Emergence of strain-rate sensitivity in Cu nanopillars: Transition from dislocation multiplication to dislocation nucleation
Introduction
A major current focus in the nanomechanical community is the investigation of single-crystalline strength at reduced dimensions through uniaxial deformation of cylindrical specimens, often referred to as micro- or nanopillars [1], [2], [3], [4], [5]. Remarkably, the results of all such experiments on face-centered cubic (fcc) crystals with non-zero initial dislocation densities indicate that their strengths depend on pillar diameter in a power-law fashion: σ ∝ D−n, where σ is the flow stress and D is the pillar diameter, with 0.5 ⩽ n ⩽ 1.0 [1], [2], [3], [4], [5]. Size-dependent strength is counter-intuitive, as crystalline strength in bulk is generally considered to be independent of sample size. In bulk metals, strength is proportional to the increasing-with-strain dislocation density via the Taylor relation: [6], whereby dislocations multiply via double-cross slip and by operation of pinned dislocation sources [7].
Recent modeling efforts have probed possible types of dislocation sources in pillars at a range of length scales from ∼1 nm up to several microns. For example, dislocation dynamics (DD) simulations performed on micron-sized pillars reveal that in these relatively large samples, single-arm, or spiral, dislocation sources generate stochastic stress–strain signatures and unambiguous size effects, in accordance with those observed experimentally [8], [9], [10], [11], [12], [13]. If the pillar dimensions are reduced by an order of magnitude, molecular dynamics (MD) simulations of nanowires then show the nucleation of partial dislocations from the surface of the wire [14], [15], [16]. An important consideration is that neither type of simulation has been able to accurately capture both mechanisms simultaneously: discrete DD simulations cannot accurately describe surface nucleation, whereas MD simulations are too computationally intensive to accurately describe the collective DD of large systems.
Major advances investigating in depth the crystalline plasticity in the sub-micron regime have also been enabled via in situ transmission electron microscopy (TEM) tensile tests. These experiments reveal two mechanisms for dislocation generation in small-scale crystals: (i) via spiral, or single-arm, sources (SASs), as is the case for ∼455 nm single-crystalline Al under tensile loading [17]; and (ii) via partial dislocation nucleation from the surface, or surface sources (SS), as revealed during uniaxial tension of ∼15 nm diameter Au nanowires by Zheng et al. [18], [19]. In the former, the dislocations are multiplied as they are generated from an already existing pinned source, while in the latter individual dislocations are nucleated stochastically, from a distribution of surface locations.
Although the precise nature of either type of source is being vigorously pursued, a general agreement exists that in micron-sized fcc pillars the dislocations multiply and form complex intertwined networks through the operation of SASs, whereas nanosized pillars are characterized by virtually non-existent dislocation multiplication or storage, and deform via dislocation nucleation at the surface (via SSs), glide and subsequent annihilation at the free surfaces. However, despite this general agreement, the possible coexistence and/or transition between these two mechanisms, as well as their strength, geometry, stability and thermal nature remain important open questions.
A previously unexplored route in nanopillar experiments is to probe the presence of a particular type of dislocation source by computing the activation volumes required for their operation. For example, atomistic simulations have predicted SSs to have an activation volume of ∼1–10b3 which would result in a significant thermal contribution to the source’s strength [20]. In contrast, a SAS, often represented as a truncated Frank–Read source (FRS), whose activation volume is relatively large, ∼100–1000b3, would make an almost negligible thermal contribution to its strength [7]. We hypothesize that this great difference in the activation volumes should manifest itself in vastly different strain-rate dependences between the two mechanisms, with SSs being more sensitive to strain rate than SASs [7], [20].
In this work, we present compressive behavior of single-crystalline Cu nanopillars with diameters between 75 and 500 nm, fabricated without the use of a focused ion beam (FIB) and deformed at different constant strain rates spanning over ∼4 orders of magnitude. The dependence of the flow stress on the strain rate is measured to determine the activation volumes for each strain rate and pillar diameter, which are then compared to theoretically determined activation volumes. Our experiments reveal a discontinuity in the measured strain-rate sensitivity and activation volume, suggesting a possible deformation mechanism transition from collective DD to surface dislocation nucleation [21].
Section snippets
Sample preparation and resulting microstructure
Single crystalline Cu nanopillars were fabricated by electroplating Cu from a Cu(II) sulfate bath under an applied voltage. A schematic of the electroplating process can be seen in Fig. 1A. The cathode, shown in a zoomed-in image in A, is a Si wafer upon which a ∼100 nm thick Au/Ti seed layer was deposited to serve as an electrical contact. The Au seed layer had a columnar grain structure with a 〈1 1 1〉 texture and an average grain width larger than all pillar diameters tested. A poly (methyl
Stress–strain behavior
Characteristic stress–strain curves are shown in Fig. 3: the four different stress–strain curves correspond to two distinct diameters, 125 and 250 nm, each of which is deformed under two different strain rates, 10−1 and 10−3 s−1. The stress plotted here is true stress following the analysis assuming a homogeneously deforming pillar while correcting for the elastic response of the pillar acting as an indenter into the substrate [3]. All of the stress–strain curves have a nearly elastic loading
Influence of sample fabrication on mechanical properties
To date, the vast majority of experiments on fcc nanopillars have been performed on samples fabricated by the use of the FIB [1]. Pillars fabricated via this methodology exhibit size-dependent strengths, and the mechanisms responsible for this strengthening have been a heavily debated topic [1]. One of the major points of contention has been the influence of the FIB fabrication technique on nanopillar strength [3], [10], [23], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38] as
Summary
We demonstrate a notable effect of both strain rate and sample size on the compressive strength of single-crystalline Cu nanostructures. By determining the activation volume for each pillar diameter and strain rate, we observe a clear transition in the slope of strength vs. strain rate for the two smallest diameters, 75 and 125 nm, while these slopes remain constant for larger pillars. Further, we report a deviation from the ubiquitously reported power-law size-dependent strength for the
Acknowledgements
The authors gratefully acknowledge Wei Cai, Chris Weinberger and Ting Zhu for useful discussions. Further, we would like to thank Wei Cai and Chris Weinberger for the SAS atomistic image. A.T.J., J.L. and J.R.G. gratefully acknowledge the financial support of the National Science Foundation through A.T.J.’s NSF Graduate Research Fellowship, J.L.’s NSF CMMI-0728069 and J.R.G.’s CAREER Grant DMR-0748267.
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