THE CRITICAL STRESS FOR TRANSMISSION OF A DISLOCATION ACROSS AN INTERFACE: RESULTS FROM PEIERLS AND EMBEDDED ATOM MODELS

Transcription

1 THE CRITICAL STRESS FOR TRANSMISSION OF A DISLOCATION ACROSS AN INTERFACE: RESULTS FROM PEIERLS AND EMBEDDED ATOM MODELS P.M. ANDERSON*, S. RAO**, Y. CHENG*, AND P.M. HAZZLEDINE** *Dept. MSE, The Ohio State University, 2041 College Rd., Columbus, OH , anderson.1@osu.edu **Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, OH UES, Inc., 4401 Dayton-Xenia Rd., Dayton, OH ABSTRACT A continuum Peierls model of a screw dislocation being pushed through an interface and an atomistic EAM study of dislocation transmission across a [0 0 1] Al-Ni interface suggest that core spreading into the interface and misfit dislocations in the interface are both potent effects that can significantly increase barrier strength of interfaces. INTRODUCTION The critical resolved shear stress, t*, to push a dislocation past an obstacle is a fundamental quantity that controls the onset of widespread plasticity in many materials. The familiar Hall- Petch analysis of a pile-up of screw dislocations against an obstacle furnishes the critical resolved shear stress, t H-P, to push the leading dislocation in the pile-up past the obstacle [1,2] t H-P = t o + k H- P d, k H-P = mbt* p Equation (1) suggests that t H-P can be increased by decreasing the spacing, d, between obstacles or increasing the strength, t*, of the obstacle. Other parameters are the elastic shear modulus m, magnitude, b, of the dislocation Burgers vector, and resistance, t o, to dislocation motion in a material free of obstacles (d = ). A characteristic of materials with strong obstacles is that k H-P is large and thus the yield strength or hardness of the material is more sensitive to a change in d. Nanolayered composite materials provide a unique opportunity to change both the spacing and strength of obstacles, which take the form of interfaces. In principle, such interface-rich materials can, in single-crystal layer form, provide a large density of obstacles in a direction normal to the interface, but a negligible density in the plane of a layer. Data for several metallic and nonmetallic nanolayered composites show that, despite the directional nature of obstacle density, hardness monotonically increases as layer thickness decreases down to ª 5nm [3]. Experimental hardness data on multilayers also suggests that the strength of interfaces varies significantly with material system and individual layer thickness. Equation (1) provides a reasonable fit to the data for layer thickness typically greater than 10nm. In that regime, the Hall-Petch slope (k H-P ) varies dramatically among material systems, from 7.0GPa nm 0.5 for epitaxial Ag/Cr multilayers to 42GPa nm 0.5 for epitaxial Fe/Pt multilayers [3]. The difference is attributed to the variation of interfacial strength to slip transmission among systems. At layer thickness less than 5nm, many systems display a strong departure from eqn. (1), in that experimental hardness values increase only modestly or even decrease with further reduction in layer thickness. This sharp change in behavior is believed to occur since semicoherent interfaces at larger layer thickness contain obstacles in the form of misfit dislocations, and these obstacles disappear when interfaces become coherent at smaller layer thickness. The blocking strength of misfit dislocations to glide dislocation movement can be estimated as [4, 5] published in MRS Symp. Proc. on Interfacial Engineering for Optimized Properties (C.B. Carter, E.L. Hall, C.L. Briant, and S Nutt, eds.), MRS Symp Proc 586, pp , Materials Res Soc: Warrendale, PA (1999). (1)

2 * t misfit = amda / a where da / a is the misfit in lattice parameter in the plane of the multilayer and a ~ 0.5 is Saada's constant [6]. Equation (2) gives a value of m/16 for the blocking strength of misfit dislocations in the Al-Ni multilayer system. This paper reports on the strength of interfaces to dislocation transmission based on two models. The first model uses a continuum, Peierls type of description of a dislocation and also permits the interface to slip according to a simple empirical relation. The second model uses the Embedded Atom Method (EAM) to study the pinning effect of misfit dislocations on the transmission of a dislocation across the interface. The basic geometry is shown in Fig. 1, where a dislocation in material 1 with elastic shear modulus m 1 and Poisson's ratio n 1 transmits across the interface into material 2 with properties m 2 and n 2. The continuum Peierls and atomistic EAM models used here extend previous efforts to model dislocation transmission. Early continuum work generally assumes that interfaces are welded, in the sense that interfaces or grain boundaries do not slip or open. Head [7] concluded that a Volterra dislocation with a tight, step-function distribution of slip is repelled from a welded interface if it lies in the lower modulus phase (m 1 < m 2 in Fig. 1) and it is attracted to the interface if it lies in the larger modulus phase. Deviations from this behavior can occur for edge dislocations, with b along the x or y-directions, if n 1 n 2 [8, 9]. For freely-slipping interfaces, Volterra dislocations of a screw or edge type with b along the y- direction are always attracted to the interface. However, an edge type with b along the x- direction is attracted to the interface only if it lies in the larger modulus phase. Recent models employing an interface with continuum frictional shear properties [10] or an interface having a linear relation between shear traction and relative shear displacement [11] have been pursued as intermediate cases to the welded and freely-slipping models. However, all of these Volterra models predict a singular force (either attractive or repulsive) on the dislocation as it is moved toward the interface. As such, they are not able to furnish a finite value of t*. The artificial singular force can be removed by using a Peierls description of the dislocation, with a smoothly varying slip distribution, or by having elastic properties vary smoothly across a diffuse interface. Pacheco and Mura [12] considered the first approach, and concluded that the critical stress to push a Peierls-type screw dislocation across a welded interface is t 1 */m 1 ª 0.2(m 2 - m 1 )/(m 2 + m 1 ). The model predicts t 1 */m 1 = for a Cu/Ni interface, with shear moduli given by m Cu = 54.6 GPa and m Ni = 94.7 GPa [13]. More recent work varies the shear traction-relative shear displacement relation on the incoming and outgoing slip planes and concludes that t* decreases when the unstable stacking fault energy on the slip planes is reduced, so that dislocation cores are wider [14]. Krznowski pursued the second approach of transmitting a Volterra dislocation across a diffuse interface, with an elastic shear modulus that varies linearly from m 1 to m 2 over a width w [15]. In that case, t* = (m 2 - m 1 )bln(w/2b)/4pw so that a Cu-Ni interface with w = 20b, for example, has t 1 */m 1 = In general, t* decreases when interfaces become more diffuse. Thus, continuum models suggest that interfaces are strong barriers to slip transmission when they are chemically sharp with large, abrupt changes in elastic moduli and when dislocation slip profiles have an abrupt, step-like shape. In principal, there are several additional contributions to interfacial resistance such as (1) dislocation core spreading into the interface, (2) unstable stacking fault energy of the incoming and outgoing planes, (3) creation of a residual dislocation at a semicoherent or an incoherent (2)

3 interface during transmission, (4) dissociation of the dislocation into partials that can be separated or contracted by non-glide components of stress (the Escaig effect), and (5) ordering mismatch across interfaces. EAM studies do reveal core spreading into the interface during transmission across Cu/Ni and Ti/Al interfaces [4]. Those studies predict t* = m 1 for transmission from Cu to Ni across either (111) or (100) epitaxial interfaces [4]. Two extensions are pursued in this paper. First, the continuum Peierls treatment by Pacheco and Mura is extended to include possible sliding of the interface, so that the effect on t* of dislocation core spreading into the interface can be modeled. Second, the EAM is used to study the effect of misfit dislocations on t* in the Al/Ni multilayer system. MODEL DEVELOPMENT Peierls Model The Peierls model adopted here is inspired by the development of Rice [16]. A screw dislocation of Burgers vector magnitude b is represented as N model screw dislocations of Burgers vector magnitude db = b/n. All model dislocations are positioned initially on the incoming slip plane in material 1 (Fig. 1), with model dislocation N at the front of the dislocation core, so that it will be the first dislocation to cross the interface. The relative slip, du z, across the slip plane in front of dislocation i is simply (N-i)db. The prescribed atomic shear stress to impose a relative slip is t atomic yz = (m 1 /2p)sin(2pDu z /b), where Du z = du z + bt atomic yz /m 1 is the relative slip between adjacent atomic planes with normal separation distance b. The corresponding elastic shear stress, t elastic yz, at each site is given by the applied shear stress plus the elastic shear stress contributed by all model dislocations and their images. Equilibrium requires that the positions of all model dislocations be adjusted until t atomic yz = t elastic yz at each model dislocation site. The corresponding energy of the mechanical system, given by the elastic energy plus core energy of the dislocation minus any Peach-Koehler work done by the applied stress on newly slipped portions of the plane, is computed. During the transmission process, model dislocations may move onto the interfacial plane, which has a sinusoidal atomic relation as discussed for the incoming slip plane, but with m int replacing m 1. The result is that the interface may slip and accommodate some of the dislocation core into it. The interface plane is permitted to slip only when the mechanical energy of the system is lowered. Model dislocations also may be moved onto the outgoing slip plane, which has a sinusoidal atomic relation as discussed earlier, but with m 2 replacing m 1. The numerical code written for this process increments the applied stress and at each value of applied stress, it determines the minimum energy of the system, by movement of model dislocations from: (1) the incoming slip plane to the outgoing slip plane, (2) the incoming slip plane to the interfacial plane, and (3) the interfacial plane to the outgoing plane. Ultimately, the analysis provides the peak shear stress and critical configuration during transmission of the dislocation from the left to the right side of the interface. Embedded Atom Model The EAM approach uses potentials developed to fit the properties of FCC Al and Ni and B2 NiAl [17]. The energy of an ensemble of atoms is [18, 19] E = Â E i = Â V ij (R ij ) + Â F i (r i ) ; r i = Â f j (R ij ) (3) i i, j,i j i j The form of V ij (R ij ) is taken to be a Morse potential [20], f j (R ij ) is taken to decrease exponentially with distance, and F i (r i ) is obtained from an exact fit to 'Rose's equation of state'

4 [21]. The EAM potential can be adjusted to fit the properties of FCC Al and Ni by varying the Morse potential parameters and the rate of decay of f j (R ij ). The Al-Ni Morse pair interaction potentials are developed by fitting to the properties of B2 NiAl [22]. The interface simulation cell is constructed with material 1 in Fig. 1 having 24,500 Al atoms with lattice parameter nm and material 2 having 37,760 Ni atoms with lattice parameter nm. The interface normal, or x-direction, corresponds to the [0 0 1] crystallographic direction. The dimensions (d x, d y, d z ) in the x, y, and z-directions are (10.1nm, 70 periodic units, 7 periodic units) for the Al side and (10.23nm, 80 periodic units, 8 periodic units) for the Ni side. Along the y and z directions in the interface, every 8 periodic units in Ni perfectly match with every 7 periodic units in Al. Atomic rearrangement at or near the interfaces is permitted by relaxing atomic positions, using periodic boundary conditions along y and z-faces of the model and perfect lattice positions for any atoms a distance 2R cut in from the x-faces of the model. Here, R cut = 0.6nm is the range of the interatomic interactions. The plot of the y-component of differential displacement [23, 22] in Fig. 2 is taken of a thin slice perpendicular to the z-direction at the center of the cell. The relaxed interface has misfit dislocations every 8 periodic units in Ni or every 7 periodic units in Al. Either a ±a/2<110> screw or 60 dislocation is introduced by displacing all atoms according to the anisotropic elastic displacement field of a straight screw dislocation in an Al crystal [24], with the dislocation center at least two periodic units into the elastically softer Al crystal. Atoms are relaxed, with periodic conditions on the z-face and fixed positions for any atoms within a distance, 2R cut, from the x and y faces of the cell. A pure shear strain, g [1-1 -1] [1 1 0], is then applied to push the dislocation toward the Al-Ni interface [4, 3] and final relaxed configurations are inspected using differential displacement plots [23, 22]. RESULTS Interfacial Slip Table I shows the continuum Peierls analysis results for the peak shear stress, t 1 */m, needed to transmit a screw dislocation through an interface with m 2 /m 1 = Two cases are considered. Both use the sinusoidal shear stress-relative shear displacement relation, t atomic yz = (m int /2p)sin(2pDu z /b), for the interface, but Case A has m int = 0.5(m 2 + m 1 ) m and Case B has m int = 0.6m. Table I shows that the peak shear stress obtained with N = 48 is within 5% Table I Peak Shear Stress (t 1 */m ) for Slip Transmission N = 12 N = 24 N = 48 N = 60 Case A Case B of the values at N = 60, so that N = 48 is considered sufficient for accuracy. In Case A, the interface is unable to accommodate core spreading within it, so that the transmission process involves propagation of slip directly from the incoming slip plane to the outgoing one. The peak

5 shear stress is t 1 */m 1 = In Case B, the interface accommodates significant core spreading within it, and the corresponding peak shear stress is t 1 */m 1 = Thus, core spreading of the dislocation into the interface dramatically increases the barrier strength of the interface. Figures 3(a, b) show the displacement profiles for a screw dislocation at the point of peak shear stress in Case A and Case B, respectively. Case A shows a welded interface for which the peak shear stress occurs when the center of the Peierls dislocation is located at the interface. In contrast, Case B shows a slipping interface for which the peak shear stress occurs when the spread core in the interface begins to move onto the outgoing slip plane. Interfacial Misfit Dislocations Differential displacement plots of the relaxed configuration of the ±a/2<110> screw dislocation with or without applied stress reveal that it cross-slips on to the interface plane without crossing the interface. However, in the case of 60 dislocations, the leading edge Shockley partial crosses the interface even though the screw component of the trailing Shockley partial cross-slips on to the interface plane. Figure 4 shows differential y-component displacement plots of an a/2<110> 60 dislocation at an applied shear stress of 0.04m 1 and 0.06m 1. At 0.06m 1, the leading Shockley partial has crossed from the elastically softer Al into the harder Ni layer. Also, some atomic rearrangement at or near the misfit dislocations occurs as the dislocation crosses the Al-Ni interface. Similar differential displacement plots of thin slices at other regions of the simulation cell show that very little bowing of the dislocation occurs before it crosses the Al-Ni interface. Previous simulation results [25] for strengthening due to modulus mismatch in the Cu-Ni system are used to estimate the modulus mismatch component of the blocking strength of the Al-Ni interface to be ~ 0.02m 1 for a a/2<110> 60 dislocation. This indicates that the misfit dislocations contribute a blocking strength of 0.02m 1 to 0.04m 1 to the Al-Ni interface in the simulation. CONCLUSIONS A continuum Peierls model of a screw dislocation being pushed through an interface reveals that interfaces which are more compliant to shearing can accommodate the core of the dislocation by slipping along the interface. In such cases, the critical remote stress to transmit the dislocation across the interface can be several times the critical value for a welded or nonslipping interface. For the slipping interface considered here, the peak stress is associated with

6 the onset of extracting the dislocation core from the interface onto the outgoing slip plane. For the welded interface considered, the peak stress occurs when the dislocation core is equally proportioned between the incoming and outgoing slip planes. An atomistic EAM approach reveals that a ±a/2<110> screw dislocation residing in Al approaches but does not cross a [0 0 1] Al-Ni interface with misfit dislocations but cross-slips on to the interface plane instead. For a 60 dislocation, the leading edge Shockley partial crosses the interface but the screw component of the trailing Shockley partial cross-slips onto the interface plane. The results indicate that the critical stress for transmission is t Al = (0.04 to 0.06)m Al and that (0.02 to 0.04)m Al of the resistance can be attributed to the role of misfit dislocations as obstacles. ACKNOWLEDGEMENTS PMH and SR acknowledge support of the AFRL Materials and Manufacturing Directorate contract F C-5258 with UES Inc. PMA and YC acknowledge the support of the Air Force Office of Scientific Research, Grant F REFERENCES 1. E.O. Hall, Proc. Roy. Soc. B64, p. 747 (1951) 2. N.J. Petch, J. Iron Steel Inst. 174, p. 25 (1953). 3. B.M. Clemens, H. King, S.A. Barnett, MRS Bulletin 24(2), p. 20 (1999). 4. S.I. Rao, P.M. Hazzledine, D.M. Dimiduk, Mater. Res. Soc. Proc. 362, p. 67 (1995). 5. P.M. Hazzledine and S.I. Rao, Mater. Res. Soc. Proc. 434, p. 135 (1996). 6. G. Saada, in Electron Microscopy and Strength of Crystals, edited by G. Thomas and J. Washburn (Interscience, New York) p. 651 (1963). 7. A.K. Head, Philos. Mag. 44, p. 92 (1953). 8. A.K. Head, Proc. Phys. Soc. (London) B66, p. 793 (1953). 9. J. Dundurs, in Math. Theory of Dislocations, edited by T. Mura (ASME, NY) p. 70 (1969). 10. J.A. Hurtado and L.B. Freund, J. Elasticity 52(2), p. 167 (1998). 11. L.E. Shilkrot and D.J. Srolovitz, Acta Mater. 46(9), p (1998). 12. E.S. Pacheco and T. Mura, J. Mech. Phys. Solids 17, p. 163 (1969). 13. J.P. Hirth and J. Lothe, Theory of Dislocations, Wiley, NY, 1982, p X. Xin and P.M. Anderson, to be published in Multiscale Fracture and Deformation in Materials and Structures: James R. Rice 60th Anniversary Volume, edited by T.-J. Chuang and J. W. Rudnicki (Kluwer, Dordrecht). 15. J.E. Krzanowski, Scripta Metall. Mater. 25(6), p (1991). 16. J.R. Rice, J. Mech. Phys. Solids 40, p. 235 (1992). 17. T.A. Parthasarathy, S.I. Rao, and D. Dimiduk, Philos. Mag. A 67, p. 643 (1993). 18. M.S. Daw and M.I. Baskes, Phys. Rev. B 29, p (1984). 19. M.W. Finnis and J.E. Sinclair, Philos. Mag. A 50, p. 45 (1984). 20. A.F. Voter and S.P. Chen, Mater. Res. Soc. Proc. 82, p. 175 (1987). 21. J.H. Rose, J.R. Smith, F. Guinea, and J. Ferrante, Phys. Rev. B. 29, p (1984). 22. S.I. Rao, C. Hernandez, J.P. Simmons, T.A. Parthasarathy, C. Woodward, Philos. Mag. A 77(1), p. 231 (1998). 23. V. Vitek, Crystal Lattice Defects 5, p. 1 (1974). 24. A.N. Stroh, Philos. Mag. 3, p. 625 (1958). 25. S.I. Rao and P.M. Hazzledine, accepted for publication in Philos. Mag. A (1999).