Original ArticlesDeformation mechanism maps for polycrystalline metallic multiplayers
Introduction
Metallic multilayers represent an ideal vehicle for the exploration of length scales in plasticity 1, 2. They also provide the opportunity to synthesize materials with controlled interfaces and structures for the production of materials close to the theoretical strength [1]. The influence of length scales has been explored by a number of authors building on the framework of the Hall-Petch and Orowan mechanisms of strengthening. Early work by Embury and Fisher [3] on drawn pearlite had shown that the Hall-Petch model for strengthening in single phase metals by grain refinement is also applicable to two-phase materials with interphase spacing taken as the obstacle distance. More recent studies, such as Embury-Hirth [4], Anderson et al. [2], Chu and Barnett [5] and Nix [6] have indicated that in nanometer-scale multilayers the mechanical behavior may be governed by single dislocation behavior (Orowan model of dislocation bowing between layers) rather than piling up of dislocations against interfaces. Another recent study by Masumura et al. [7] reveals that below a critical grain size in single-phase nanostructured materials, diffusion-based mechanisms such as Coble creep may be operative and may lead to softening with grain refinement. Often these models, for simplicity, are developed either for multilayers with single crystal constituent layers or for single-phase fine-grain materials. Understanding the mechanical behavior of polycrystalline multilayers poses an additional complexity since both layer thickness and in-plane grain size may influence the yield strength. Although the in-plane grain size may scale with the layer thickness, there is no universal relation that allows us to calculate grain size knowing only the layer thickness and vice-versa. The relation between these parameters is usually determined through detailed microstructural characterization. Hence for a given polycrystalline metallic multilayer, how does one obtain insight on the operative deformation mechanism for different combinations of layer thickness and grain size values?
In the present investigation, we present a simple analysis that allows us to obtain limiting values of microstructural scales at which these different mechanisms operate. We present the results in the form of two-dimensional maps of layer thickness and grain size ranges over which different deformation mechanisms operate. These maps are intended to be guidelines for interpreting the scale-dependent strengthening or softening mechanisms in multilayers, in the same manner as Ashby’s deformation mechanism maps for temperature and stress dependent deformation behavior of metals [8]. An attempt to extend Ashby’s deformation mechanism maps to Al thin films by Frost [9] revealed that the predicted strain rates were several orders of magnitude higher than the observed rates due to the higher flow stresses of thin films. Hence, more work is needed to incorporate the fundamental differences in the deformation behavior of thin films and bulk polycrystals to map the mechanisms as a function of stress, temperature and microstructural scale. In this article we only consider changes in deformation mechanisms with decreasing length scales at constant temperature and strain rates.
In this section we briefly describe the approach used to obtain limits to pile-up behavior in polycrystalline metallic multilayers. Consider a multilayer structure of two metals A and B. Let the layer thickness (h) of the two layers be equal and the in-plane grain size of layer A be d, as shown schematically in Fig. 1. Let the initial state be a stress-free multilayer with an array of misfit-compensating edge dislocations at the interface to compensate differences in lattice parameters. We further assume that layer A has significantly lower yield stress than B. If we let additional dislocations be generated in layer A, then added dislocations build up in the interface and transfer load to layer B until the stress is sufficient for B to begin to flow, as discussed by Embury and Hirth [4]. At this stage the structure consists of additional interface dislocations with spacing λ and an effective screw dislocation pile-up at a grain boundary in layer A. A comparison of the stress concentrations in the center of a grain in layer A associated with the added interface dislocations and that due to the screw pile-ups gives an indication of which controls further flow. We thus link the number of dislocations in the screw pile-up (n) to d and h by equating the back stresses of the two dislocation arrays shown in Fig. 1. The back stress of the interface edge array at the center of the grain is given by the following equation: where G is shear modulus, υ is Poisson’s ratio and b is Burgers vector. The back stress of the screw dislocation pile-up is as follows: By equating back stresses 1 and 2, we obtain, This equation gives the combination of d and h values for which the two back stresses are equal. The above analysis assumes the following: (i) eq. (1) is not significantly affected by the position of cores of edge dislocation array at the interface (e.g. whether the core is in layer A or layer B), and (ii) eq. (1) which is derived for an infinitely long array still applies, although the periodicity of the array may be disrupted by grain boundaries terminating at the A/B interface. Equation (3) can be plotted as d versus h for different values of n, if λ is known. No significant difference was noted by incorporating an h-dependence of λ in the above equations, especially for the saturation value of d with increasing h and hence, we only present results for constant λ that is given as b/ϵm. The results are shown in Fig. 2(a) and (b) for multilayers with 2.5% and 10.5% misfits (ϵm) respectively at n = 1 and 2. The results for 2.5% misfit may be compared with systems such as Cu-Cr where {110} bcc and {111} fcc planes define the misfit strain at the semi-coherent interface, and Cu-Ni where {100} fcc planes form the interface; while the results for 10.5% misfit may be compared with experimental data on Cu-Nb multilayers which have interface crystallography same as Cu-Cr.
Note from Fig. 2 that for a given misfit, all combinations of d and h values below the n = 1 locus correspond to length scales where pile-up behavior is not expected. With decreasing λ, the boundary between the pile-up and no pile-up region is shifted to lower values of d at a given h. The boundaries that mark the region where no more than 2 dislocations can be accommodated in a pile-up are also shown in Fig. 2. These will be used to separate the continuum pile-up and discrete pile-up regions as discussed later.
A similar situation exists for multilayers with d ≫ h (in the limit, this would correspond to single crystal layers), i.e., there will be a critical h below which pile-ups will not form (Fig. 3). This value of h, in the most simple case, can be obtained by considering a pile-up of edge dislocations at the interface and putting n = 1 in the equation relating shear stress, n and length of pile-up [10]: where τ is the applied shear stress acting on the slip plane, and can be approximated by where m is Schmid factor and σ is the normal stress in the layer. A lower bound for σ is the misfit stress σm. The misfit stress may be related to the strain, ϵm, as follows [11]: Combining , , and putting m−1 = 0.5, we obtain For Cu, ht ≈ 80 Å for misfit of 2.5% and ∼20 Å for misfit of 10.5% at n = 1. This would plot as vertical lines on Fig. 2 for different values of n. The plot is modified as shown in Fig. 4(a) for the plot of Fig. 2(a) and in Fig. 4(b) for Fig. 2(b) respectively, implying no pile-up to the left of the line for n = 1 and pile-up to the right. Similarly, for h ≫ d (Fig. 5) pile-ups will not form below a critical value of d. With a similar approach to that shown in , , , , it follows that dc is approximately equal to ht and will plot as a horizontal line in the map of d vs h . The value of dc is equal to the saturation value of d from eq. (3), within a few atomic spacings.
Thus, in Fig. 4, the boundary between pile-up region and no pile-up region may simply be obtained by the vertical line (eq. 7) and horizontal line (saturation value of d from eq. 3). Further, we note that a very small d or h, plasticity may not involve dislocation-based mechanisms. This is the microstructural scale below which softening is usually observed with decreasing grain size (or layer thickness) in nanocrystalline materials. This transition from hardening to softening with decreasing microstructural scale may be due to the onset of grain boundary sliding or Coble creep type deformation mode [7]. Alternately, deformation may involve shearing of atomic planes at stresses approaching the theoretical strength independent of microstructural scale. Each will give a slightly different value of the microstructural scale that marks the onset of non-dislocation type deformation modes. For the case of the Orowan stress approaching the theoretical strength of say G/10, we have Again, a value of either d or h below this value will indicate the onset of deformation mechanisms other than bulk slip within the layers.
Section snippets
Results and discussion
The concepts described above of dislocation-based pile-up or non pile-up mechanisms, and non-dislocation type deformation modes can be combined to give a “deformation mechanism map” for polycrystalline multilayers. An example is shown in Fig. 6 for multilayers with misfit of ∼2.5% (e.g. Cu-Ni, Cu-Cr). At larger misfits, the pile-up region would shift to lower values of d and h, scaling almost linearly with the misfits strain. A vertical line, labelled hc, indicates the approximate value of
Summary
We have presented an approach to map the deformation mechanisms in columnar-grain metallic multilayers on plots of in-plane grain size versus layer thickness. These maps can be constructed if the misfit strain, and the Burgers vector and Poisson’s ratio of the softer layer are known and serve as guidelines to the deformation mechanisms that may operate at varying length scales. While these simple maps are in good agreement with the experimental observations, the effects of other factors such as
Acknowledgements
The authors acknowledge discussions with Prof. M.F. Ashby, Dr. T.E. Mitchell and Dr. M. Nastasi. This research is funded by DOE-OBES.
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