential functions between atoms, since first principle calculations provide exact predictions of the chemical, physical and also mechanical properties of the Ni/Al2O3 interface. Normally, when considering the interactions between dissimilar atoms, like here Y–Al and Y–O, the potential functions given in the literature are often inadequate in reproducing its mechanical property. Hence, a task having to face for a mechanics person is to modify or even re-construct such functions, which cannot be an easy work. Then, the mechanical responses of the interface under external loadings should be considered as inputs in the fitting procedure of these functions. In this sense, the first principle result of tensile simulation is of great value. It is the present authors’ opinion that, using the interatomic potential function derived in this way, a reasonable prediction of the interface mechanics from a molecular dynamics simulation could be guaranteed. As mentioned above, first principle tensile simulation results are conducive to the determination of a cohesive zone model with a proper constitutive law. To date, a variety of cohesive zone models are still popular in analyzing the complicated fracture processes, and conventional cohesive finite element method in a continuum mechanics framework is under a continuous development, such as Refs. [27,28]. It is pointed out [29] that, an accurate cohesive zone model should capture the characteristics of atomic-level bond breaking between adjacent atomic layers of, here, the interface region. Thus, an ideal cohesive finite element method would be based on the interface potential that is linked to the atomistic potential obtained from first principle calculations, according to Ref. [27]. This kind of cohesive zone modeling would naturally have some merits in advancing our understanding of fracture. 4. Summary The fracture characteristics of Ni(1 1 1)/a-Al2O3(0 0 0 1) interface with yttrium dopant have been investigated at atomic and electronic levels through a DFT-based first principle calculation. Tensile simulations are performed to extract the fundamental mechanical properties of this interface, including the theoretical tensile strength and the work of separation. Relative to the ideal clean Ni/Al2O3 interface, both the distinct enhancements of the tensile strength (69% higher) and the work of separation (165% higher) are seen for the Ni/Al2O3 interface with Y dopant. The variations of the atomic bonds and valence charge densities near interface region during tension clarify the failure and fracture characteristics of the interface. We found that the Y doped Ni/Al2O3 interface breaks along the Y–Al and Y–O atomic bonds at a relatively higher level of tensile stress compared to the clean interface, tends to fracture in a ductile way, and separates along the interlayer between the initial Ni slab and Al2O3 slab. We also discussed the approaches to use these first principle results to assess the TBC adhesion, including prediction of macroscopic interface toughness, development of a reliable interatomic potential function, and determination of an accurate cohesive zone model. All these aspects are critical in studying the interfacial fracture behavior of a TBC system. Acknowledgements The studies reported here have been supported by NSAF (Grant No. U1330116), NSFC (Grant No. 11272243), and Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110201110019). We thank Dr. Xiaoyuan Wang from CAEP for fruitful discussions on first principle study. References [1] Mumm DR, Evans AG. On the role of imperfections in the failure of a thermal barrier coating made by electron beam deposition. Acta Mater 2000;48:1815–27. [2] Evans AG, Mumm DR, Hutchinson JW. Mechanisms controlling the durability of thermal barrier coatings. Prog Mater Sci 2001;46:505–53. [3] Padture NP, Gell M, Jordan EH. Thermal barrier coatings for gas-turbine engine applications. Science 2002;296:280–4. [4] Mao WG, Dai CY, Zhou YC, Liu QX. 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Please cite this article in press as: Bao Z et al. An atomistic investigation into the nature of fracture of Ni/Al2O3 interface with yttrium dopant under tension. Engng Fract Mech (2015), http://dx.doi.org/10.1016/j.engfracmech.2015.05.040