Materials Transactions, Vol. 53, No. 1 (2012) pp. 156 to 160 Special Issue on Advanced Materials Science in Bulk Nanostructured Metals © 2011 The Japan Institute of Metals
Grain Size Dependence of Creep in Nanocrystalline Copper by Molecular Dynamics Yun-Jiang Wang+, Akio Ishii and Shigenobu Ogata+ Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan The grain size dependence of creep is critical to understand the plastic deformation mechanism of nanoscale metals. Here we used molecular dynamics to study the stress-induced grain size exponent transition in creep of nanocrystalline copper. The grain size exponent was found to initially increase with increasing stress, then decrease after some critical stress. The derived grain size exponents are in agreement with experimental results for diffusional and grain boundary sliding creep at low stress. While, the founded decreasing grain size exponent with increasing stress for dislocation nucleation creep in nanocrystal is in contrast with conventional materials. We propose a constitutive equation for dislocation nucleation governed creep in nanocrystal to explain its grain size dependence transition with stress. [doi:10.2320/matertrans.MD201122] (Received July 29, 2011; Accepted September 5, 2011; Published October 19, 2011) Keywords: creep, nanocrystal, grain size dependence, molecular dynamics
1.
Introduction
Creep is a time-dependent plastic deformation of materials under a constant stress at elevated temperature. It is a critical property of materials for numerous applications, such as microelectronic devices, superalloys used in gas turbines, and nuclear reactors.1) Almost half a century ago, pioneers had established the classical constitutive equation for steadystate creep, which is known as the MukherjeeBirdDorn equation:2) AD0 Gb b P · n Q exp ¾_ ¼ ð1Þ kB T d G kB T where A is constant, D0 is a frequency factor, G is the shear modulus, b is the Burgers vector, kB is the Boltzmann’s constant, and (·, T, d) are the applied stress, absolute temperature, and grain size, respectively. (P, n) are the grain size and stress exponents, respectively. ¦Q is the activation energy for specific thermal-activated process. The deformation map in the conventional materials has been well identified previously.3) At low stress, diffusional mechanisms such as Coble4) and NabarroHerring5,6) creep dominate plastic deformation. The grain boundary (GB) diffusion, and lattice-diffusion creep leads to grain size exponent of P = 3 and P =2, respectively.46) While GB sliding can be also served as a creep mechanism at a bit higher stress,7) in particular when the grain size is down to nanoscale. P = 3 is a typical grain size exponent for GB sliding creep. At very high stress, dislocation behaviors govern creep. There exists a power-law relation between creep rate and stress in dislocation creep (with stress exponent usually larger than 4). However, the power-law creep has no grain size dependence in the conventional coarse-grained materials. This is because the inside grain collective dislocation activities in normal material do not feel too much constraints from GB geometry. As we know, reducing the grain size recently opens novel pathway to balance the trade-off between strength and ductility.813) Size effect is critical to understand plastic deformation mechanism in nanocrystalline (nc) materials. +
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The volume fraction of GB becomes significant when the grain size reaches well below 100 nm. The intragrain dislocations do not have large free space to move anymore in such confined volume. Therefore creep in nanoscale material should present some new characteristics compared with that of conventional material.14) Because of the difficulties in experimental techniques, there were not many studies on creep in actual nc materials.15,16) Instead, the classical molecular dynamics (MD) played an important role in exploring the creep mechanisms in nanocrystal during the last decades.1721) These MD studies have revealed the diffusional mechanisms for creep in the vicinity of melting temperature. The grain size dependence was qualitatively shown to be consistent with Coble or NabarroHerring creep. However, there lacks a description of the grain size dependence of creep in nanocrystal over wide stress range dominated by different mechanisms. In this study, we will show the grain size dependence of nanocreep dominated by different thermal-activated mechanisms. 2.
Methodology
In order to determine the grain size dependence of nanocreep, we adopted four nanocrystalline copper models with different grain size of 8 nm, 10.7 nm, 13.3 nm, and 16 nm, respectively. The 16 identical 3-dimensional Voronoi grains with random orientation were arranged in a bodycentered cubic distribution in the supercells (see the inset in Fig. 1). The biggest sample with d =16 nm has dimension of 32.5 © 32.5 © 32.5 nm, tolerating about 2,850,000 atoms. We imposed a periodic boundary condition in all the three directions. The embedded-atom method potential was used to describe the atomic interaction.22) All the simulation were performed using NoséHoover thermostat.23,24) Stress was controlled by ParrinelloRahman method.25) The initial structures were firstly fully relaxed, and then thermally equilibrated to the targeted temperature for 200 picoseconds before loadings. Then we applied uniaxial stress · with different magnitude on the samples within a N·T ensemble. In order to observe creep in the very limited MD timescale, we carried out the calculations at high temperature to accelerate the processes. Figure 1 shows a typical creep
Grain Size Dependence of Creep in Nanocrystalline Copper by Molecular Dynamics
Fig. 1 A typical creep curve of nc copper with grain size of 10.7 nm under stress 1.33 GPa, and at temperature 960 K. The initial nc model before being thermally equilibrated is shown as the inset.
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Fig. 2 Creep curves of samples with varied grain size under (0.66 GPa, 960 K).
Fig. 3 Snapshots of sample with grain size 10.7 nm during steady-state creep for (a)(d) 0.66 GPa, 960 K, and (e)(h) 1.66 GPa, 960 K. The applied uniaxial stress is · = ·xx. Atoms are colored by centro-symmetry parameter,26) which can distinguish the atoms in perfect lattice sites, and those in GBs and stacking-fault.
curve of the sample with grain size d = 10.7 nm at 960 K under constant stress of 1.33 GPa. The creep rate is changing with increasing plastic strain or time. It quickly reaches a constant value after initial adjustment according to the applied uniaxial stress based on the elastic relation. The stage with constant strain rate is termed as the steady-state creep. The steady-state creep rate could be determined by the slope of creep curve in our calculations. Based on these creep data under a (T, ·, d) condition mesh, we can derive the important parameters shown in eq. (1). Here, we mainly focus on the grain size exponent P of different mechanisms governed creep in nanocrystal. 3.
Results and Discussions
First, we carry out the creep of samples with different grain size at several applied stress levels. Figure 2 is one example of them at 0.66 GPa, and 960 K. The smaller the grain size is, the higher the strain is at the same time. So that the creep rate of sample with smaller grain size possesses higher creep rate.
Creep at other stress levels holds the same trend. This is reasonable to creep, leading to a positive grain size exponent P. Next, we will provide data for ¾_ d, so that the grain size log ¾_ exponent could be reached as P ¼ @@ log d according to eq. (1). Before discussing the grain size dependence of creep in nanocrystal, we would like to distinguish the different deformation mechanisms at low and high stress level. Thus we can specify clearly that what kind of mechanism is corresponding to the obtained different grain size exponent. Figure 3 shows the MD snapshots of samples during steadystate creep processes at stress of 0.66 GPa and 1.66 GPa, respectively. At low stress level, snapshots (a)(d) demonstrate clearly that there is no dislocation behaviors during creep. On the other hand, because of the significant GB volume fraction in nanostructured Cu, GB diffusion and sliding may be possible candidates for the deformation mechanisms. Table 1 shows that the stress exponent of _ log · is determined as 1.08 « 0.19 within creep n ¼ @ log ¾=@ stress interval 66133 MPa. On the other hand, n becomes
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Table 1
Creep rate ¾_ and stress exponent n at different stress levels.
stress (MPa)
¾_ (©106 s¹1)
n
66 100
1.01 1.74
1.08 « 0.19
133
2.11
166
3.10
332
14.18
498
26.08
664
53.38
2.01 « 0.10
Fig. 5 Grain size exponent P of creep in nanocrystalline copper as a function of applied stress.
Fig. 4 Log-log plot of the steady-state creep rate against grain size at different stress level.
2.01 « 0.10 at stress from 166 MPa to 664 MPa. The derived stress exponents are corresponding to Coble4) and GB sliding7) creep, respectively. Furthermore, we notice that grain morphologies change with time during creep at stress 0.66 GPa. Both of these information propose GB sliding as the dominating mechanism of nanocreep at 0.66 GPa. However, dislocation activities begin to dominate creep at high stress level, as shown in Fig. 3(e)(h). Here the dislocation activities are quite different from the conventional metals, where dislocations have plenty of free space to move and multiply inside grain. The dislocation creep in these material is grain size independent. But for nanostructured metals in our case, intragrain dislocation activities are constricted a lot by the confined volume in the extreme small grains. So that nanocreep could be anticipated to be a source-controlled plastic mechanism. Many previous works have suggested dislocation nucleation as the controlling deformation mechanism in nanocrystalline metals.2729) However, the grain size dependence of such GB dislocation nucleation governed creep is still unclear. In particular, there lacks knowledge about the stress-coupled grain size dependence of power-law creep in nanocrystals, and its origin for this dependence. The log-log relationships of creep rate versus grain size are shown in Fig. 4 at stress levels of 0.50, 0.66, 0.83, 1.50, 1.66, and 1.83 GPa, respectively. All the plots lie in a well-defined linear relation. This illustrates that the grain size exponent is a constant number within the grain size range considered. The slopes of these linear relation shifts with increasing stress, attributing to the deformation mechanism transition. At stress
0.50 GPa, the dominating mechanism is competing diffusion and GB sliding. While the GB sliding begins to overwhelm at the stress of 0.66 GPa. With stress above 0.83 GPa, dislocation nucleation is served as the dominating mechanism, although GB sliding still exists here as an assistant mechanism. Next, we will discuss the grain size dependence of nanocreep at varied stress levels. Figure 5 apparently shows the calculated grain size exponent of creep in nc copper as a function of stress. At stress · = 0.50 GPa, the grain size exponent is determined as P = 2.21 « 0.44. As we has mentioned previously, the deformation mechanism at this stress is competing diffusion and GB sliding. Coble and GB sliding creep both have the grain size exponent P = 3.4,7) Whereas it is P = 2 for NabarroHerring creep.5,6) So that the obtained P µ 2.2 is actually an effective value for both diffusional and GB sliding creep. Maybe the contribution from lattice-diffusion here is larger compared to that from GB diffusion and sliding. Because the derived grain size exponent is near to that of NabarroHerring creep. When the stress increases, the reliance of creep on grain size reaches to be P = 3.02 « 0.43 at · = 0.66 GPa. This is exactly corresponding to the classical ¾_ d relation for GB sliding creep. Snapshots in Fig. 3(a)(d) also support the derived grain size exponent P µ 3 as that for GB sliding creep. Of course GB sliding itself is provided by boundary diffusion. As a result, we can not exclude Coble creep as a mechanism at stress of 0.66 GPa. The higher the stress is, the more contribution GB sliding does. Because GB sliding is a collective activity of a group of atoms, it needs more external work to activate compared to that of diffusion. It is very interesting to find a decrease in grain size exponent from stress 0.66 GPa to 0.83 GPa. As we know, this transition is corresponding to a mechanism transition from GB sliding to dislocation nucleation. The classical power-law creep dominated by dislocation glide, multiplication, or climb has stress exponent n = 46 and no grain size dependence.1) This is because the pre-existing dislocation activities are inside of grain in the conventional coarse-grained materials. These dislocations feel little constraint from GB geometry. Therefore the power-law creep in normal material has no grain size dependence. However, we find a grain size
Grain Size Dependence of Creep in Nanocrystalline Copper by Molecular Dynamics
exponent of P = 2.65 « 0.23 at stress 0.83 GPa. Nevertheless, the exponent is decreasing with increasing stress almost linearly. It turns to be P = 1.4 « 0.32 when the stress reaches 1.83 GPa. Why power-law creep is grain size dependent in nanocrystal? To answer this question, let’s first consider the difference between conventional and nanostructured materials. As we have mentioned, the conventional power-law creep is governed by the collective dislocation activities in the interior of the grains. Some of the pre-existing dislocations are ready to move with minor hinderance from GBs. In contrast, new dislocation emission from grain boundary requires more local stress concentrated. However, the mobility of dislocations are constricted significantly by GBs in nanocrystalline materials. FrankRead sources do not have enough space to bowing out dislocations in confined volume. Instead, dislocation nucleation on boundary becomes necessary and possible in the small clean nanograins. Therefore, the plastic deformation in nanocrystals is dominated by GB dislocation nucleation, a source-controlled mechanism. The GB dislocation nucleation is a strongly size dependent activity.27) As a result, the nanocreep dominated by dislocation nucleation has size effect. Next, we will reveal the variation in grain size exponent of creep in nanocrystal. According to eq. (1), P can be derived as: P¼
1 _ ðn ln · ln ¾ð·ÞÞ þ const: ln d
ð2Þ
at a constant temperature. Here the grain size is ranging from 8 to 16 nm in our calculations. ln d does not change too much within this interval, so that it can be regarded as a parameter. The stress exponent n is a constant number for specific mechanism dominated creep within required stress regime. For example, the Coble or NabarroHerring creep rate is proportional to stress. Whereas n is equal to 2 for grain boundary sliding creep. The n will increase to a value usually larger than 4 when the dominating deformation mechanism involves with dislocation activities. It turns to be even larger value to power-law breakdown creep. Anyhow, n always increases with increasing stress. As a result, n ln · may increase with increasing stress. However, creep rate is always increased by the increasing applied uniaxial stress ·. Consequently, there should be competition between n ln · and ln ¾_ to determine P. This probably results in the trend of transition in grain size exponent with stress, namely, it initially increasing with stress, and then decreases after a critical point. Actually, the grain size dependence of creep dominated by diffusion and GB sliding were well understood. For lattice diffusion, the vacancy flux is proportional to exp(«·³/kBT)/ d. Consequently, the global strain rate could be determined as the flux divided by the grain size d. This leads to a 1/d 2 dependence of creep rate on grain size. But in Coble or GB sliding creep, there is another factor ¤/d describing the ratio between grain boundary cross section area and grain itself, where ¤ is the grain boundary width. Instead of having grain size exponent P = 2, now it is P = 3 for mechanisms involving with boundary diffusion. However, to the best of our knowledge, there exists no clear theory about creep in
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nanocrystal governed by dislocation nucleation. Physically, the creep rate accommodated by dislocation nucleation could be suggested as: 3¤ b G ¯0 exp ð3Þ ¾_ ’ N d d kB T where N is the atomic number of each grain plus grain boundary unit, 3¤/d describes the volume fraction of GB atoms, b/d is the unit strain that each dislocation nucleation from GBs contributes to macroscopic strain, ¯0 is a trial frequency, ¦G = ¦Q ¹ ·³ ¹ T¦S is the activation Gibbs free energy. Here we define the apparent activation volume as _ @G=@· ¼ kB T @ ln ¾=@·. ¦S is the activation entropy for creep. Whereas expð kG Þ is proportional to the BT probability of dislocation nucleation occurs. Naturally, there is a grain size dependence of ¾_ / d 2 for GB dislocation nucleation governed creep in nanocrystal. But ¦G is a function of grain size. As a result, if we apparently derive _ ln d based on eq. (3), it grain size exponent by P ¼ @ ln ¾=@ will lead to the formula P ¼2
· @ 1 @Q þ kB T @ ln d kB T @ ln d
ð4Þ
at the same temperature and stress. Generally speaking, activation energy ¦Q should be a function of grain size d in nanoscale metals. However, here we only focus on the relation between grain size exponent P and stress ·. Within small interval of grain size in our calculations, the term 1 @Q kB T @ ln d maybe do not change too much with response to stress. Thus it can be regarded as a parameter here. As a result, the stress dependence of P mainly comes from the second term on the right side of eq. (4). It is well-known that activation volume for dislocation nucleation is a function of grain size for nanoscale metals. ³ increases with increasing grain size.27) So that @³/@ ln d is a positive quantity. Consequently, P decreases with increasing stress for dislocation nucleation governed creep in nanocrystal. This is consistent with the trend shown in Fig. 5 at high stress. 4.
Summary
In summary, we use classical molecular dynamics to check the grain size dependence of creep in nanocrystal. The grain size exponent has been shown clearly from P µ 2 for diffusional creep, to P µ 3 for GB sliding creep, and then decreases with increasing stress for dislocation nucleation governed creep. The obtained grain size exponents are consistent with experimental values. Besides, the finding of size effect in nanocreep is in contrast with that in conventional coarse-grained material. This is because the preexisting dislocation activities governed plasticity in normal materials are replaced by GB dislocation nucleation in nanocrystalline counterparts. In order to explain the decrease of grain size exponent with increasing stress, we propose a constitutive equation for GB dislocation nucleation governed creep in nanocrystal. This successfully explains the decrease of P with increasing stress in dislocation creep of nanoscale metals. Our results provide intuitive understanding on the novel size effect of creep in nanocrystals.
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Acknowledgements We would like to thank Professor Hajime Kimizuka for helpful discussions and guidance in MD simulations. This work is partially supported by the “Grant-In-Aid” project of JSPS Postdoctoral Fellowships for Foreign Researchers (Grant No. P10370), and Scientific Research on Innovative Area, “Bulk Nanostructured Metals” (Grant No. 22102003), Scientific Research (A) (Grant No. 23246025), and Challenging Exploratory Research (Grant No. 22656030). REFERENCES 1) M. E. Kassner: Fundamentals of Creep in Metals and Alloys, (Elsevier, Amsterdam, 2009). 2) A. K. Mukherjee, J. E. Bird and J. E. Dorn: Trans. ASM 62 (1964) 155. 3) M. F. Ashby and R. A. Verrall: Acta Metall. 21 (1973) 149. 4) C. Coble: J. Appl. Phys. 34 (1963) 1679. 5) F. R. Nabarro: Report of a Conference on Strength of Solids, (The Physical Society, London, 1948) p. 75. 6) C. Herring: J. Appl. Phys. 21 (1950) 437. 7) H. Lüthy, R. A. White and O. D. Shereby: Mater. Sci. Eng. A 39 (1979) 211. 8) J. Schiøtz and K. W. Jacobsen: Science 301 (2003) 1357. 9) L. Lu, Y. Shen, X. Chen, L. Qian and K. Lu: Science 304 (2004) 422. 10) S. Yip: Nature 391 (1998) 532.
11) E. Ma: Science 305 (2004) 623. 12) Z. W. Shan, E. A. Stach, J. M. K. Wiezorek, J. A. Knapp, D. M. Follstaedt and S. X. Mao: Science 305 (2004) 654. 13) T. Zhu and J. Li: Prog. Mater. Sci. 55 (2010) 710. 14) B. Wilshire and M. T. Whittaker: Acta Mater. 57 (2009) 4115. 15) Z. H. Jiang, X. L. Liu, G. Y. Li, Q. Jiang and J. S. Lian: Appl. Phys. Lett. 88 (2006) 143115. 16) P. Huang, F. Wang, M. Xu, K. W. Xu and T. J. Lu: Acta Mater. 58 (2010) 5196. 17) H. Van Swygenhoven and A. Caro: Phys. Rev. B 58 (1998) 11246. 18) V. Yamakov, D. Wolf, S. R. Phillpot and H. Gleiter: Acta Mater. 50 (2002) 61. 19) V. Yamakov, D. Wolf, S. R. Phillpot and H. Gleiter: Nat. Mater. 3 (2004) 43. 20) D. Wolf, V. Yamakovb, S. R. Phillpot, A. Mukherjee and H. Gleiter: Acta Mater. 53 (2005) 1. 21) P. C. Millett, T. Desai, V. Yamakov and D. Wolf: Acta Mater. 56 (2008) 3688. 22) Y. Mishin, M. J. Mehl, D. A. Papaconstantopoulos, A. F. Voter and J. D. Kress: Phys. Rev. B 63 (2001) 224106. 23) S. Nosé: Mol. Phys. 52 (1984) 255. 24) W. G. Hoover: Phys. Rev. A 31 (1985) 1695. 25) M. Parrinello and A. Rahman: J. Appl. Phys. 52 (1981) 7182. 26) J. Li: Model. Simul. Mater. Sci. Eng. 11 (2003) 173. 27) R. J. Asaro and S. Suresh: Acta Mater. 53 (2005) 3369. 28) T. Zhu, J. Li, A. Samanta, A. Leach and K. Gall: Phys. Rev. Lett. 100 (2008) 025502. 29) X. Y. Li, Y. J. Wei, L. Lu, K. Lu and H. J. Gao: Nature 464 (2010) 877.
