Size-dependent transition of deformation mechanism, and nonlinear elasticity in Ni3Al nanowires Yun-Jiang Wang, Guo-Jie J. Gao, and Shigenobu Ogata Citation: Appl. Phys. Lett. 102, 041902 (2013); doi: 10.1063/1.4789528 View online: http://dx.doi.org/10.1063/1.4789528 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v102/i4 Published by the American Institute of Physics.
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APPLIED PHYSICS LETTERS 102, 041902 (2013)
Size-dependent transition of deformation mechanism, and nonlinear elasticity in Ni3Al nanowires Yun-Jiang Wang,1,2,a) Guo-Jie J. Gao,1 and Shigenobu Ogata1,2,b) 1
Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan 2 Elements Strategy Initiative for Structural Materials, Kyoto University, Kyoto 606-8501, Japan
(Received 23 November 2012; accepted 10 January 2013; published online 28 January 2013) A size-dependent transition of deformation mechanism is revealed in Ni3Al nanowire under atomistic uniaxial tension. Deformation twinning is replaced by phase transformation when the diameter of Ni3Al nanowire reduces to a critical value near 4 nm. Enhanced size-dependent nonlinear elasticity is observed in the nanowires, in comparison to their bulk counterpart which is benchmarked by combined density functional and atomistic study. This study provide fundamental C 2013 understanding on the size-dependent deformation mechanisms of nanostructured alloys. V American Institute of Physics. [http://dx.doi.org/10.1063/1.4789528] Ni3Al is well-known as the c0 phase, which is the strengthening phase of nickel-based superalloys. It is a L12 (ordered fcc structure) intermetallic compound with superior mechanical properties, especially at elevated-temperature.1 It is intrinsically brittle with properties similar to both a ceramic and a metal. Therefore, it should be embedded in c phase (nickel-based solid solution) to present high strength and reasonable ductility simultaneously. Deformation twinning has been found by experiment as a significant deformation mechanism of c0 -Ni3Al at intermediate temperature.2 Smaller is usually stronger.3 There is great interest to understand how Ni3Al is deformed, and what kind of mechanical properties should be anticipated when the characteristic size is smaller than 100 nm, e.g., in a nanowire form. In confined volume, the conventional plastic carriers-dislocations behave anomalously as those coarse-grained polycrystals.4 Small characteristic length of nanostructures leads to unique mechanical properties via different deformation mechanisms in contrast with their bulk counterparts.5,6 Quite recently, sizedependent ultrahigh strength is detected in dislocation-free Ni3Al nanocubes since surface dislocation nucleation replaces Frank-Read source dislocation “forest” interaction as the dominating deformation mechanism.7 Besides Ni3Al, in situ mechanical testings show that surface-related deformation twinning governs the deformation of magnesium and titanium alloy single-crystalline nanowires.8,9 However, most of the experiments and molecular dynamics (MD) have focused on mono-elemental ductile metals.10–15 Little knowledge exits on the fundamental deformation mechanisms of nanowires of brittle alloys. Deformation mechanisms in nanowires must have deep origin on the increasing significant role of surface. Recently, experiments16 and atomistic statics17 discovered an obvious nonlinear elasticity in nanowires of fcc metals, which implies not only bulk nonlinear elasticity becomes perceptible due to the wide elastic deformation range in defect starved nanospecimen but also there exists a stronger anharmonic elastic behavior of surface compared with bulk. Here, we would a)
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b)
0003-6951/2013/102(4)/041902/5/$30.00
like to explore the size effect on the plastic deformation mechanism, and the elastic nonlinearity in Ni3Al nanowires. MD reveals that the phase transformation becomes the dominating deformation mode instead of deformation twinning when the diameter of nanowire is extremely small. They found that deformation twinning, phase transformation, and elastic nonlinearity are in agreement with experimental observations in different kinds of nanowires. Based on both density functional theory (DFT) and MD analyses on the size-dependence of second- and third-order elastic moduli, anharmonicity is found to become considerably enhanced at smaller size. The MD uniaxial tensile simulations are performed with embedded-atom method (EAM) potential for Ni-Al system developed by Mishin.18 This potential gives very reasonable lattice properties and planar faults energetics of L12 Ni3Al as compared to experiments. It also predicts ab initio comparable formation energies of Ni3Al in D03 and D019 structures. Those points enable us to explore reliable phase transformation and dislocation behaviors in Ni3Al nanowires. All the wires are created in a cylindrical shape with the ratio of length and diameter at least larger than roughly 3. The diameter of free nanowires range from 2 nm to 20 nm, with h100i as the longitudinal direction. Here, we focus on h100i direction in order to provide some comparison of deformation mechanisms between nanostructured Ni3Al and its bulk form in Nickel-based superalloys in service.2 Periodic boundary condition is applied to the wire axis, while keeping considerable vacuum layer outside of the wires. The models contain up to 2.7 106 atoms depending on diameter of nanowires. The surface atoms of the wire are initially relaxed by the conjugate gradient algorithm to their local stable sites. Before loading, the wires are thermally equilibrated for 200 ps with a isothermal-isostress ensemble, which ensures stress-free condition along the longitudinal direction. Then, we apply tension with a constant engineering strain rate of 108 s1 to the wires. The true stress is derived by summing up atomic stress, and divided by the initial wire volume within Virial theorem.19 Nose-Hoover thermostat is used to keep constant temperature 300 K.20 Atoms in all the snapshots are recognized by common neighbor analysis (CNA) and visualized by ATOMEYE.21 The DFT calculations of ideal tensile strength
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of bulk Ni3Al are carried out by VASP code.22 The generalized gradient approximation (GGA) of projector augmented wave (PAW)23 is adopted for parametrization of the exchange-correlation functional. A unit cell of Ni3Al with 4 atoms are described by a 21 21 21 k-points with a regular Monkhorst-Pack scheme. Details about the first principles ideal strength calculations could be found elsewhere.24–26 We first perform a series of atomistic tensile tests on Ni3Al nanowires to distinguish their intrinsic deformation mechanism. The diameter d of cylindrical nanowires ranges from 2 nm to 20 nm. The length is kept as 35.4 nm when the diameter is smaller than 10 nm, while the rest is 53.8 nm. Stress-strain curves are shown in Fig. 1(a). Four features are noticed in those curves: (i) All the wires fails at similar critical strain of e 0:08; (ii) wires with diameter greater than 4 nm experience an abrupt drop after the critical strain; (iii) wires with diameter d 4 nm evolve to a steady-state after the critical strain; and (iv) there exists strain stiffening in elastic deformation. A strong size effect is found relating to the critical stress and wire diameter d, as shown in Fig. 1(b). The critical stress increases with increasing diameter, saturating at 12.2 GPa when d 8 nm because surface effect becomes ignorable. The peak stress is strain rate sensitive in nanostructured wires. However, we could extrapolate the present result at strain rate of 108 s1 to experimental rate of 103 s1 if we consider the frequency of rate-controlling defect nucleation.27 The extrapolated data could be 1 GPa lower in order, and is in good agreement with experimental value around 10 GPa when the diameter of wire is as small as 200 nm.7 The size effect in stress could drive a transition in deformation mechanisms from phase transformation to deformation twinning with increasing d. The typical deformation modes of those nanowires are presented in Fig. 1(c). For the smallest wire with d ¼ 2 nm, two type of phase transformations successively take place with increasing imposed strain. They
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are L12 ! D03 , and then D03 ! D019 , which is corresponding to the two steady-states shown in the stress-strain curve. The atoms in a D03 structure sit on the sites of a bcc lattice, shown as the light green zone of 2 nm wire in Fig. 1(c). While D019 is a lattice with hcp structure. The space group of L12, D03, and D019 are Pm3m, Fm3m, and P63/mmc, respectively (see Fig. S1 in supplementary material28). Thus, the found phase transformation is actually fcc ! bcc ! hcp. The former fcc ! bcc phase transformation has been observed in sub-10-nm-sized gold nanowire through in situ highresolution transmission electron microscopy observations.5 The wire with d ¼ 3 nm also experiences such a two-stage phase transformation. However, only a L12 ! D019 phase transformation happens in d ¼ 4 nm nanowire. The deformation mechanism changes to deformation twinning from d ¼ 6 nm. Successive 1=6f111gh112i Shockley partial dislocations nucleate from the surface of wires, forming deformation twinning with different thicknesses, which is shown in Fig. 1(c) as the blue color stacking faults in wires with d ¼ 6 – 16 nm. Transition of deformation mechanisms in free Ni3Al nanowires are more detailed in movie S1 and S2 (see supplementary material28). Deformation twinning also leads to sharp fracture surface, see Fig. S2 (in supplementary material28). It corresponds to the brittle nature of Ni3Al. The discovered deformation twinning in Ni3Al nanowires is in agreement with that of c0 -Ni3Al in Ni-based superalloy at intermediate temperature.2 We should note that similar inelastic deformations have been also observed in a BCC (B2)-type NiTi nanostructure.29 Besides, nanowires of other metallic systems also present deformation twinning in experiments.8,9 What is the driving force underlying the size-dependent transition of deformation mechanisms in nanowires? Regarding to the large surface-volume ratio in these small wires, surface stress could be a possible candidate. It can be large enough to initiate phase transformation.19 Actually, Zheng et al. has already indicated that phase transformation is
FIG. 1. Transition of deformation mechanisms with varying diameter of Ni3Al nanowire. (a) Stress-strain curves. (b) Critical stress as a function of diameter. The inset shows the normalized critical stress by the corresponding Young’s modulus. (c) Deformation mechanism changes from deformation twinning to phase transformation when reducing the diameter. Atoms colored by blue, light blue, and brown represent those in fcc, hcp, and disordered lattice, respectively, for 2 nm and 4 nm wires. For the rest, blue, and brown atoms means hcp, and disordered lattice, while the fcc atoms are removed for clarity. Movie S1 and S2 are provided in the supplementary material28 demonstrating the deformation mechanisms of nanowires with diameter 2 nm and 8 nm.
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FIG. 2. Nonlinear elasticity of free wires. (a) E and D as a function of wire diameter d. (b) Strain-expanded nonlinearity parameter b varies with d.
size-dependent, and induced by surface stress in their experiments.5 Interestingly, before the onset of plastic deformation, nonlinear elastic behavior has been noticed in those clean nanowires both in our atomistic simulations and previous experiments.7,16 In the inset of Fig. 1(b), we normalize the critical stress r by its Young’s modulus E. r=E is almost a constant value of 0.07 regardless of diameter, implying a large elastic regime before catastrophic failure. Nonlinear behavior could be expected even prior to plasticity with so large elastic region. As we have mentioned in Fig. 1(a), strain stiffening is an evidence of nonlinear elasticity. Strain stiffening in h100i direction we find here is consistent with experiment on Ni3Al nanocubes.7 In the next part, we will analyze the interesting elastic nonlinearity observed in those clean nanowires via both MD and DFT. From the continuum mechanics, the response of elastic strain energy density to strain is expanded in a Taylor serious30 UðgÞ 1 1 ¼ Cij gi gj þ Cijk gi gj gk þ ; V0 2! 3!
(1)
where UðgÞ is the strain energy, V0 is the reference volume before deformation, Cij and Cijk are the second- and thirdorder elastic constants, respectively, and gi is a component of Green-Lagrangian (G-L) strain tensor expressed by Voigt notation. The 2nd Piola-Kirchhoff (P-K) stress tensor which is energy conjugate to the G-L finite strain tensor could be derived as rP-K ¼ V10 @UðgÞ @g . In the case of uniaxial tension, one could simplify the P-K stress scalar rP-K as a function of normal G-L strain in tensile direction 1, eG-L ð¼ g1 Þ, i.e., r
P-K
¼ Ee
G -L
þ Dðe
G -L 2
Þ þ ;
(2)
where F is the deformation gradient tensor and J ¼ detF is the Jacobian determinant. eGL is related to engineering strain eE through eGL ¼ eE þ ðeE Þ2 =2, where eE ¼ F1 1 is the normal engineering strain in tensile direction 1. In Fig. 2, we plot Eh100i ; Dh100i , and bh100i as a function of wire diameter, respectively. It is noteworthy that Dh100i is positive, which is corresponding to strain stiffening of tension in h100i direction. This is in contrast with tension in h111i direction, which is strain softening (see Table I). Tension along h111i direction will be demonstrated by both DFT and MD in the next part. Both Eh100i and Dh100i are sizedependent, saturating at around 6 nm. These behaviors in moduli lead to a strong size-dependent nonlinearity parameter bh100i , which decreases from about 6 to 2 with increasing diameter. The smaller the wire is, the stronger the anharmonicity will be. bh100i 2 agrees qualitatively with that of fcc Cu,17 and could be regarded as the contribution from the bulk-like inner core of wires (bh100i ¼ 3:3 estimated from bulk Ni3Al at 300 K). The obvious anharmonicity found in metallic nanowires is in agreement of a recent experiment on Pd wires as thin as 30 nm.16 Anharmonicity has been also demonstrated as a strong entropic effect on dislocation nucleation frequency from a wire surface.31,32 It is instructive to compare the deformation mechanisms with that of variation in bh100i . When nonlinearity parameter starts to increase with decreasing d at 6 nm, deformation twinning is replaced by phase transformation. Larger bh100i means more significant surface stress effect and more enhanced anharmonicity. It seems that variation in anharmonicity accompanies the TABLE I. Nonlinear elasticity of bulk Ni3Al along different directions from different methods: DFT, EAM, and experiment. Method
T (K)
E (GPa)
[100]
DFT EAM EAM Exp.a
0 1 300 5
112.7 6 2.6 125.6 6 0.2 122.6 6 0.3 117
179.8 6 36.2 484.0 6 2.5 410.2 6 3.1
1.60 3.85 3.35
[111]
DFT EAM EAM Exp.b
0 1 300 5
293.4 6 3.1 296.7 6 0.3 277.8 6 0.2 285
926.5 6 29.4 997.3 6 2.3 980.1 6 1.9
3.16 3.36 3.53
Direction
where E is the second-order elastic constant or Young’s modulus, while D is the third-order modulus, which implies the magnitude of nonlinearity in elasticity. There is a socalled strain-expanded nonlinearity parameter b ¼ E/D, which directly scales the level of anharmonicity since it is related to the third-order derivative of strain energy to strain.16 Since all the stress-strain curves shown in this study is a Cauchy stress-engineering strain scheme, we translate them to the P-K stress-G-L strain scheme before doing leastsquares quadratic fitting. The 2nd P-K stress tensor rPK is related to Cauchy stress tensor rC via rPK ¼ JF1 rC FT ,
a
Reference 33. Reference 33.
b
D (GPa)
b
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FIG. 3. (a) Mechanical responses of bulk Ni3Al along different directions by DFT and EAM at different temperature. (b)(d) Failure modes of bulk Ni3Al, which are corresponding to the points labeled in (a). Atoms are colored by CNA as that of Fig. 1.
transition in deformation mechanisms in free wires. Moreover, anharmonicity turns to be less pronounced at higher temperature, which is demonstrated Fig. S3 in supplementary materials.28 In order to provide some benchmarks for the future study on nonlinear elasticity of nanostructured metals and alloys, we show in Fig. 3 about the stress-strain curves and deformation patterns of bulk Ni3Al by both DFT and MD along h100i, as well as h111i direction. Comparison with DFT enables us to know the validity of the present empirical EAM study on nonlinear elasticity. The stress-strain curves from EAM at 1 K agree well with those from DFT, in particular, along h111i direction. EAM overestimates the stain stiffening of tension along h100i direction. However, EAM could predict a qualitatively correct trend about strain stiffening or softening in comparison to DFT. It is also interesting to notice that bulk Ni3Al fails at smaller strain than the ideal critical strain given by DFT. It is because other deformation mode happens prior to ideal yielding, as illustrated by the snapshots just after failure in Figs. 3(b)–3(e). For h100i direction, MD predicts a global deformation twinning at 1 K happens at the critical point (Fig. 3(b)). However, deformation twinning turns to be more localized at 300 K (see the nanotwin networks in Fig. 3(c)). Fig. 3(d) shows a catastrophic fracture behavior at 1 K under h111i tension, which is corresponding to the intrinsically brittle nature of Ni3Al alloy. Instead, homogenous dislocation nucleation prevails at 300 K. In Table I, we summarize the elastic properties of bulk Ni3Al by different studies, such as DFT, EAM, and experiment.33 EAM predicts the Young’s modulus Eh100i to be 125.6 GPa and Eh111i to be 296.7 GPa at extremely low temperature, which is comparable to both DFT and experimental data. As we have mentioned previously, EAM overestimates the magnitude of Dh100i . But at least EAM potential is reliable to shown a correct sign of bh100i and bh111i , and therefore, a qualitatively right trend in the nonlinear elasticity of different direction. DFT here benchmarks the magnitude of elastic nonlinearity of bulk Ni3Al to be bh100i ¼ 1:60 and bh111i ¼ 3:16 direction, respectively. This could be regarded as the intrinsic property of fcc metals directly from the electronic scale atomic bonding nature. Therefore, the anharmonicity we found in those nanowires is actually from the inner core bulk-like region, but further enhanced by the small size and large surface of nanowires. It is in agreement with previous conclusions from both experiment and simulation.16,17
In summary, large scale molecular dynamics predict a transition of deformation mechanism in nanoscaled Ni3Al nanowires. Phase transformation replaces deformation twinning as the dominating deformation modes when the diameter of nanowire reduces from 20 nm to as small as 4 nm. Deformation twinning of Ni3Al nanowires agrees with that of their bulk counterpart which is served as the strengthening phase in Nickel-based superalloys, as well as those nanowires of other alloys demonstrated by experiments.2,8,9 Transition to phase transformation is also reported in gold nanowires by in situ transmission electron microscopy observations when the diameter is in the scale of sub-10-nm.5 The transition in deformation mechanism induces a size effect on the critical stress of nanowires. Besides transition in deformation modes, strong size-dependent nonlinear elasticity or anharmonicity is found in the elastic deformation of those wires. It implies a increasing important role of surface stress on the deformation of nanostructured metals and alloys. We should note that although strong anharmonicity is found in Ni3Al nanowires, it is also supported by recent experiments on Ni3Al nanocubes,7 Pd,16 and Cu nanowires.15 Combined DFT and EAM studies on bulk Ni3Al validate the present atomistic investigations on Ni3Al nanowires, and provide benchmark for the magnitude of nonlinearity. This work was partially supported by a Grant-In-Aid for JSPS Fellows (No. P10370), Scientific Research on Innovative Area “Bulk Nanostructured Metals” (No. 22102003), Scientific Research (A) (No. 23246025), Challenging Exploratory Research (No. 22656030), the Elements Strategy Initiative for Structural Materials (ESISM), and JST under Collaborative Research Based on Industrial Demand (Heterogeneous Structure Control). 1
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