- Email: [email protected]

Generalized stacking fault energy in FCC metals with MEAM Xiu-Mei Wei a, Jian-Min Zhang a,*, Ke-Wei Xu b b

a College of Physics and Information Technology, Shaanxi Normal University, Xian 710062, Shaanxi, PR China State-Key Laboratory for Mechanical Behavior of Materials, Xian Jiaotong University, Xian 710049, Shaanxi, PR China

Received 31 January 2007; accepted 3 July 2007 Available online 17 July 2007

Abstract The second nearest-neighbor modified embedded atom method (2NN-MEAM) is used to investigate the generalized stacking fault (GSF) energy surfaces of eight FCC metals Cu, Ag, Au, Ni, Pd, Pt, Al and Pb. An offset is observed in all the metals for the displacement dus of unstable stacking fault energy from the geometrically symmetric displacement point d0us . The offset value is the greatest for Al and the smallest for Ag. By analyzing the stable stacking fault energy gsf and unstable stacking fault energy gusf, it can be predicted that stacking fault is more favorable in Cu, Ag, Au, and especially in Pd than the other metals, while it is most preferred to create partial dislocation for Ag and to create full dislocation for Al. # 2007 Elsevier B.V. All rights reserved. PACS : 61.43.Bn; 61.72.Mm; 98.38.Bn Keywords: Stacking fault energy; MEAM; Slide

1. Introduction For nanocrystalline materials, the plastic deformation is no longer dominated by dislocation motion as in coarse grains but is instead carried by atomic sliding in grain boundaries [1]. It is claimed that the nature of slip in nanocrystalline metals cannot be described in terms of an absolute value of the stacking fault energy—a correct interpretation requires the generalized stacking fault (GSF) energy curve, involving both stable and unstable stacking fault energies [2]. For those metals with low stable stacking fault energy gsf, plenty of stacking fault can be observed in experiments. Yamakov et al. reported that the transition with decreasing grain size from a dislocation-based to a grain-boundary-based deformation mechanism in nanocrystalline FCC metals results in a maximum yield strength at a grain size that depends strongly on the stacking-fault energy and the elastic properties of the metal as well as the magnitude of the applied stress. A decreasing in stacking fault energy will result in an increasing in the ‘strongest’ grain size [3]. Furthermore, it is also believed that the brittle and ductile behavior of the materials is related with the surface energy gs

* Corresponding author. Tel.: +86 29 85308456. E-mail address: [email protected] (J.-M. Zhang). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2007.07.078

and the unstable stacking fault energy gusf, which is the maximum energy barrier encountered in blocklike sliding along a slid plane [4]. At present, an experimental determination of gusf is not possible [2], although various methods have been used to measure gsf, large scattering values, up to a factor of three, have been obtained even if the same method is used for the same metal [5]. Therefore, it is necessary to make an accurate investigation of the GSF energy surface. In frame of the embedded atom method (EAM) [6–8], Zimmerman et al. used six potentials to calculate the GSF energy and obtained an expected behavior for Cu and Ni but a very unphysical behavior for Al [9]. The reason may be that, in the EAM used by them, the energy of a given atom is taken as the energy in two-body bonds with its neighboring atoms plus the energy to embed the atom in the electron density at its site arising from the other atoms, but the electron density is a simple sum of radially dependent contributions from the other atoms. Considering directional bonding, Baskes et al. developed two forms of the modified EAM (MEAM) [10–15]. Compared with the first nearest-neighbor (1NN) MEAM [10,11], the secondnearest-neighbor (2NN) MEAM considers the pair potential and the background electron density contributed by farther to the 2NN atoms [12–15]. In this paper, by using the 2NNMEAM the GSF energy surfaces have been constructed for eight FCC metals Cu, Ag, Au, Ni, Pd, Pt, Al and Pb.

1490

X.-M. Wei et al. / Applied Surface Science 254 (2007) 1489–1492

2. Calculation method In the MEAM, the total energy of a system is expressed as Etotal ¼

X

Fðr¯ i Þ þ

i

1X fðr i j Þ 2 j 6¼ i

(1)

where r¯ i is the background electron density at site i, f(rij) is the pair interaction between atoms i and j separated by a distance rij. F is the embedding function and is given by Fðr¯ i Þ ¼ AEc

r¯ i r¯ ln 0i r¯ 0 r¯

(2)

in which, A is an adjustable parameter, Ec is the sublimation energy, and r¯ 0 is the background electron density in the reference structure in which individual atoms are on the exact lattice positions without deviation. The background electron density at a specific site r¯ i is composed of a spherically ð0Þ symmetric partial electron density ri and three angular ð1Þ ð2Þ ð3Þ dependent contributions ri , ri and ri which are expressed just as in reference [12]. In the 2NN-MEAM, as mentioned above, the pair interactions and partial electron densities are extended to the second nearest-neighbor atoms. The equation for energy per atom is modified as Eu ðRÞ ¼ F½r¯ 0 ðRÞ þ

Z1 Z2S fðaRÞ fðRÞ þ 2 2

(3)

Here, Z1 and Z2 are the number of the 1NN atoms and that of the 2NN atoms in the reference structure, R is the 1NN distance in the reference structure, a is the ratio between the second and the first nearest-neighbor distance, S is the screening function on the second nearest-neighbor interactions. By introducing w(R), the Eq. (3) can be rewritten as Eu ðRÞ ¼ F½r¯ 0 ðRÞ þ

Z1 ’ðRÞ 2

(4)

in which ’ðRÞ ¼ fðRÞ þ

Z2S fðaRÞ Z1

(5)

the pair potential f(R) can be calculated by the following equation fðRÞ ¼ ’ðRÞ þ

N X n¼1

ð1Þn

Z2S Z1

n

’ðan RÞ

(6)

The value of N can be determined when the summation is performed until the correct value of energy is obtained for the equilibrium reference structure. For the FCC metals studied here, N = 5 is sufficiently enough. For any structure, the 1NN distance R in the reference structure should be instead by

practical atomic distance rij n N X Z2S fðr i j Þ ¼ ’ðr i j Þ þ ð1Þn ’ðan r r j Þ Z 1 n¼1 n N X n Z2S ¼ ð1Þ ’ðan r i j Þ Z1 n¼0

(7)

Then Eq. (3) can be rewritten as Ei ¼ Fðr¯ i Þ þ

1X Si j fðr i j Þ 2 j 6¼ i

(8)

The screening function Sij can be calculated for any atom j around atom i by the method supplied in reference [12]. Thus, the contribution of atom i to the total energy is shown by Ei + Ec. Summing all the contributions of the atoms in the concerned region, we get the total energy E X E¼ ðEi þ Ec Þ (9) i

3. Results and discussion The above model is used in molecular statics calculation in which the simulation region is a cuboid with two opposite ¯ and [1 1 1] directions faces perpendicular to the ½1 1¯ 0, ½1 1 2 respectively. Periodic boundary conditions are used in the ¯ directions and the lattice is divided in half by a ½1 1¯ 0 and ½1 1 2 (1 1 1) plane. Then the average energy per unit area can be calculated when the upper block is displaced relative to the down block parallel the (1 1 1) slip plane. g¼

E As

(10)

As is the area of the calculated region projected onto the (1 1 1) plane. For convenient description, we define a relative translation vector in the (1 1 1) plane. ~f ¼ x 1½110 ¯ þ y 1½1 1 2 ¯ 2 2

(11)

By using the above equations, the GSF energy surface can be obtained and are shown in Fig. 1(a) for Al as one example. The vectors with any point in the rectangular area of Fig. 2 are the ~f vectors given by equation (11), which preserves the underlying translational and rotational symmetry of the FCC lattice, so the calculated zone are defined by 0 x 1 and 0 y 1. The ¯ direction energy versus displacement curves along the ½1 1 2 with x = 0 and along the ½1 1¯ 0 direction with y = 0 are shown in Fig. 1(b and c), respectively. Three peaks can be observed in Fig. 1(a), corresponding to the stacking fault configuration ABC/CABC, while the stable intrinsic stacking fault energy corresponds to the ABC/BCABC configuration, which has been broadly recognized and is expected from geometrical considerations. But we can see from Fig. 1(b), the unstable stacking fault energy gusf does not reach at one-half of the partial Burgers pﬃﬃﬃ vector, a displacement of d0us at y = 1/6 (that is 6a0 =12, a0 is

X.-M. Wei et al. / Applied Surface Science 254 (2007) 1489–1492

1491

¯ direction with Fig. 1. (a) The GSF energy surface for displacements along a (1 1 1) plane in Al (in unit of J/m2). (b) The energy vs. displacement along the ½1 1 2 x = 0. (c) The energy vs. displacement along the ½1 1¯ 0 direction with y = 0.

the lattice constant). We have simulated the stacking fault energy curves along x = 0 for the metals Cu, Ag, Au, Ni, Pd, Pt, Al and Pb, whose fitted parameters are presented in reference [13], and the results are depicted in Fig. 3. It is found that for all the metals considered, gusf is difficult to reach exactly at the geometrically symmetric displacement point d0us . More or less, the actual displacements dus for all the metals are lager than d0us . Except a somewhat unsmooth curve for Pd, which makes it difficult to give the certain value of dus, the offset of dus from d0us , ðdus d0us Þ=d0us , are given in Table 1. The result shows that the offset degree is different widely, from the greatest ðdus d0us Þ=d0us value of 16% for Al, to 1.5 and 1% for Cu and Ag, which means that the offsets for the latter may be neglectable. However, the maximum energies gm (also the energy peak in Fig. 1(a)) are observed just at the displacement of y = 2/3 for all the metals. The stable stacking fault energies gsf are listed in Table 1, which are consistent with experimental data [16–18]. From the

Fig. 2. The area of displacement vectors for the energy calculation of Fig. 1.

result, we can expect that stacking fault is more favorable in Cu, Ag, Au, and especially in Pd than the other metals. However, it has been known that the deformation mechanism in nanocrystallines cannot be explained by the absolute value of gsf alone [2], so the ratios of g usf =g sf are also listed in Table 1. For a metal with low value of g usf =g sf , which is the case for Al, although the stacking fault energy is the highest by this method, full dislocation will be observed more easily. Contrarily, in spite of

Fig. 3. The stacking fault energy curve with the displacement along the ¯ for the eight FCC metals. direction of ½1 1 2

1492

X.-M. Wei et al. / Applied Surface Science 254 (2007) 1489–1492

Table 1 Offset value of ðdus d0us Þ=d0us , stable stacking fault energy gsf and ratio of g usf =g sf Metals

d0us Þ=d0us 2

ðdus gsf (J/m ) g usf =g sf

Cu

Ag

Au

Ni

Pd

Pt

Al

Pb

1.5% 0.043 7.22

1.0% 0.021 10.42

6.6% 0.041 4.79

5.6% 0.126 4.14

– 0.101 –

4.4% 0.111 5.37

16% 0.150 1.76

3.2% 0.009 7.16

the relatively low gsf value of Ag, it may be the most impossible one to create full dislocation for its highest g usf =g sf value, but the easiest one to create partial dislocation. It has been verified in simulations by Ref. [1] in which extended partial dislocations in Cu have been observed as the predominant deformation mechanism at nanocrystalline grains, while in our result, the g usf =g sf value of Cu is the second largest one. Contrasting the value of g usf =g sf with ðdus d0us Þ=d0us , we find that for most metals, they are inversely related. For example, the g usf =g sf value is the biggest for Ag with the smallest ðdus d0us Þ=d0us value, on the contrary, for Al, the g usf =g sf value is the smallest with the biggest ðdus d0us Þ=d0us value. Therefore, we can conclude that the possibility of partial dislocation in a metal may be determined by the displacement point of the unstable stacking fault energy. In addition, it is interesting to note that for the congeners in the eight metals, the curves of their GSF energy kink together to form a group. It is indicated obviously in Fig. 3 for two groups, one contains Cu, Ag and Au, the other one contains Ni, Pd and Pt. 4. Conclusion The 2NN-MEAM is applied to study the GSF energy surfaces of the FCC metals. The displacement offset of unstable stacking fault energy from geometrically symmetric displacement point is the greatest for Al and the smallest for Ag. Using the GSF energy surface determined from the method, it predicts that stacking fault is more favorable in the metals Cu, Ag, Au and Pd, and that partial dislocation for Ag and full dislocation for Al are most preferred to create.

Acknowledgments The authors would like to acknowledge the State Key Development for Basic Research of China (grant no. 2004CB619302) and the National Natural Science Foundation of China (grant no. 50271038) for providing financial support of this research. References [1] J. Schiøtz, K.W. Jacobsen, Science 301 (2003) 1357. [2] H.V. Swygenhoven, P.M. Derlet, A.G. Frøseth, Nature 3 (2004) 399. [3] V. Yamakov, D. Wolf, S.R. Phillpot1, A.K. Mukherjee, H. Gleiter, Nature 3 (2004) 43. [4] J. Rice, J. Mech. Phys. Solids 40 (1992) 239. [5] R.P. Reed, R.E. Schramm, J. Appl. Phys. 45 (1974) 4705. [6] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 50 (1983) 1285. [7] M.S. Daw, M.I. Baskes, Phys. Rev. Lett. 29 (1984) 6443. [8] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. Lett. 33 (1986) 7983. [9] J.A. Zimmerman, H. Gao, F.F. Abraham, Modelling Simul. Mater. Sci. Eng. 8 (2000) 103. [10] M.I. Baskes, Phys. Rev. Lett. 59 (1987) 2666. [11] M.I. Baskes, Phys. Rev. B 46 (1992) 2727. [12] B.J. Lee, M.I. Baskes, Phys. Rev. B 62 (2000) 8564. [13] B.J. Lee, J.H. Shim, M.I. Baskes, Phys. Rev. B 62 (2003) 144112. [14] B.J. Lee, M.I. Baskes, H. Kim, Y.K. Cho, Phys. Rev. B 64 (2001) 184102. [15] J.M. Zhang, D.D. Wang, K.W. Xu, Appl. Surf. Sci. 252 (2006) 8217. [16] L.E. Murr, Interfacial Phenomena in Metals and Alloys, Addison-Wesley, Reading, MA, 1975. [17] J.P. Hirth, J. Lothe, Theory of Dislocations, Wiley-Interscience, New York, 1982. [18] C.S. Barrett, T.B. Massalski, Structure of Metals, McGraw-Hill, New York, 1966.

Copyright © 2021 COEK.INFO. All rights reserved.