Emergence of strain-rate sensitivity in Cu nanopillars: Transition from dislocation multiplication to dislocation nucleation

Abstract

We demonstrate strain-rate sensitivity emerging in single-crystalline Cu nanopillars with diameters ranging from 75 up to 500 nm through uniaxial deformation experiments performed at different constant strain rates. In the range of pillar diameters and strain rates tested, we find that the size dependence of the pillar strength deviates from the ubiquitously observed power law to a relatively size-independent flow strength, markedly below the predicted theoretical strength for strain rates slower than 10⁻¹ s⁻¹. We find this transition diameter, D_t, to be a function of strain rate, where faster strain rates shift the transition diameter to smaller pillar diameters: D_t ∼ 150 nm at 10⁻³ s⁻¹ and D_t ∼ ⩽75 nm at 10⁻¹ s⁻¹. We compute the activation volumes, Ω, as a function of pillar diameter at each strain rate and find that for pillar diameters below D_t, the activation volumes are relatively small, Ω < 10b³. This range agrees favorably with atomistic simulations for dislocation nucleation from a free surface. We postulate a plasticity mechanism transition from dislocation multiplication via the operation of truncated dislocation sources, also referred to as single-arm sources, in pillars with diameters greater than D_t to dislocation nucleation from the surface in the smaller samples.

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: $\sigma \propto \mu b\sqrt{\rho }$ [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].