The dissociation behavior of dislocation arrays in face centered cubic metals

Computational Materials Science 124 (2016) 384–389

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The dissociation behavior of dislocation arrays in face centered cubic metals K.Q. Li, Z.J. Zhang, L.L. Li, P. Zhang, J.B. Yang ⇑, Z.F. Zhang ⇑ Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 23 June 2016 Received in revised form 22 July 2016 Accepted 24 July 2016 Available online 24 August 2016 Keywords: Stacking fault Dislocation Face centered cubic Atomistic modeling

a b s t r a c t Dislocations h1 1 0i/2 are usually dissociated into two h1 1 2i/6 partials with a stacking fault in face centered cubic metals. Their behavior depends strongly on the stacking fault width (SFW) in plastic deformation. However, there is no quantitative study to correlate the SFW with the dislocation configuration when these dislocations are grouped together. In this work, the SFW for different dislocation arrays is analyzed within the framework of the elasticity theory of dislocations and then verified by atomistic simulations. The results demonstrate that the spacing of dislocation arrays has to be taken into account for the SFW variation besides the dislocation character. In addition, the SFW variation with the dislocation spacing seems to be independent to temperature. Our approach can also provide a basis for the accurate estimate of the influence of stacking faults on cross-slip, the competition between slip and twinning during plastic deformations in face centered cubic metals. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Stacking faults (SFs) are usually formed by the dissociation of perfect dislocations h1 1 0i/2 into two Shockley partials h1 1 2i/6 in face centered cubic (FCC) metals [1]. They often dominate the behaviors of plastic deformations of these materials to a large extent. For example, dislocations h1 1 0i/2 have to glide on the plane {1 1 1} parallel to these SFs because of their restriction. If a dislocation h1 1 0i/2 tends to cross slip, its SF have to shrink to a minimum distance on the original glide plane to facilitate its redissociation on the new glide plane [2,3]. This process plays an important role in the onset of stage III work-hardening for single crystals and the deformation textures of polycrystals [4]. Besides, SFs significantly impact some more elementary processes, such as synchronous improvement of strength and ductility [5–7], the transition from slip to twinning of the dominant deformation mode in FCC metals including transformation- and twinninginduced plasticity alloys [8], the cracking behaviors of the coherent twin boundary [9] and the formation of various configurations of dislocations including persistent slip bands, labyrinth and cell structures in cyclically deformed FCC metals [10]. In the quantitative study of these phenomena, it is now recognized that the SF width (SFW) is the most important factor that should be taken into account when describing the influence of SFs on the mechanical properties of FCC metals. ⇑ Corresponding authors. E-mail addresses: [email protected] (J.B. Yang), [email protected] (Z.F. Zhang). http://dx.doi.org/10.1016/j.commatsci.2016.07.027 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.

Numerous theoretical and experimental attempts [11–17] have been undertaken to analyze the variation of the SFW (termed by d) because of its importance mentioned above. In dissociation of an isolated dislocation h1 1 0i/2 without any external loading, the elasticity theory of dislocations successfully predicts that the SFW is simply inversely proportional to the stacking fault energy (SFE denoted by cI ), and it is also a function of the orientation of a dislocation line. These predictions have been confirmed in the accurate measurement of SFW by the weak-beam technique of transmission electron microscopy (TEM) [4,11,12]. On the other hand, when an isolated dislocation h1 1 0i/2 is subjected to an applied loading, the elasticity theory of dislocations demonstrates that the SFW depends on the external loading [13,17]. However Li et al. [16] found that the equilibrium separation of partial dislocations in a wall of extended edge dislocations is a function of the misfit angle of the wall. This means that the dissociation of dislocation arrays or groups is different from that of an isolated dislocation. Unfortunately, this model is concentrated on symmetrical tilt boundaries in which each dislocation has its own glide plane so that it is not valid to the common experimentally observed dislocation arrays in association with cross slip, dislocation multiplications, dislocation pile-ups and so on [18– 26]. Because in the latter case all dislocations in each array should glide on a single plane {1 1 1} and it is found that there usually exist two sets of Shockley partial dislocations h1 1 2i/6 in each dislocation array, which are formed by the dissociation of a single set of perfect dislocations h1 1 0i/2. Therefore, it is meaningful to provide a quantitative description of the SFW variation for dislocation arrays.

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For this purpose, our task is two-fold: (i) to derive an analytical equation to correlate the variation of the SFW d with the dislocation spacing D (the d-D relation or equation hereafter) for different dislocation arrays by applying the elasticity theory of dislocations, and (ii) to perform systematical atomistic simulations to verify the formulae in (i). In (ii), Cu and Ag are selected to conduct atomistic simulations because of their large difference of SFE. At the same time, different empirical potentials of Cu and Ag [27–31] are used to consider the possible influences from the construction of potentials on the simulation results. And the influence of temperature on SFW is also discussed. Finally, the application and generality of the d-D equation for different dislocation types, and its influence on many behaviors of FCC metals during plastic deformations are discussed in detail. 2. Theoretical consideration In this section, the isotropic elasticity theory of dislocations is employed to derive the d-D relation. Fig. 1(a) sketches an array of perfect dislocations with the Burgers vector b ¼ ½1 1 0=2 on a single glide plane. It represents a typical dislocation array gliding on a single plane observed in the experiments [18–26]. We assume that these dislocations in Fig. 1(a) can be placed into a coordinate system {xyz}, where x is parallel to b (x||b), and z is parallel to the normal of their glide plane (1 1 1). The dislocation line direction l is restricted to the x-y glide plane, and it makes an angle h with x. The angle h is restricted in the range from 0 to 90°, because it suffices to model the change from screw to edge dislocations (imagining b is fixed along x). This coordinate system {xyz} will be also selected for the atomistic simulations in the next section. According to the isotropic elasticity theory of dislocations [1], P the stress tensor s arising from a single dislocation b at the origin in Fig. 1(a) has two non-zero components: " # 2 Gðx sin h þ y cos hÞ b sin h ðx sin h þ y cos hÞ  z2 i bs cos h þ e rsxz ¼ h 2 1  m ðx sin h þ y cos hÞ2 þ z2 2p ðxsin h þ y coshÞ þ z2 ð1Þ

r

s yz

Gðxsin h þ ycoshÞ

"

2

b cosh ðxsin h þ y coshÞ  z2 i bs sinh  e ¼ h 2 1  m ðxsin h þ y coshÞ2 þ z2 2 2p ðxsinh þ ycoshÞ þ z

Pa

#

Fig. 1. (a) An array of perfect dislocations with the Burgers vector b. Their glide plane is parallel to the x-y plane, in which the dislocation line direction l makes an angle h with x. (b) Each perfect dislocation in (a) dissociates into two Shockley partials b1 and b2. Only five dislocations are sketched for clarity. In (b), h = 90° is selected as an example. See parameters in the text.

ple coordinate transformation, e.g. Rsb1 and Rab1 (or Rsb2 and Rab2 ) represent the stress tensors of an isolated dislocation and a dislocation array with the Burgers vector b1 (or b2) by replacing x by x ± dsin h/2 and y by y ± dcos h/2. In this case, the two partials b1 and b2 nearest to the origin are selected to establish the d-D relation. The stress tensor of these two partial dislocation arrays excluding the selected two partials b1 and b2 nearest to the origin can be easily derived as:

Rr ¼ Rrb1 þ Rrb2

ð5Þ

where

Rrb1 ¼ Rab1  Rsb1

ð6Þ

Rrb2 ¼ Rab2  Rsb2

ð7Þ

Here the stress tensor Rrb1 (or Rrb2 ) denotes the stress tensor of the partial dislocation b1 (or b2) without the one nearest to the origin in Fig. 1(b). Besides the stress in Eq. (5), the interaction and SF

ð2Þ

Correspondingly, the stress tensor of the dislocation array in Fig. 1(a) can be derived by superposing the stresses of each dislocation in Eqs. (1) and (2) on the basis of Refs. [1,32,33] and its nonzero components are:

raxz ¼

  G sinðX sin h þ Y cos hÞ be sin h A  Z sinh Z bs cos h þ 2DA 1m A

rayz ¼ 

ð3Þ

  G sinðX sin h þ Y cos hÞ be cos h A  Z sinh Z bs sin h  2DA 1m A ð4Þ

Note that all the stresses in the above equations are expressed in {xyz} and A ¼ cosh Z  cosðX sin h þ Y cos hÞ; X ¼ 2px=D; Y ¼ 2py=D; Z ¼ 2pz=D: In this work, G, b, m denote the shear modulus, the length of the Burgers vector b, and the Poisson’s ratio, respectively. The partial dislocation arrays b1 and b2 in Fig. 1(b) come from the dissociation of the dislocation array b in Fig. 1(a). It is noted that a translation occurs in the x-y plane associated with the dislocation dissociation for each partial pair relative to their original position before dissociation. This translation can be involved when applying Eqs. (1)–(4) to the partial dislocations b1 and b2 by a sim-

Fig. 2. Schematic presentation of simulated crystallite with M: region of mobile atoms, P: regions where periodic boundaries are applied, F: lower fixed block and R: upper rigid mobile block. See other parameters in the text.

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Fig. 3. Comparison of the variations of the stacking fault width with the dislocation spacing when Lx = D and Lx = 2D in the range of D from 22a to 184a. The simulated dislocations after relaxation are shown as the inset for clarity.

between the selected partials b1 and b2 have to considered to obtain the force balance at these two selected dislocations b1 and b2, respectively, as given below:

½ðRr1  b1 Þ  l  m  cI þ F int  F f ¼ 0

ð8Þ

Fig. 5. (a) The generalized SFE curves along 1/6h1 1 2i on the (1 1 1) plane. (b) The stacking fault width changing with the dislocation spacing from three different potentials in Cu.

Table 1 Generalized SFE (mJ/m2) for different EAM potentials of Cu. Parameters

Cu_A97

Cu_M01

Cu_M12

cI cU cU  cI

36.6 314.5 279.9

44.4 162.6 118.2

44.1 234.5 190.4

½ðRr2  b2 Þ  l  m þ cI  F int  F f ¼ 0

ð9Þ

Pr

Here Rr1 and Rr2 represent at the location of the partial dislocation b1 and b2, respectively. The dislocation line direction l = (cos h, sin h, 0), the unit vector m = (sin h, cos h, 0) is perpendicular to l on the glide plane, and the scalar parameter bp is the length of the Burgers vectors b1 or b2. Each part in Eqs. (8) and (9) is described in the following for clarity. The first term in Eqs. (8) or (9) represents the force due to the remaining partial dislocations when the two partials b1 and b2 nearest to the origin in Fig. 1(b) are assumed to be moved to infinity. The second term cI , which is equal to the SFE, represents the force arising from the SF. The third part, F int ¼

Fig. 4. (a) The generalized SFE curves sliding along 1/6h1 1 2i on the (1 1 1) plane for Cu_M12 and Ag_W06. The peak and local minima indicate the unstable SFE (cU ) and intrinsic SFE (cI ), respectively. (b) Comparison between atomistic simulations and the prediction from Eq. (11) about the variations of the stacking fault width with the dislocation spacing in Cu and Ag.

Gb2p ð2m2m cos 2hÞ , 8pð1mÞd

is

the repulsive force between these two partial dislocations b1 and b2 nearest to the origin. The last term Ff is the lattice resistance, which is assumed to be the same in Eqs. (8) and (9) because it represents the forces exerted on the leading and trailing partials that tend to glide as a whole in Fig. 1(b). By adding Eqs. (8) and (9), it is obtained that:

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Fig. 6. The dissociation of an array of perfect dislocations at 0 K, 150 K and 300 K for Cu_M12 and Ag_W06.

Ff ¼ 0

ð10Þ

On the other hand, if subtract Eqs. (8) from (9), we have:

d¼

D

p

 arctan

Here w ¼

w DcI



Gb2p ð2m2m cos 2hÞ . 8ð1mÞ

ð11Þ It is apparent that w depends only on

the nature of the material and the character of dislocation arrays. Eq. (10) indicates that the partial dislocations are under force balance, without any tendency of motion. Eq. (11) is the d-D relation we need in this work. When D becomes infinite, this relation will become the same as Eq. 15 in Chapter 10 in Ref. [1] for the dissociation of a single dislocation. It is obvious that the dissociation of dislocation arrays depends on the dislocation spacing D, and thus each dislocation in arrays cannot be simply treated as an isolated dislocation. This interesting phenomenon has not been discovered previously until this work to our knowledge. In the next section, Eq. (11) will be systematically verified by atomistic modeling.

3. Methodology The atomistic modeling technique has been previously described in details [34], but its main features are presented here for clarity. Fig. 2 shows the simulation box in this work. The coordinate system of the simulation box contains three axes, x|| ½1 1 0, y|| ½1 1 2 and z|| ½111. As mentioned previously, this coordinate system is the same as that given in Fig. 1. Without specification

all the vectors including directions and plane normal are expressed in the crystallographic coordinate system, denoted by {xe, ye, ze}, where the axes are xe = [001], ye = [010] and ze = [001], respectively. First of all, an edge dislocation b is generated at the center of the box, with a glide plane (1 1 1) and a line direction ½1 1 2. Periodic boundary conditions are applied along the x and y directions to generate the dislocation array in Fig. 1(b). The length along x Lx = nD, where D is the spacing between the perfect dislocations as defined in the foregoing section and n is an integer. At the same time, Lx should be the periodic length along x and thus pffiffiffi Lx ¼ ðnx  0:5Þ 2a=2, where nx is an integer from 6 to 250, where the constant a represents the lattice parameters. The size along y pffiffiffi is fixed such that Ly ¼ 3 6a. In this work, the large size along z (H  208a) is selected to exclude the influence due to the free boundary condition along z. The relaxation calculation is performed with a molecular statics method [34]. After this relaxation, the perfect dislocation could dissociate into two Shockley partial dislocations b1 and b2 in Fig. 1(b). The partial dislocations are simply visualized by the analysis of excess energy of atoms [35] and further characterized by our method on the basis of the Nye tensor [36]. The cores of a pair of partial dislocations can be precisely located at atomic scale in this procedure, and thus their accurate spacing d can be simply determined from the simulation results. The stress field of a dislocation is calculated by the Virial theorem in simulations [37]. The chosen embedded-atom-method (EAM) potentials include three Cu potentials: Cu_A97 [27], Cu_M12 [28] and Cu_M01 [29], and two Ag potentials: Ag_A87 [30] and Ag_W06 [31]. These notations will be followed in all the tables and figures in this work.

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4. Results In this work, the periodic boundary condition along x is utilized to create the images of the single dislocation at the center of the simulation box. Then, the dislocation in the box and its images seem to be equivalent to a dislocation array as sketched in Fig. 1 (b), having an infinite number of dislocations. It is apparent that such a treatment has to be carefully tested in the following. First of all, the smallest size of the simulation box along x has to be determined to ensure that all the simulations can capture the dissociation of dislocation arrays and give the change of d with D with high accuracy. For this purpose, Lx is selected to be different values, e.g., Lx = D, 2D, 3D and so on. As a special example, Fig. 3 shows the change of the SFW d with D for two different Lx. Here the potential is selected as Cu_M12. It is seen from the left top in Fig. 3 that in both cases the initial perfect dislocations b at the box center are always dissociated into two partial Shockley dislocations with the Burgers vectors b1 and b2. Here the Nye tensor was calculated to determine the core of partials: brown atoms represent the b2 ¼ 6a ½2 1 1, dark green ones denote b1 ¼ 6a ½1 2 1. More importantly, as seen from Fig. 3, the box size has no influence on the two d-D curves when D increases from a very small value 22a. All our simulation results indicate that Lx can be safely selected to be its minimum value D. Now atomistic simulations are performed in Cu and Ag to obtain d for various D. Their SFE values can be found from the generalized SFE (GSFE) curve in Fig. 4(a), calculated along the displacement b1 + b2 on plane (1 1 1) according to the procedure proposed by Vitek [38]. Then the d-D pairs are compared with the curves from the analytical d-D relation in Eq. (11) with h = 90°. In Fig. 4 (b), the simulation data show that in both Cu and Ag d increases rapidly with D in the range of small D, and then becomes almost constant when D increases to 100a–150a. All these features agree well with the curves predicted from Eq. (11), as shown in Fig. 4 (b). It is impressive that the analytical d-D relation also works well in comparison with the simulation data when d is extremely small, about 2a (2.7a) for Cu_M12 (Ag_W06) in Fig. 4(b). Many previous researches [27–31] have demonstrated that the selection of potential functions plays an important role in the reliability of atomistic simulations. For this purpose, three Cu EAM potentials are chosen with almost the same SFE, as given in Fig. 5(a). It is seen that they exhibit a large difference in view of cU , cU  cI (see Table 1 for details) and their transition topography from cI to cU in Fig. 5(a). Except the potential Cu_A97, all the other two Cu potentials give the results in agreement with Eq. (11), as shown in Fig. 5(b). The relatively large discrepancy from the potential Cu_A97 in Fig. 5(b) may be attributed to the fitting procedure and database of the potential Cu_A97 [27], because the Ag_A87 potential [30], developed in the same manner as the potential Cu_A97, has an identical drawback. Therefore, the potentials to simulate dislocations in FCC metals have to be carefully and thoroughly tested to ensure that they can yield reliable simulation results.

5. Discussion A simple analytical equation (Eq. (11)) to correlate the SFW d with the dislocation spacing D of different arrays has been developed on the basis of the elasticity theory of dislocations. Its validity has been systematically testified by atomistic simulations in Cu and Ag with largely different SFE values. Our results show that the elasticity theory of dislocations is suited to describe the main features of the SFW variation of dislocation arrays. All our simulation results support that the dislocation spacing D could cause the variation of the SFW d, and that the d-D relation Eq. (11) can be

Fig. 7. Continuous curves predicted from Eq. (11) of: (a) the variations of the stacking fault width with h for four dislocation spacing D = 10a, 20a, 40a and 150a, and (b) the variations of the stacking fault width with the dislocation spacing D for h = 0°, 30°, 60°, and 90°.

applied in a large range of SFE. One could argue that the SFW could also change with temperature besides D. In fact, the influence of temperature can be safely neglected here because the SFW almost does not change in the range from 0 to 300 K (the experimental temperature range) in both Ag and Cu, as shown in Fig. 6. In addition to the dislocation spacing D discussed above, d depends on the character of dislocation arrays. This can be seen from the angle h between the lines and the Burgers vector of dislocations, as seen from Eq. (11). Fig. 7(a) shows the d-h curve for a given D by solving Eq. (11), in which D = 10a, 20a, 40a and 150a. It is apparent that d for edge dislocations (h = 90°) is always larger than screw dislocations (h = 0°) for a given D, the same as the rule for an isolated dislocation [1]. Therefore, it is more convenient to choose edge dislocations in the experimental measurement of d. When D approaches 150a, d becomes close to that of an isolated dislocation. This result can be used to select the proper size of the simulation box when modeling the behavior of an isolated dislocation with a periodic boundary condition. On the other hand, the d-D curves can also be predicted for a given h. Fig. 7(b) shows the d-D curves for four different dislocation line orientations with h = 0°, 30°, 60° and 90°, respectively. Fig. 7(b) indicates that the d-D curves have a similar shape for different h, e.g. d always increase rapidly in the low value range of D. As a consequence, this will facilitate the Shockley partial pair of screw dislocations to merge and cross slip, and then cause the transition from planar to wavy slip, which dominates the evolution of dislo-

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cation pattern during fatigue [10,41]. Furthermore, it is also necessary to utilize the present d-D relation to describe the behavior of dislocation arrays in association with twin boundary cracking [9] and the transition of deformation mode from slip to twinning [39,40]. This could provide a better understanding of the mechanical properties of FCC materials, and the related results will be given in our future work. 6. Conclusions In this study, the dislocation dissociation behavior has been investigated for dislocation arrays in FCC metals by the combination of the elasticity theory of dislocations and atomistic modeling. It is found that the SFW is strongly affected by the dislocation spacing of a dislocation array. This relation is neglected in most previous studies because dislocations are usually treated as isolated ones. An analytical equation has been established to correlate the SFW with the dislocation spacing for different dislocation types, and then it has been fully testified by atomistic simulations of FCC metals with both low and high SFE. Our follow-on studies will investigate cross slip, dislocation pile-up, dislocation-boundary interaction and boundary cracking in FCC metals, all of which are tightly related to the quantitative relation of the d-D relations. Acknowledgement This research was funded by the Program of ‘‘One Hundred Talented People” of the Chinese Academy of Sciences (JBY) and the National Natural Science Foundation of China (NSFC) under grant No. 51331007. References [1] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed., Wiley, New York, 1982. [2] A.M. Hussein, S.I. Rao, M.D. Uchic, D.M. Dimiduk, J.A. El-Awady, Microstructurally based cross-slip mechanisms and their effects on dislocation microstructure evolution in fcc crystals, Acta Mater. 85 (2015) 180–190. [3] T. Rasmussen, K.W. Jacobsen, T. Leffers, O.B. Pedersen, S.G. Srinivasan, H. Jónsson, Atomistic determination of cross-slip pathway and energetics, Phys. Rev. Lett. 79 (1997) 3676–3679. [4] D.J.H. Cockayne, M.L. Jenkins, I.L.F. Ray, Measurement of stacking-fault energies of pure face-centred cubic metals, Philos. Mag. 24 (1971) 1383–1392. [5] Z.J. Zhang, Q.Q. Duan, X.H. An, S.D. Wu, G. Yang, Z.F. Zhang, Microstructure and mechanical properties of Cu and Cu-Zn alloys produced by equal channel angular pressing, Mater. Sci. Eng., A 528 (2011) 4259–4267. [6] X.H. An, Q.Y. Lin, S.D. Wu, Z.F. Zhang, R.B. Figueiredo, N. Gao, T.G. Langdon, The influence of stacking fault energy on the mechanical properties of nanostructured Cu and Cu-Al alloys processed by high-pressure torsion, Scripta Mater. 64 (2011) 954–957. [7] S. Qu, X.H. An, H.J. Yang, C.X. Huang, G. Yang, Q.S. Zang, Z.G. Wang, S.D. Wu, Z.F. Zhang, Microstructural evolution and mechanical properties of Cu-Al alloys subjected to equal channel angular pressing, Acta Mater. 57 (2009) 1586– 1601. [8] D.T. Pierce, J.A. Jiménez, J. Bentley, D. Raabe, C. Oskay, J.E. Wittig, The influence of manganese content on the stacking fault and austenite/e-martensite interfacial energies in Fe-Mn-(Al-Si) steels investigated by experiment and theory, Acta Mater. 68 (2014) 238–253. [9] Z.J. Zhang, P. Zhang, L.L. Li, Z.F. Zhang, Fatigue cracking at twin boundaries: effects of crystallographic orientation and stacking fault energy, Acta Mater. 60 (2012) 3113–3127. [10] P. Li, S.X. Li, Z.G. Wang, Z.F. Zhang, Fundamental factors on formation mechanism of dislocation arrangements in cyclically deformed fcc single crystals, Prog. Mater Sci. 56 (2011) 328–377. [11] C.B. Carter, I.L.F. Ray, Stacking-fault energies of copper alloys, Philos. Mag. 35 (1977) 189–200.

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