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An overview of interface-dominated deformation mechanisms in metallic multilayers
Current Opinion in Solid State and Materials Science 15 (2011) 20–28
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An overview of interface-dominated deformation mechanisms in metallic multilayers J. Wang, A. Misra ⇑ Los Alamos National Laboratory, Los Alamos, NM 87545, United States
a r t i c l e
i n f o
Article history: Received 8 July 2010 Accepted 13 September 2010
Keywords: Multilayer Interface Dislocation Atomistic simulations Experiments
a b s t r a c t Recent advances in the fundamental understanding of the deformation mechanisms in metallic multilayers are reviewed. The strength of metallic multilayers increases with decreasing layer thickness and reaches a maximum at layer thickness of a couple of nanometers. The unit processes of slip transmission across the interphase boundary, without the mechanical advantage of a dislocation pile-up, are critical in determining the maximum flow strengths of multilayers. For the case of non-coherent fcc–bcc nanolayered composites such as Cu–Nb, we show that the atomic structure of the interface leads to low interface shear strength. The stress field of a glide dislocation approaching the interface locally shears the interface, resulting in dislocation core spreading and trapping in the interface plane. Glide dislocation trapping at the weak interface via core spreading is thus the key unit process that determines the interface barrier to slip transmission. The maximum strength achieved in a non-coherent multilayer can be tailored by the shear strength of the interface. The role of the atomic structure of the interface in promoting room temperature climb at interfaces and its implications in dislocation recovery is highlighted. Experimental validation of the model predictions is discussed. Published by Elsevier Ltd.
1. Introduction Grain boundaries and interphase interfaces in metals have been shown to play a fundamental role in material properties such as strength, fracture, work hardening, and damage evolution under irradiation and shock. Overall, the role of interfaces in plastic deformation of metals encompasses interfaces acting as: (i) sources of defects; (ii) sinks of defects via absorption and annihilation; (iii) barriers to defects; and (iv) storage sites for defects. Previous studies in the literature have often treated the role of interfaces in a phenomenological way, yielding scaling laws such as Hall–Petch that relate strength to the spacing between interfaces that act as barriers to glide dislocations [1]. The formation of dislocation pile-ups at grain boundaries [2–4], interphase boundaries [5,6] and slip transmission mechanisms have been investigated [7,8]. However, the intrinsic role of atomic structures of interfaces remains poorly understood and essentially unquantified. Metallic multilayers are good model systems to explore the role of interfaces in metal plasticity. Earlier work focused on investigating length scale effects on the yield strength [9–20]. The flow strength of multilayers increases with decreasing thickness of the
⇑ Corresponding author. E-mail address: [email protected] (A. Misra). 1359-0286/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.cossms.2010.09.002
individual layers, following different unit processes that are schematically illustrated in Fig. 1 [20,21]. The dislocation pile-up based Hall–Petch scaling law, r / h1/2, (r is the flow strength and h is the layer thickness), is applicable at h greater than 50–100 nm but over-estimates the strength at lower h [22–29]. For h in the range of approximately 5–100 nm, the deformation mechanism is confined layer slip (CLS) that involves propagation of single dislocation loops parallel to the interfaces in both layers [16,30–35]. At h 2–5 nm, experimental data indicate that the hardness of metallic multilayers reaches a maximum. This behavior has been interpreted as a change in the key unit process from CLS to interface crossing of single dislocations. The interface barrier strength to the transmission of a single glide dislocation, without the mechanical advantage of a dislocation pile-up, is largely independent of the layer thickness but may drop at h< 1 nm when the dislocation core dimension is on the order of h [7,8,36–44]. This maximum in multilayer strength, achieved at h 2–5 nm, can be extremely high (e.g., 2.5 GPa for Cu–Nb), and typically, on the order of one-half to one-third of a lower bound estimate of the theoretical strength of perfect crystals [45,46]. The interface barrier stress to the transmission, without the stress concentration of a pile-up, of a single dislocation across interfaces must depend on the atomic structure, geometry and energetics of the interface [7,8,36–44]. The focus of this article is to overview recent investigations that have explored the atomic structure of the interface and the
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Fig. 1. Schematic illustration of the key unit processes that determine the yield mechanism in metallic multilayers.
interaction of a single dislocation with the interface to elucidate the role of the interface atomic structure and properties on the mechanism of slip transfer across the interface, and therefore, determine the maximum interface barrier stress to slip transmission in the absence of a pile-up. The role of atomic structure of the interface on dislocation storage and recovery at interfaces is also discussed. Atomistic modeling is compared with experimental results on the same interfaces, and an outlook for future work is presented.
2. Coherent vs semi-coherent interfaces The difficulty of dislocation transmission through interfaces is ascribed to different mechanisms with respect to interface types [7,8,37–41]. For coherent interfaces in multilayers, coherency stresses play crucial role in defining the maximum strength that can be achieved. The strength model suggested by Hoagland and coworkers [7] is based on the simple idea that a dislocation cannot traverse the composite unless the net forces on the dislocation in all layers are the same sign. Thus, a stress must be applied that at least cancels the coherency stress in one of the two constituents. In systems such as Cu–Ni with equal layer thickness, coherency stresses are on the order of 2 GPa that is comparable to the experimentally measured strengths. For coherent interfaces in single-phase metals, such as a coherent twin boundary, the presence of twin boundaries causes the change of crystal orientations between matrix and twinned crystals [47], resulting in the discontinuity of slip systems across twin boundaries [48,49]. Consequently a high resolved shear stress is required to transmit a single dislocation across twin interfaces, thereby strengthening materials [50–58]. For semi-coherent interfaces with small misfit (<5%), misfit dislocations relax only the long-range coherency stresses and the interface between the misfit dislocations remains coherent and, therefore, a glide dislocation that intersects the coherent segment of interface in this region still encounters very large stresses that must be overcome if the dislocation is to move through the composite. In addition, the misfit dislocations at interfaces must be cut for single dislocation transmission across the interface [41,59]. For interfaces between non-isostructural phases (such as fcc and bcc) with large misfit (>5%), there are no coherency stresses even for the thinnest layers (1 nm). Atomic relaxations in the interface lead to local patches of high and low atomic coordination and periodic arrays of defects. The structures and properties of
these high misfit semi-coherent interfaces and dislocation interactions with interfaces is overviewed in the following sections, with a focus on fcc–bcc systems such as Cu–Nb.
3. Atomistic simulations 3.1. Interface structures A bilayer system of Cu and Nb crystals adopts the Kurdjumov– Sachs (KS) crystallographic orientation relation [8,40,41]. It corresponds to {1 1 1}Cu||{1 1 0}Nb|| interface plane and h1 1 0iCu||h1 1 1iNb in the interface plane. Atomistic simulations were performed using embedded atom method empirical interatomic potentials [8]: Cu developed by Voter and Chen [60] and Nb developed by Johnson and Oh [61]. The interaction potentials designed to model bonding between these two elements are based on the usual form used for other metal pairs [43]. In the potentials, the dilute heat of mixing corresponding to one Cu atom in bcc Nb crystal was varied, keeping the lattice parameters constant [43]. Interface formation energies were 770 mJ/m2, 573 mJ/m2, 485 mJ/m2, 228 mJ/m2 and 176 mJ/m2 with respect to different dilute heats of mixing, 1.40 eV, 1.03 eV, 0.76 eV, 0.26 eV, and 0.39 eV, respectively. Atomistic simulations revealed that the atomic structures of Cu/ Nb interfaces could have multiple states with nearly degenerate energies [40,41,62–64], such as the KS1 atomic structure formed by directly combining two semi-infinite perfect crystals according to the KS orientation relation [64], and another two types of atomic structures, KS2 and KSmin formed by inserting an intermediate Cu monolayer into the KS1 interface [40,41,62,63]. For KS2 interface, the intermediate Cu monolayer is a strained Cu {1 1 1} atomic layer that is strained by applying an in-plane displacement gradient [40]. As a result, the strained Cu {1 1 1} atomic layer has a low areal density of 17.58 atoms/nm2, a ratio of 0.991 compared to a normal Cu {1 1 1} atomic layer. For KSmin interface, the intermediate Cu monolayer is formed as a result of introducing 23 vacancies at the vortex regions (Fig. 2), resulting in a lower areal density of 16.82 atoms/nm2, a ratio of 0.948 compared to a normal Cu {1 1 1} atomic layer. It should be mentioned that the KS orientation relation between Cu and Nb crystals is still maintained [40]. One example of the atomic structure of Cu–Nb interface, referred to KS1, is shown in Fig. 2. Two important features are revealed regardless of the dilute heats of mixing [8,40–43,64]. First, a quasi-repeating pattern corresponding to a two-dimensional
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Fig. 2. Atomic structure of Cu/Nb interface with Kurdjumov–Sachs (KS) orientation relation. Dashed lines outline the quasi-repeatable patterns (forming a conventional boundary-unit-cell, BUC). The circle outline the vortex region, at the center a Cu atom sits over a Nb atom. Atoms are colored according to their excess potential energies. The top atoms are Cu and the lower atoms are Nb.
boundary-unit-cell (BUC), is shown by the parallelogram. Second, groups of atoms around the center labeled in Fig. 2, contained within a circular region of about 15 Å in diameter, undergo displacements tangential to the circle. The magnitudes of the displacements are zero at/near circle center and increase with increasing distance from the center, analogous to a vortex [8,40]. The vortex-like displacement patterns that appear in the KS1 interface perturb the locations of neighboring atoms in a way reminiscent of the creation of a defect with a well-defined elastic strain field. These interface relaxations can be described geometrically in terms of two intersecting sets of interface dislocations [41,62–68]. For KS1 interface, the presence of dislocations can be demonstrated by analyzing the disregistry across the interface plane upon relaxation [7,8], in which a reference state for the CuNb bilayer must be chosen in which the interface between the neighboring elements is coherent, i.e., both of the adjoining interface atom planes have identical crystal structures [41,63]. The analysis of interfacial dislocation content was then carried out based on the Frank–Bilby equation [69,70]. One set of dislocations has the line sense along aCu ½0 1 1 Cu and a Burgers vector of 2 ½0 1 1, and is a screw-type; 1 and the other set of dislocations has the line sense along ½1 2 Cu aCu and a Burgers vector of 2 ½1 0 1, and is an edge-type. Corresponding to KS2, the two sets of interface dislocations are separated by the intermediate Cu monolayer layer, one set between the 1st and 2nd Cu layer and the other between Nb and the 1st Cu layer [1,41]. For KSmin, both sets of interface dislocations are still present in the interface between Cu and Nb crystals but with vacancies distributed at their intersections [1,41]. 3.2. Interface shear strengths and sliding mechanisms The shear strength of an interface is the critical shear stress at which irreversible sliding of one crystal with respect to the other commences along the interface. This is determined by gradually increasing a homogeneous shear strain applied to bilayer models of Cu/Nb [40,41]. The in-plane shear resistances for three types of interfaces with a given dilute heat of mixing (1.03 eV) is plotted in Fig. 3a and for the KS1 interface with respect to the different
Fig. 3. Shear response of interfaces, showing: (a) two-dimensional flow strength of interfaces KS1, KS2 and KSmin for the dilute heat of mixing, 1.03 eV, (b) twodimensional flow strength of interfaces KS1 with respect to the dilute heats of mixing, and (c) the relative displacements of the KS1 interface when it is subjected direction. The arrowed lines represent the magnitudes and to the shear along ½1 1 2 directions of the relative displacements. The blue dashed lines are interfacial dislocations separating the slipped regions from the non-slipped regions.
dilute heats of mixing are plotted in Fig. 3b. The results revealed that the shear strength of a Cu/Nb interface: (i) is significantly lower than the theoretical estimates of shear strengths on glide planes in perfect crystals of Cu and Nb (>2.0 GPa); (ii) is strongly dependent on the atomic structures of the interfaces, with KS2 the weakest and KSmin the strongest; (iii) increases with decreasing heat of mixing; and (iv) is anisotropic (i.e., dependent on the applied inplane shear direction). Details of the atomic-scale sliding in interfaces are revealed by disregistry analysis [40]. When the applied shear stress reaches a
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critical value, irreversible sliding occurs nonuniformly, beginning within the vortex regions in the interface, as shown in Fig. 3c, implying that shear resistance of the interface is spatially non-uniform in the interface plane. The spatially non-uniform sliding in the interface suggests an interface sliding mechanism in which dislocation loops nucleate in interfaces and subsequently expand by gliding in the interface plane. For convenience in describing the interface sliding, the dashed lines in Fig. 3c represent dislocation lines that are a result of the onset of sliding in the weak shear regions. As sliding commences, the dislocation loops nucleate at the vortex regions, then expand, coalesce, and glide in the interface. The dashed lines in Fig. 3c approximate the locations of interfacial dislocations, separating the slipped and non-slipped regions in the interface [40,41,71]. It is worth pointing out that the shear response observed in this study of Cu–Nb interfaces could be general features for fcc/bcc interfaces with the KS orientation relation [39,71]. The spatial non-uniformity of shear strength in the interface is related to the quasi-repeating pattern that is determined by crystal structures, independent of the dilute heats of mixing. As a result, the vortex regions with the lower shear resistance act as nucleation sites for interfacial dislocations.
3.3. Interaction of lattice dislocations with interfaces As a consequence of the low shear resistance of the interface, the interface will shear in response to the stress field of a nearby dislocation and attract the dislocation into the interface wherein the core of the dislocation spreads [8,39,41,64,71]. Using atomistic simulations, we showed that a leading Shockley partial dislocation (dissociated from a mixed full dislocation), no matter what type or
Fig. 4. Core spreading of a glide dislocation within the interface, showing (a) the vector field plot of disregistry across the interface plane when an edge Shockley partial dislocation enters the interface, and (b) the magnitude plot of two in-plane components of the interface shear. The dashed line indicates the trace of glide plane with the interface plane. The core width is determined in (b) between the two thin black lines that are the half of the magnitude.
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sign, spontaneously enters the interface, from Cu due to shear of the interface [8,41]. The core of the lattice glide dislocation readily spreads within the Cu–Nb interface, as shown in Fig 4a, resulting in a non-planar core structure. The width of the spread core of an edge partial for the KS1 interface (for the dilute heat of mixing equal to 1.03) is about 4 nm, Fig 4b, more than 15 times the Burgers vector. Atomistic simulations show that the width of core spreading increases with increasing heat of mixing (or with decreasing shear strength) [71]. In addition, the relative displacements in the sheared region are not parallel to the direction of the in-plane component of the Burgers vector of the partial dislocation because of the spatial non-uniformity of shear resistance [39]. For a screw dislocation initially gliding on Cu (1 1 1), it readily cross-slips onto the interface plane due to the low shear resistance [39]. We also studied the core spreading of lattice dislocations in Nb within interfaces. For a mixed dislocation, the in-plane component of Burgers vector readily spreads within the interface, resulting in a non-planar core structure [41]. For a screw dislocation, again it cross-slips onto the interface plane, with a planar core structure differing from the non-planar core structure in a Nb crystal [8,41]. The same study has been performed for KS2 and KSmin interfaces. Similar phenomena are observed: mixed dislocations (in Cu or in Nb) enter interfaces, accompanied with the core spreading of the in-plane component of the Burgers vector within interfaces; screw dislocations cross-slip onto the interface plane. Since KS2 interface has the lowest shear strength among the three types of interfaces, the extent of core spreading is greater than the other two interfaces. The results indicate that the extent of interface shear accompanied with core spreading of lattice dislocations within the interface is greater for interfaces with the lower shear strength. As a consequence, slip transmission across interface becomes difficult due to the non-planar core structure. In addition, screw dislocations after cross-slip onto the interface are still mobile, resulting in interface sliding. Furthermore, glide of screw dislocations on the interface plane facilitates the reaction of dislocations within interfaces that may lead to dislocation recovery [39]. As a glide dislocation approaches the interface, its stress field locally shears the weak interface that in turn results in an attractive force that attracts the lattice dislocation towards the interface. With respect to different shear strength, the attraction force can be derived as the first order derivative of the potential energy with respect to the distance from the interface. The change of potential energy is due to two factors, one arising from the Koehler force and the other from the sheared interface [8]. The Koehler force on glide dislocations is computed from the anisotropic solution [72], and then the Koehler force is integrated as a function of the distance from the interface. The potential energy due to the sheared interface is determined from atomistic simulations as a function of the distance from the interface [8,71]. For glide dislocations in a Cu crystal, the potential energy always decreases as a glide dislocation approaches the interface [8]. For glide dislocations in a Nb crystal, the potential energy increases until a critical distance is reached, and then decreases [8]. With respect to interface shear strength, the relative potential energy due to interface shear is plotted in Fig. 5 as a function of the distance of a dislocation in Nb from interface. The result reveals that the interface with the lower shear strength results in the quick drop of energy as the dislocation approaches the interface, implying the stronger attraction force on the dislocation. The critical distance for a dislocation in Nb, dc, at which the first derivative of the total relative potential energy with respect to the distance is zero, is around 1.5 nm, and increases with the decrease of interface shear strength [8]. More importantly, the existence of a critical distance suggests that dislocations cannot stay in a Nb layer if the thickness of the Nb layer is less than 2dc. This result is in
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Fig. 5. The change of potential energies associated with a mixed dislocation 1/2h1 1 1i{1 1 0} in Nb crystal approaching the KS1 interface for different heats of mixing.
agreement with experimental observations, in which transmission electron microscopy (TEM) studies on cold-rolled Cu–Nb multilayers showed that the films exhibit large plastic deformation without dislocation cell structure formation inside layers [73]. 3.4. Slip transmission of a single dislocation across fcc/bcc interface For the case of dislocation trapping in the interface plane via core spreading, slip transmission is achieved through three unit processes: (a) a glide dislocation is attracted into the interface in
association with the core spreading within the interface; (b) the spread core has to shrink in order to nucleate a glide dislocation in the adjacent crystal; and (c) the nucleated dislocation loop is bowed out from the interface under stress, overcoming the attraction force due to the interface shear and the residual dislocation at interface. This transmission process is shown schematically in Fig. 6. For certain slip systems, the slip transmission process is studied using the chain-of-state method [74]. First, a straight dislocation is introduced into one crystal in the bicrystal model close to the interface. After fully relaxing the dislocated configuration, the equilibrium configuration containing a lattice dislocation is obtained, and this acts as the initial configuration. Second, the final configuration containing a dislocation loop in the adjacent crystal is created and relaxed at a certain applied stress. Finally, molecular dynamics is performed for the final configuration while reducing the applied stress to obtain a series of configurations between the initial and the final configurations. The change of potential energy is then calculated as a function of the size of the dislocation loop [75]. Fig. 7 shows three snapshots: initial nucleus, one intermediate configuration, and the final configuration. Corresponding to the different states, the change of potential energy of the bicrystal is calculated. The critical stress corresponding to the stabilized dislocation loop is determined from the first derivative of the potential energy with respect to the dislocation loop and found to be on the order of 0.8 GPa in terms of the resolved shear stress. Corresponding to different interfaces, this critical stress will increase as the decrease of interface shear resistance due to the wider spreading of dislocation core within the weaker interface [71,75]. In addition, the critical stress varies corresponding to slip systems with parallel or non-parallel traces, higher for non-parallel traces because of the smaller size of the stabilized loop [75].
Fig. 6. Schematic illustration of slip transmission process for composites with low shear strength interfaces. (a) Interface shear, accompanied with core spreading of a lattice dislocation within the interface plane, (b) a dislocation loop nucleates in association with the shrinking of the spread core, and finally, (c) dislocation escapes from the interface by overcoming the attraction due to the interface shear.
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Fig. 8. The dissociation of a mixed dislocation b1 aNb ½1 1 1 in the Cu/Nb interface. 2 The schematic plots show (a) a mixed dislocation b1 situated at the interface, and (b) the dissociation into an interfacial dislocation b2 a3Cu ½1 1 1 and an interfacial aNb aCu discontinuity b3 2 ½1 1 1 3 ½1 1 1 left near the trace of the Nb slip; (c) atomic structures after the dissociation of the dislocation in the interface. The red arrows indicate the diffusion of vacancies and the black arrows indicate the counter diffusion of Cu atoms. The yellow lines outline Cu (1 1 1) planes, and the black lines Nb (1 1 0) planes. (d) Atomic structures of the interfacial Cu plane. The dashed lines indicate locations of the two dislocations. The regions with low areal density of Cu atoms are outlined in gray atoms and the gray shadow, and the regions with the areal density same as Cu (1 1 1) in yellow atoms (from Ref. [44]).
Fig. 7. Atomistic simulation of slip transmission across a Cu/Nb interface from Cu to Nb when the two slip planes have parallel traces with interface plane. The top crystal is Cu and the lower Nb. A full dislocation with Burgers vector of 1/2h1 1 0i on {1 1 1} plane initially approaches the Cu/Nb interface, resulting in (a) the nucleation of a dislocation loop in Nb with Burgers vector of 1/2h1 1 1i on {1 1 0} plane, (b) the propagation of the dislocation loop, and (c) the final stabilized loop at tensile stress 2.0 GPa. Atoms are colored according to the magnitude of the projected disregistry on each slip plane. The blue atoms in Nb outline the dislocation loop in Nb.
3.5. Dislocation climb in interfaces The glide dislocations entrapped, with spread core, in the interface can move along the interface plane through glide and climb. Atomistic simulations have shown that the low shear resistance of Cu–Nb interface enables the in-plane component of dislocations to glide with a high mobility [44]. The out-of-plane component of dislocations can climb in the interface through absorption and emission of vacancies, as shown in Fig. 8. Consequently, a patch of extra Cu (1 1 1) forms at the end of the half plane of Nb (1 1 0). The rate of climb depends on the concentration and diffusivity of vacancies. The equilibrium vacancy concentration is determined to be around 5.8% using atomistic simulations, which is far higher than that in perfect crystals of Cu and Nb [39,44]. The diffusivity of
a vacancy depends on the formation energy and the migration energy of a vacancy in the interfacial plane. This diffusivity is very high. One reason is that the equilibrium vacancy concentration is quite high, since the defect energy with respect to removal or insertion of atoms in the interfacial Cu layer is only 0.12 eV/atoms at 300 K, which is one order of magnitude smaller than the formation energies of a vacancy (1.26 eV) and an interstitial (3.24 eV) in Cu. The second reason is that the kinetic barriers of ‘‘vacancy” diffusion in interfaces are low. Using the Nudged Elastic Band method [74], the kinetic barriers associated with the migration of one Cu ‘‘vacancy” in the interface plane are computed and in the range of 0.03–0.10 eV [44]. This small kinetic barrier is comparable with a Cu adatom diffusing on a flat Cu(111) surface [76–78]. Finally, the dislocation acts as a perfect source for vacancies, and the interface acts as a sink for vacancies [79]. At room temperature, the climb velocity is around 0.5 m/s by solving Fick’s second law with respect to a moving source of vacancies, about five orders of magnitude larger than the climb velocity in a Cu crystal. Dislocation climb could play a crucial role during mechanical deformation. First, this climb enables slip transmission: glide dislocations can reassemble the spread cores in the interface plane via climb, and dislocation debris scattered in interfaces can reassemble into lattice glide dislocations, facilitating slip transmission. Secondly, reactions between interfacial dislocations assisted by climb could lead to
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pile-up. This implies that unless a long pile-up can form, the slip transmission across Cu–Nb interfaces will not be assisted by the mechanical advantage of the pile-up. Thus, crossing of single dislocations across the interface in nanolayered materials is the critical unit process and may be assisted by climb. Finally, dislocation climb enables the interfacial plane to be in an equilibrium state with respect to concentrations of point defects because of absorption and/or emission of vacancies at dislocation cores. Thus, these interfaces act as ideal sinks for vacancies and interstitials, allowing Cu–Nb interfaces to persist in a damage-free steady state even when driven out of equilibrium by intense particle radiation.
4. Experimental validations
Fig. 9. A cross-section HRTEM image taken close to an interface (marked by the dotted line) showing a Burgers circuit in the Cu layer defining a dislocation with b||h2 1 1i. (From Ref. [81]).
annihilation of dislocation content (recovery). Thirdly, discrete pile-ups can be absorbed in the interface plane, assisted by climb in the interface, thereby blunting the stress concentration of the
Experimental measurements performed on sputter deposited fcc–bcc multilayers have shown results consistent with the predictions of the atomistic simulations. With regard to interface crystallography and structure, X-ray diffraction [80] and transmission electron microscopy [81] show the KS orientation relationship and interface dislocations. Fig. 9 is a high-resolution cross-section TEM image of a Cu–Nb multilayer showing the KS orientation relationship and a dislocation with Burgers vector in the interface plane. The sub-Å scale atomic displacements that appear as vortices in the interface plane in the atomistic simulations [41,64] are not resolved in the cross-section TEM images such as Fig. 9. Hardness and pillar compression tests in a nanoindenter were used to measure flow stresses of Cu–Nb and Al–Nb multilayers with layer thickness on the order of 5 nm or less. For Cu–Nb multilayers, tests with compression axis at 90° to the interface, measured flow stress of 2.3 GPa [46] suggesting a high barrier stress
Fig. 10. HR-TEM snapshots of the dislocation annihilation process at an interface captured at different instants during in situ nanoindentation. (a) At 0 s the two dislocations were separated by 2.4 nm. (b) After 2 s the separation distance was reduced to 1.7 nm via dislocations climb. (c) At 2.5 s the two dislocations annihilated each other, and dislocation b2 inside the Nb glided towards the interface. (d) At 3.5 s the interface became nearly perfect after complete annihilation of the dislocations (from Ref. [85]).
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for slip transmission. Compression tests in Cu–Nb with interfaces oriented at 45° to the loading axis resulted in shear along interfaces at relatively low resolved shear stress of 550–750 MPa [82]. These experimental measurements validate the atomistic simulations that Cu–Nb interfaces are weak in shear but have a high interface barrier stress to single dislocation transmission. Indentation tests normal to interfaces in Al–Nb resulted in a much lower flow stress than in Cu–Nb [83]. Note that Al–Nb has a negative heat of mixing as compared to Cu–Nb which has a positive heat of mixing. These results are also consistent with atomistic models that predict a decrease in interface barrier stress as heat of mixing becomes less positive (more negative). Compression tests also show strain softening that may be due to shear localization across interfaces (for interfaces that are strong in shear, Ref. [84]) or due to rotation of interface plane towards the plane of maximum shear (for interfaces that are weak in shear, Ref. [82]). Finally, the atomistic modeling prediction of climb at interfaces was verified by in situ indentation in a TEM in Al–Nb multilayers. The edge components of two glide dislocations trapped at interfaces were observed to annihilate via climb along the interface. This is shown in Fig. 10 [85]. Climb along interfaces can lead to recovery of stored dislocation content during plastic deformation. 5. Summary The current work on multilayers shows that the atomic structure of the interface may lead to specific properties of the interface (such as low shear strength) that control the interaction of glide dislocations with the interface. The interface barrier stress for slip transmission, in the absence of a dislocation pile-up, depends on how strongly a glide dislocation interacts (i.e., pinning stress) with the interface, which can be correlated with the defect structure of the interface, that in turn depends on factors such as crystallographic orientation relationships, lattice misfit and heats of mixing. In future, this framework can be extended to other kinds of interfaces besides fcc–bcc to interpret the maximum strength that may be achieved in a nanoscale multilayer. In addition to atomistic modeling, other methods such as discrete dislocation dynamics, and crystal plasticity modeling are being used and this will lead to multi-scale models that predict the complete stress–strain curve using atomic-scale deformation mechanisms at interfaces [39,86–92]. The field of research in metallic multilayers continues to grow with exploration of other properties such as fatigue and fracture [93–98], creep [99,100], and other systems such as metal–ceramic [100–105], bcc–hcp [106], and bulk fabrication of multilayers [107,108]. Acknowledgements The authors acknowledge support from DOE, Office of Science, Office of Basic Energy Sciences, and fruitful collaborations with R.G. Hoagland, J.P. Hirth, J.D. Embury, M.J. Demkowicz, N.A. Mara, N. Li, D. Bhattacharyya, X. Zhang, X.Y. Liu, Q. Wei, O. Anderoglu, Y.C. Wang that resulted in the published literature that is reviewed here. References [1] Hirth JP. The influence of grain boundaries on mechanical properties. Metall Trans 1972;3:3047–67. [2] Lee TC, Robertson IM, Birnbaum HK. TEM in situ deformation study of the interaction of lattice dislocations with grain boundaries in metals. Philos Mag A 1990;62:131–53. [3] Lee TC, Robertson IM, Birnbaum HK. An in Situ transmission electron microscope deformation study of the slip transfer mechanisms in metals. Metal Trans A 1990;21A:2437–47. [4] Shen Z, Wagoner RH, Clark WAT. Dislocation and grain boundary interactions in metals. Acta Metall 1988;36:3231–42.
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Current Opinion in Solid State and Materials Science journal homepage: www.elsevier.com/locate/cossms
An overview of interface-dominated deformation mechanisms in metallic multilayers J. Wang, A. Misra ⇑ Los Alamos National Laboratory, Los Alamos, NM 87545, United States
a r t i c l e
i n f o
Article history: Received 8 July 2010 Accepted 13 September 2010
Keywords: Multilayer Interface Dislocation Atomistic simulations Experiments
a b s t r a c t Recent advances in the fundamental understanding of the deformation mechanisms in metallic multilayers are reviewed. The strength of metallic multilayers increases with decreasing layer thickness and reaches a maximum at layer thickness of a couple of nanometers. The unit processes of slip transmission across the interphase boundary, without the mechanical advantage of a dislocation pile-up, are critical in determining the maximum flow strengths of multilayers. For the case of non-coherent fcc–bcc nanolayered composites such as Cu–Nb, we show that the atomic structure of the interface leads to low interface shear strength. The stress field of a glide dislocation approaching the interface locally shears the interface, resulting in dislocation core spreading and trapping in the interface plane. Glide dislocation trapping at the weak interface via core spreading is thus the key unit process that determines the interface barrier to slip transmission. The maximum strength achieved in a non-coherent multilayer can be tailored by the shear strength of the interface. The role of the atomic structure of the interface in promoting room temperature climb at interfaces and its implications in dislocation recovery is highlighted. Experimental validation of the model predictions is discussed. Published by Elsevier Ltd.
1. Introduction Grain boundaries and interphase interfaces in metals have been shown to play a fundamental role in material properties such as strength, fracture, work hardening, and damage evolution under irradiation and shock. Overall, the role of interfaces in plastic deformation of metals encompasses interfaces acting as: (i) sources of defects; (ii) sinks of defects via absorption and annihilation; (iii) barriers to defects; and (iv) storage sites for defects. Previous studies in the literature have often treated the role of interfaces in a phenomenological way, yielding scaling laws such as Hall–Petch that relate strength to the spacing between interfaces that act as barriers to glide dislocations [1]. The formation of dislocation pile-ups at grain boundaries [2–4], interphase boundaries [5,6] and slip transmission mechanisms have been investigated [7,8]. However, the intrinsic role of atomic structures of interfaces remains poorly understood and essentially unquantified. Metallic multilayers are good model systems to explore the role of interfaces in metal plasticity. Earlier work focused on investigating length scale effects on the yield strength [9–20]. The flow strength of multilayers increases with decreasing thickness of the
⇑ Corresponding author. E-mail address: [email protected] (A. Misra). 1359-0286/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.cossms.2010.09.002
individual layers, following different unit processes that are schematically illustrated in Fig. 1 [20,21]. The dislocation pile-up based Hall–Petch scaling law, r / h1/2, (r is the flow strength and h is the layer thickness), is applicable at h greater than 50–100 nm but over-estimates the strength at lower h [22–29]. For h in the range of approximately 5–100 nm, the deformation mechanism is confined layer slip (CLS) that involves propagation of single dislocation loops parallel to the interfaces in both layers [16,30–35]. At h 2–5 nm, experimental data indicate that the hardness of metallic multilayers reaches a maximum. This behavior has been interpreted as a change in the key unit process from CLS to interface crossing of single dislocations. The interface barrier strength to the transmission of a single glide dislocation, without the mechanical advantage of a dislocation pile-up, is largely independent of the layer thickness but may drop at h< 1 nm when the dislocation core dimension is on the order of h [7,8,36–44]. This maximum in multilayer strength, achieved at h 2–5 nm, can be extremely high (e.g., 2.5 GPa for Cu–Nb), and typically, on the order of one-half to one-third of a lower bound estimate of the theoretical strength of perfect crystals [45,46]. The interface barrier stress to the transmission, without the stress concentration of a pile-up, of a single dislocation across interfaces must depend on the atomic structure, geometry and energetics of the interface [7,8,36–44]. The focus of this article is to overview recent investigations that have explored the atomic structure of the interface and the
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Fig. 1. Schematic illustration of the key unit processes that determine the yield mechanism in metallic multilayers.
interaction of a single dislocation with the interface to elucidate the role of the interface atomic structure and properties on the mechanism of slip transfer across the interface, and therefore, determine the maximum interface barrier stress to slip transmission in the absence of a pile-up. The role of atomic structure of the interface on dislocation storage and recovery at interfaces is also discussed. Atomistic modeling is compared with experimental results on the same interfaces, and an outlook for future work is presented.
2. Coherent vs semi-coherent interfaces The difficulty of dislocation transmission through interfaces is ascribed to different mechanisms with respect to interface types [7,8,37–41]. For coherent interfaces in multilayers, coherency stresses play crucial role in defining the maximum strength that can be achieved. The strength model suggested by Hoagland and coworkers [7] is based on the simple idea that a dislocation cannot traverse the composite unless the net forces on the dislocation in all layers are the same sign. Thus, a stress must be applied that at least cancels the coherency stress in one of the two constituents. In systems such as Cu–Ni with equal layer thickness, coherency stresses are on the order of 2 GPa that is comparable to the experimentally measured strengths. For coherent interfaces in single-phase metals, such as a coherent twin boundary, the presence of twin boundaries causes the change of crystal orientations between matrix and twinned crystals [47], resulting in the discontinuity of slip systems across twin boundaries [48,49]. Consequently a high resolved shear stress is required to transmit a single dislocation across twin interfaces, thereby strengthening materials [50–58]. For semi-coherent interfaces with small misfit (<5%), misfit dislocations relax only the long-range coherency stresses and the interface between the misfit dislocations remains coherent and, therefore, a glide dislocation that intersects the coherent segment of interface in this region still encounters very large stresses that must be overcome if the dislocation is to move through the composite. In addition, the misfit dislocations at interfaces must be cut for single dislocation transmission across the interface [41,59]. For interfaces between non-isostructural phases (such as fcc and bcc) with large misfit (>5%), there are no coherency stresses even for the thinnest layers (1 nm). Atomic relaxations in the interface lead to local patches of high and low atomic coordination and periodic arrays of defects. The structures and properties of
these high misfit semi-coherent interfaces and dislocation interactions with interfaces is overviewed in the following sections, with a focus on fcc–bcc systems such as Cu–Nb.
3. Atomistic simulations 3.1. Interface structures A bilayer system of Cu and Nb crystals adopts the Kurdjumov– Sachs (KS) crystallographic orientation relation [8,40,41]. It corresponds to {1 1 1}Cu||{1 1 0}Nb|| interface plane and h1 1 0iCu||h1 1 1iNb in the interface plane. Atomistic simulations were performed using embedded atom method empirical interatomic potentials [8]: Cu developed by Voter and Chen [60] and Nb developed by Johnson and Oh [61]. The interaction potentials designed to model bonding between these two elements are based on the usual form used for other metal pairs [43]. In the potentials, the dilute heat of mixing corresponding to one Cu atom in bcc Nb crystal was varied, keeping the lattice parameters constant [43]. Interface formation energies were 770 mJ/m2, 573 mJ/m2, 485 mJ/m2, 228 mJ/m2 and 176 mJ/m2 with respect to different dilute heats of mixing, 1.40 eV, 1.03 eV, 0.76 eV, 0.26 eV, and 0.39 eV, respectively. Atomistic simulations revealed that the atomic structures of Cu/ Nb interfaces could have multiple states with nearly degenerate energies [40,41,62–64], such as the KS1 atomic structure formed by directly combining two semi-infinite perfect crystals according to the KS orientation relation [64], and another two types of atomic structures, KS2 and KSmin formed by inserting an intermediate Cu monolayer into the KS1 interface [40,41,62,63]. For KS2 interface, the intermediate Cu monolayer is a strained Cu {1 1 1} atomic layer that is strained by applying an in-plane displacement gradient [40]. As a result, the strained Cu {1 1 1} atomic layer has a low areal density of 17.58 atoms/nm2, a ratio of 0.991 compared to a normal Cu {1 1 1} atomic layer. For KSmin interface, the intermediate Cu monolayer is formed as a result of introducing 23 vacancies at the vortex regions (Fig. 2), resulting in a lower areal density of 16.82 atoms/nm2, a ratio of 0.948 compared to a normal Cu {1 1 1} atomic layer. It should be mentioned that the KS orientation relation between Cu and Nb crystals is still maintained [40]. One example of the atomic structure of Cu–Nb interface, referred to KS1, is shown in Fig. 2. Two important features are revealed regardless of the dilute heats of mixing [8,40–43,64]. First, a quasi-repeating pattern corresponding to a two-dimensional
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Fig. 2. Atomic structure of Cu/Nb interface with Kurdjumov–Sachs (KS) orientation relation. Dashed lines outline the quasi-repeatable patterns (forming a conventional boundary-unit-cell, BUC). The circle outline the vortex region, at the center a Cu atom sits over a Nb atom. Atoms are colored according to their excess potential energies. The top atoms are Cu and the lower atoms are Nb.
boundary-unit-cell (BUC), is shown by the parallelogram. Second, groups of atoms around the center labeled in Fig. 2, contained within a circular region of about 15 Å in diameter, undergo displacements tangential to the circle. The magnitudes of the displacements are zero at/near circle center and increase with increasing distance from the center, analogous to a vortex [8,40]. The vortex-like displacement patterns that appear in the KS1 interface perturb the locations of neighboring atoms in a way reminiscent of the creation of a defect with a well-defined elastic strain field. These interface relaxations can be described geometrically in terms of two intersecting sets of interface dislocations [41,62–68]. For KS1 interface, the presence of dislocations can be demonstrated by analyzing the disregistry across the interface plane upon relaxation [7,8], in which a reference state for the CuNb bilayer must be chosen in which the interface between the neighboring elements is coherent, i.e., both of the adjoining interface atom planes have identical crystal structures [41,63]. The analysis of interfacial dislocation content was then carried out based on the Frank–Bilby equation [69,70]. One set of dislocations has the line sense along aCu ½0 1 1 Cu and a Burgers vector of 2 ½0 1 1, and is a screw-type; 1 and the other set of dislocations has the line sense along ½1 2 Cu aCu and a Burgers vector of 2 ½1 0 1, and is an edge-type. Corresponding to KS2, the two sets of interface dislocations are separated by the intermediate Cu monolayer layer, one set between the 1st and 2nd Cu layer and the other between Nb and the 1st Cu layer [1,41]. For KSmin, both sets of interface dislocations are still present in the interface between Cu and Nb crystals but with vacancies distributed at their intersections [1,41]. 3.2. Interface shear strengths and sliding mechanisms The shear strength of an interface is the critical shear stress at which irreversible sliding of one crystal with respect to the other commences along the interface. This is determined by gradually increasing a homogeneous shear strain applied to bilayer models of Cu/Nb [40,41]. The in-plane shear resistances for three types of interfaces with a given dilute heat of mixing (1.03 eV) is plotted in Fig. 3a and for the KS1 interface with respect to the different
Fig. 3. Shear response of interfaces, showing: (a) two-dimensional flow strength of interfaces KS1, KS2 and KSmin for the dilute heat of mixing, 1.03 eV, (b) twodimensional flow strength of interfaces KS1 with respect to the dilute heats of mixing, and (c) the relative displacements of the KS1 interface when it is subjected direction. The arrowed lines represent the magnitudes and to the shear along ½1 1 2 directions of the relative displacements. The blue dashed lines are interfacial dislocations separating the slipped regions from the non-slipped regions.
dilute heats of mixing are plotted in Fig. 3b. The results revealed that the shear strength of a Cu/Nb interface: (i) is significantly lower than the theoretical estimates of shear strengths on glide planes in perfect crystals of Cu and Nb (>2.0 GPa); (ii) is strongly dependent on the atomic structures of the interfaces, with KS2 the weakest and KSmin the strongest; (iii) increases with decreasing heat of mixing; and (iv) is anisotropic (i.e., dependent on the applied inplane shear direction). Details of the atomic-scale sliding in interfaces are revealed by disregistry analysis [40]. When the applied shear stress reaches a
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critical value, irreversible sliding occurs nonuniformly, beginning within the vortex regions in the interface, as shown in Fig. 3c, implying that shear resistance of the interface is spatially non-uniform in the interface plane. The spatially non-uniform sliding in the interface suggests an interface sliding mechanism in which dislocation loops nucleate in interfaces and subsequently expand by gliding in the interface plane. For convenience in describing the interface sliding, the dashed lines in Fig. 3c represent dislocation lines that are a result of the onset of sliding in the weak shear regions. As sliding commences, the dislocation loops nucleate at the vortex regions, then expand, coalesce, and glide in the interface. The dashed lines in Fig. 3c approximate the locations of interfacial dislocations, separating the slipped and non-slipped regions in the interface [40,41,71]. It is worth pointing out that the shear response observed in this study of Cu–Nb interfaces could be general features for fcc/bcc interfaces with the KS orientation relation [39,71]. The spatial non-uniformity of shear strength in the interface is related to the quasi-repeating pattern that is determined by crystal structures, independent of the dilute heats of mixing. As a result, the vortex regions with the lower shear resistance act as nucleation sites for interfacial dislocations.
3.3. Interaction of lattice dislocations with interfaces As a consequence of the low shear resistance of the interface, the interface will shear in response to the stress field of a nearby dislocation and attract the dislocation into the interface wherein the core of the dislocation spreads [8,39,41,64,71]. Using atomistic simulations, we showed that a leading Shockley partial dislocation (dissociated from a mixed full dislocation), no matter what type or
Fig. 4. Core spreading of a glide dislocation within the interface, showing (a) the vector field plot of disregistry across the interface plane when an edge Shockley partial dislocation enters the interface, and (b) the magnitude plot of two in-plane components of the interface shear. The dashed line indicates the trace of glide plane with the interface plane. The core width is determined in (b) between the two thin black lines that are the half of the magnitude.
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sign, spontaneously enters the interface, from Cu due to shear of the interface [8,41]. The core of the lattice glide dislocation readily spreads within the Cu–Nb interface, as shown in Fig 4a, resulting in a non-planar core structure. The width of the spread core of an edge partial for the KS1 interface (for the dilute heat of mixing equal to 1.03) is about 4 nm, Fig 4b, more than 15 times the Burgers vector. Atomistic simulations show that the width of core spreading increases with increasing heat of mixing (or with decreasing shear strength) [71]. In addition, the relative displacements in the sheared region are not parallel to the direction of the in-plane component of the Burgers vector of the partial dislocation because of the spatial non-uniformity of shear resistance [39]. For a screw dislocation initially gliding on Cu (1 1 1), it readily cross-slips onto the interface plane due to the low shear resistance [39]. We also studied the core spreading of lattice dislocations in Nb within interfaces. For a mixed dislocation, the in-plane component of Burgers vector readily spreads within the interface, resulting in a non-planar core structure [41]. For a screw dislocation, again it cross-slips onto the interface plane, with a planar core structure differing from the non-planar core structure in a Nb crystal [8,41]. The same study has been performed for KS2 and KSmin interfaces. Similar phenomena are observed: mixed dislocations (in Cu or in Nb) enter interfaces, accompanied with the core spreading of the in-plane component of the Burgers vector within interfaces; screw dislocations cross-slip onto the interface plane. Since KS2 interface has the lowest shear strength among the three types of interfaces, the extent of core spreading is greater than the other two interfaces. The results indicate that the extent of interface shear accompanied with core spreading of lattice dislocations within the interface is greater for interfaces with the lower shear strength. As a consequence, slip transmission across interface becomes difficult due to the non-planar core structure. In addition, screw dislocations after cross-slip onto the interface are still mobile, resulting in interface sliding. Furthermore, glide of screw dislocations on the interface plane facilitates the reaction of dislocations within interfaces that may lead to dislocation recovery [39]. As a glide dislocation approaches the interface, its stress field locally shears the weak interface that in turn results in an attractive force that attracts the lattice dislocation towards the interface. With respect to different shear strength, the attraction force can be derived as the first order derivative of the potential energy with respect to the distance from the interface. The change of potential energy is due to two factors, one arising from the Koehler force and the other from the sheared interface [8]. The Koehler force on glide dislocations is computed from the anisotropic solution [72], and then the Koehler force is integrated as a function of the distance from the interface. The potential energy due to the sheared interface is determined from atomistic simulations as a function of the distance from the interface [8,71]. For glide dislocations in a Cu crystal, the potential energy always decreases as a glide dislocation approaches the interface [8]. For glide dislocations in a Nb crystal, the potential energy increases until a critical distance is reached, and then decreases [8]. With respect to interface shear strength, the relative potential energy due to interface shear is plotted in Fig. 5 as a function of the distance of a dislocation in Nb from interface. The result reveals that the interface with the lower shear strength results in the quick drop of energy as the dislocation approaches the interface, implying the stronger attraction force on the dislocation. The critical distance for a dislocation in Nb, dc, at which the first derivative of the total relative potential energy with respect to the distance is zero, is around 1.5 nm, and increases with the decrease of interface shear strength [8]. More importantly, the existence of a critical distance suggests that dislocations cannot stay in a Nb layer if the thickness of the Nb layer is less than 2dc. This result is in
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Fig. 5. The change of potential energies associated with a mixed dislocation 1/2h1 1 1i{1 1 0} in Nb crystal approaching the KS1 interface for different heats of mixing.
agreement with experimental observations, in which transmission electron microscopy (TEM) studies on cold-rolled Cu–Nb multilayers showed that the films exhibit large plastic deformation without dislocation cell structure formation inside layers [73]. 3.4. Slip transmission of a single dislocation across fcc/bcc interface For the case of dislocation trapping in the interface plane via core spreading, slip transmission is achieved through three unit processes: (a) a glide dislocation is attracted into the interface in
association with the core spreading within the interface; (b) the spread core has to shrink in order to nucleate a glide dislocation in the adjacent crystal; and (c) the nucleated dislocation loop is bowed out from the interface under stress, overcoming the attraction force due to the interface shear and the residual dislocation at interface. This transmission process is shown schematically in Fig. 6. For certain slip systems, the slip transmission process is studied using the chain-of-state method [74]. First, a straight dislocation is introduced into one crystal in the bicrystal model close to the interface. After fully relaxing the dislocated configuration, the equilibrium configuration containing a lattice dislocation is obtained, and this acts as the initial configuration. Second, the final configuration containing a dislocation loop in the adjacent crystal is created and relaxed at a certain applied stress. Finally, molecular dynamics is performed for the final configuration while reducing the applied stress to obtain a series of configurations between the initial and the final configurations. The change of potential energy is then calculated as a function of the size of the dislocation loop [75]. Fig. 7 shows three snapshots: initial nucleus, one intermediate configuration, and the final configuration. Corresponding to the different states, the change of potential energy of the bicrystal is calculated. The critical stress corresponding to the stabilized dislocation loop is determined from the first derivative of the potential energy with respect to the dislocation loop and found to be on the order of 0.8 GPa in terms of the resolved shear stress. Corresponding to different interfaces, this critical stress will increase as the decrease of interface shear resistance due to the wider spreading of dislocation core within the weaker interface [71,75]. In addition, the critical stress varies corresponding to slip systems with parallel or non-parallel traces, higher for non-parallel traces because of the smaller size of the stabilized loop [75].
Fig. 6. Schematic illustration of slip transmission process for composites with low shear strength interfaces. (a) Interface shear, accompanied with core spreading of a lattice dislocation within the interface plane, (b) a dislocation loop nucleates in association with the shrinking of the spread core, and finally, (c) dislocation escapes from the interface by overcoming the attraction due to the interface shear.
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Fig. 8. The dissociation of a mixed dislocation b1 aNb ½1 1 1 in the Cu/Nb interface. 2 The schematic plots show (a) a mixed dislocation b1 situated at the interface, and (b) the dissociation into an interfacial dislocation b2 a3Cu ½1 1 1 and an interfacial aNb aCu discontinuity b3 2 ½1 1 1 3 ½1 1 1 left near the trace of the Nb slip; (c) atomic structures after the dissociation of the dislocation in the interface. The red arrows indicate the diffusion of vacancies and the black arrows indicate the counter diffusion of Cu atoms. The yellow lines outline Cu (1 1 1) planes, and the black lines Nb (1 1 0) planes. (d) Atomic structures of the interfacial Cu plane. The dashed lines indicate locations of the two dislocations. The regions with low areal density of Cu atoms are outlined in gray atoms and the gray shadow, and the regions with the areal density same as Cu (1 1 1) in yellow atoms (from Ref. [44]).
Fig. 7. Atomistic simulation of slip transmission across a Cu/Nb interface from Cu to Nb when the two slip planes have parallel traces with interface plane. The top crystal is Cu and the lower Nb. A full dislocation with Burgers vector of 1/2h1 1 0i on {1 1 1} plane initially approaches the Cu/Nb interface, resulting in (a) the nucleation of a dislocation loop in Nb with Burgers vector of 1/2h1 1 1i on {1 1 0} plane, (b) the propagation of the dislocation loop, and (c) the final stabilized loop at tensile stress 2.0 GPa. Atoms are colored according to the magnitude of the projected disregistry on each slip plane. The blue atoms in Nb outline the dislocation loop in Nb.
3.5. Dislocation climb in interfaces The glide dislocations entrapped, with spread core, in the interface can move along the interface plane through glide and climb. Atomistic simulations have shown that the low shear resistance of Cu–Nb interface enables the in-plane component of dislocations to glide with a high mobility [44]. The out-of-plane component of dislocations can climb in the interface through absorption and emission of vacancies, as shown in Fig. 8. Consequently, a patch of extra Cu (1 1 1) forms at the end of the half plane of Nb (1 1 0). The rate of climb depends on the concentration and diffusivity of vacancies. The equilibrium vacancy concentration is determined to be around 5.8% using atomistic simulations, which is far higher than that in perfect crystals of Cu and Nb [39,44]. The diffusivity of
a vacancy depends on the formation energy and the migration energy of a vacancy in the interfacial plane. This diffusivity is very high. One reason is that the equilibrium vacancy concentration is quite high, since the defect energy with respect to removal or insertion of atoms in the interfacial Cu layer is only 0.12 eV/atoms at 300 K, which is one order of magnitude smaller than the formation energies of a vacancy (1.26 eV) and an interstitial (3.24 eV) in Cu. The second reason is that the kinetic barriers of ‘‘vacancy” diffusion in interfaces are low. Using the Nudged Elastic Band method [74], the kinetic barriers associated with the migration of one Cu ‘‘vacancy” in the interface plane are computed and in the range of 0.03–0.10 eV [44]. This small kinetic barrier is comparable with a Cu adatom diffusing on a flat Cu(111) surface [76–78]. Finally, the dislocation acts as a perfect source for vacancies, and the interface acts as a sink for vacancies [79]. At room temperature, the climb velocity is around 0.5 m/s by solving Fick’s second law with respect to a moving source of vacancies, about five orders of magnitude larger than the climb velocity in a Cu crystal. Dislocation climb could play a crucial role during mechanical deformation. First, this climb enables slip transmission: glide dislocations can reassemble the spread cores in the interface plane via climb, and dislocation debris scattered in interfaces can reassemble into lattice glide dislocations, facilitating slip transmission. Secondly, reactions between interfacial dislocations assisted by climb could lead to
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pile-up. This implies that unless a long pile-up can form, the slip transmission across Cu–Nb interfaces will not be assisted by the mechanical advantage of the pile-up. Thus, crossing of single dislocations across the interface in nanolayered materials is the critical unit process and may be assisted by climb. Finally, dislocation climb enables the interfacial plane to be in an equilibrium state with respect to concentrations of point defects because of absorption and/or emission of vacancies at dislocation cores. Thus, these interfaces act as ideal sinks for vacancies and interstitials, allowing Cu–Nb interfaces to persist in a damage-free steady state even when driven out of equilibrium by intense particle radiation.
4. Experimental validations
Fig. 9. A cross-section HRTEM image taken close to an interface (marked by the dotted line) showing a Burgers circuit in the Cu layer defining a dislocation with b||h2 1 1i. (From Ref. [81]).
annihilation of dislocation content (recovery). Thirdly, discrete pile-ups can be absorbed in the interface plane, assisted by climb in the interface, thereby blunting the stress concentration of the
Experimental measurements performed on sputter deposited fcc–bcc multilayers have shown results consistent with the predictions of the atomistic simulations. With regard to interface crystallography and structure, X-ray diffraction [80] and transmission electron microscopy [81] show the KS orientation relationship and interface dislocations. Fig. 9 is a high-resolution cross-section TEM image of a Cu–Nb multilayer showing the KS orientation relationship and a dislocation with Burgers vector in the interface plane. The sub-Å scale atomic displacements that appear as vortices in the interface plane in the atomistic simulations [41,64] are not resolved in the cross-section TEM images such as Fig. 9. Hardness and pillar compression tests in a nanoindenter were used to measure flow stresses of Cu–Nb and Al–Nb multilayers with layer thickness on the order of 5 nm or less. For Cu–Nb multilayers, tests with compression axis at 90° to the interface, measured flow stress of 2.3 GPa [46] suggesting a high barrier stress
Fig. 10. HR-TEM snapshots of the dislocation annihilation process at an interface captured at different instants during in situ nanoindentation. (a) At 0 s the two dislocations were separated by 2.4 nm. (b) After 2 s the separation distance was reduced to 1.7 nm via dislocations climb. (c) At 2.5 s the two dislocations annihilated each other, and dislocation b2 inside the Nb glided towards the interface. (d) At 3.5 s the interface became nearly perfect after complete annihilation of the dislocations (from Ref. [85]).
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for slip transmission. Compression tests in Cu–Nb with interfaces oriented at 45° to the loading axis resulted in shear along interfaces at relatively low resolved shear stress of 550–750 MPa [82]. These experimental measurements validate the atomistic simulations that Cu–Nb interfaces are weak in shear but have a high interface barrier stress to single dislocation transmission. Indentation tests normal to interfaces in Al–Nb resulted in a much lower flow stress than in Cu–Nb [83]. Note that Al–Nb has a negative heat of mixing as compared to Cu–Nb which has a positive heat of mixing. These results are also consistent with atomistic models that predict a decrease in interface barrier stress as heat of mixing becomes less positive (more negative). Compression tests also show strain softening that may be due to shear localization across interfaces (for interfaces that are strong in shear, Ref. [84]) or due to rotation of interface plane towards the plane of maximum shear (for interfaces that are weak in shear, Ref. [82]). Finally, the atomistic modeling prediction of climb at interfaces was verified by in situ indentation in a TEM in Al–Nb multilayers. The edge components of two glide dislocations trapped at interfaces were observed to annihilate via climb along the interface. This is shown in Fig. 10 [85]. Climb along interfaces can lead to recovery of stored dislocation content during plastic deformation. 5. Summary The current work on multilayers shows that the atomic structure of the interface may lead to specific properties of the interface (such as low shear strength) that control the interaction of glide dislocations with the interface. The interface barrier stress for slip transmission, in the absence of a dislocation pile-up, depends on how strongly a glide dislocation interacts (i.e., pinning stress) with the interface, which can be correlated with the defect structure of the interface, that in turn depends on factors such as crystallographic orientation relationships, lattice misfit and heats of mixing. In future, this framework can be extended to other kinds of interfaces besides fcc–bcc to interpret the maximum strength that may be achieved in a nanoscale multilayer. In addition to atomistic modeling, other methods such as discrete dislocation dynamics, and crystal plasticity modeling are being used and this will lead to multi-scale models that predict the complete stress–strain curve using atomic-scale deformation mechanisms at interfaces [39,86–92]. The field of research in metallic multilayers continues to grow with exploration of other properties such as fatigue and fracture [93–98], creep [99,100], and other systems such as metal–ceramic [100–105], bcc–hcp [106], and bulk fabrication of multilayers [107,108]. Acknowledgements The authors acknowledge support from DOE, Office of Science, Office of Basic Energy Sciences, and fruitful collaborations with R.G. Hoagland, J.P. Hirth, J.D. Embury, M.J. Demkowicz, N.A. Mara, N. Li, D. Bhattacharyya, X. Zhang, X.Y. Liu, Q. Wei, O. Anderoglu, Y.C. Wang that resulted in the published literature that is reviewed here. References [1] Hirth JP. The influence of grain boundaries on mechanical properties. Metall Trans 1972;3:3047–67. [2] Lee TC, Robertson IM, Birnbaum HK. TEM in situ deformation study of the interaction of lattice dislocations with grain boundaries in metals. Philos Mag A 1990;62:131–53. [3] Lee TC, Robertson IM, Birnbaum HK. An in Situ transmission electron microscope deformation study of the slip transfer mechanisms in metals. Metal Trans A 1990;21A:2437–47. [4] Shen Z, Wagoner RH, Clark WAT. Dislocation and grain boundary interactions in metals. Acta Metall 1988;36:3231–42.
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