Cross-Split of Dislocations: An Athermal and Rapid Plasticity Mechanism

The pathways by which dislocations, line defects within the lattice structure, overcome microstructural obstacles represent a key aspect in understanding the main mechanisms that control mechanical properties of ductile crystalline materials. While edge dislocations were believed to change their glide plane only by a slow, non-conservative, thermally activated motion, we suggest the existence of a rapid conservative athermal mechanism, by which the arrested edge dislocations split into two other edge dislocations that glide on two different crystallographic planes. This discovered mechanism, for which we coined a term “cross-split of edge dislocations”, is a unique and collective phenomenon, which is triggered by an interaction with another same-sign pre-existing edge dislocation. This mechanism is demonstrated for faceted α-Fe nanoparticles under compression, in which we propose that cross-split of arrested edge dislocations is resulting in a strain burst. The cross-split mechanism provides an efficient pathway for edge dislocations to overcome planar obstacles.

Transmission Electron Microscopy (TEM) images of a cross-section of a nanoparticle (Fig. S1) shows that such a thin-layer of hydroxide is formed and its thickness varies in a range of a few nanometers. We estimate its thickness to be around 5 nm on average.
To check the strength of the hydroxide layer that forms during exposure of Fe particles to the air, several samples were annealed at 400 ˚C for 4 h in air in order to form fully oxidized nanoparticles (see Fig. S2A). For comparison, we show in Fig Representative force-displacement curves of the indentation of faceted and oxidized nanoparticles are given in Fig. S2C. The force on Fe nanoparticles is increasing with the displacement and a strain burst occurs at an indentation depth of about 10 nm. The resulting curve of the oxidized nanoparticle is monotonously increasing and do not exhibit displacements burst as in the case of Fe particles. Moreover, for the same applied load, the indenter penetrates much deeper into the oxidized nanoparticle than into the metallic Fe particle, indicating that Fe oxide is much softer than Fe. This fact implies that the hydroxide on the surface of Fe nanoparticles deforms much easier than the underlying crystal. Indeed, this difference in strength is reflected in the curve of the faceted Fe nanoparticle in Fig. S2C. The curves of the nonoxidized and oxidized Fe nanoparticles converge only at low penetration depths below 5 nm (the thickness of the oxide layer), which means that the mechanical response of the Fe particles is dominated by the surface hydroxide layer at the lower depths.
Detailed examination of the AFM images acquired after the indentation of Fe particles demonstrated that some particles did not yield plastically under applied load. The shallow residual imprints on the upper facet of such particles have a form of the triangle but the depth of the imprints is only about 6-7 nm instead of tens of nanometers, while the maximum penetration depth of the tip was a few tens of nanometers. We relate these shallow indents with Fe oxide that was easily pushed to the sides of the imprint during indentation. In contrast, the underlying Fe crystal completely recovered its initial form after unloading.
Based upon these observations we conclude that the oxide layer is much weaker than Fe and that the contribution of the oxide layer to the force measured during indentation of Fe non-oxidized nanoparticles is negligible. Thus, we can approximate the indentation force of the Fe nanoparticles to be elastic and to fit a Hertzian contact model to the indentation force F in Fig.   S2C ( ) = * √ 3 (S1) where E* is the reduced elastic modulus, d is the indentation depth and R is radius of the tip. For comparison, we fit the similar model to the oxidized nanoparticles. We note that the strength of the oxidized nanoparticle is low and the deformation is not elastic but plastic. The large plastic deformation of the thin oxide layer of the nanoparticles leads to the conclusion that the elastic part is negligible in the curves of the oxidized nanoparticle. Therefore, the reduced modulus is not an elastic modulus but an estimated effective plastic modulus that relates plastic stress and strain. This modulus is the one important for our discussion, since this is the relevant modulus when compressing the Fe nanoparticles with a flat punch.
The Hertzian fit to the results yields the following values for the Fe and oxidized nanoparticles: * √ = 926 ± 118 The reduced moduli are defined as where x stands for Fe or oxide. The diamond tip has an elastic modulus of Etip=1141 GPa and Poisson's ratio of tip=0.07 1 . We also consider that the elastic properties of Fe fulfill EFe=211 GPa and = 0.29 2 . Since the strength of the hydroxide layer is low, we postulateoxide to be equal to 0.5, which is a typical value for a non-compressible solids. Employing these elastic properties in equations (S2)-(S4) we find that R=23.3 nm and Eoxide=20.5 GPa.
The radius of the tip obtained in this analysis is in good agreement with experimental observations. Detailed AFM scans of the residual indents in Al single crystal was performed and the radius of curvature at the residual indent was in the range of 20-30 nm. More importantly, the effective elastic modulus of the oxide is an order of magnitude smaller than that of Fe.

II. The Pseudo-Elastic Deformation
At the early stages of the deformation, the deformation is expected to be concentrated in the hydroxide layer since it is much softer than the Fe part. Since the hydroxide layer is about 5 nm thick, his part of the deformation corresponds to a compressive strains of up to 1.5% in Fig. S3.
Indeed, the slope at this stage agrees with elastic modulus of 25 GPa, which is an order of magnitude lower than the elastic modulus of Fe. This value is in a very good agreement with the estimated values from the nanoindentation given in Sec. II in this Supplemental File.
For compressive strains larger than 1%, it is fairly reasonable to assume that the mechanical response is mainly from the Fe part, due to the large differences in the compliances of the hydroxide layer and the Fe nanoparticle. Nonetheless, even for strains above 1%, the slopes in the stress-strain curve are substantially smaller than the expected elastic response. To demonstrate the theoretical curve, we plot in Fig where h is the particle height and E is the elastic modulus of Fe in the <110> direction. The faceted shape of the nanoparticle is complex and for simplicity we assume the diameter is increasing linearly along the height,

(S6)
A(0) is the cross section of the top facet and A(h) is the cross section area of the bottom surface just above the substrate. In the faceted nanoparticle, the largest cross-section area is three times larger than the area of the top facet. Therefore, we shall consider that Combining equations (S5)-(S7) we find that The effective modulus E' is defined as the applied compressive stress F/A(0) divided by the engineering strain /ℎ and it satisfys the relation From equations (S7) and (S8) we find that In this pseudo-elastic deformation, isolated dislocation nucleation events will not be visible in the experimental stress-strain curves, since the nucleation itself does not release a substantial amount of elastic energy. This is clearly demonstrated in MD simulation, where the load-control interpretation of the results leads to small load bursts (the upper envelope in Fig. S4), followed by an increase in the compressive stresses. However, each nucleation will contribute to total strain and will decrease the effective slope of the stress-strain curve. For instance, the average slope of the stress-strain curves in the MD simulations presented in Fig. S4, once dislocations are being nucleated into the pile-up, decreases by a factor of 2.4 to 164 GPa.
The MD simulations suggest that two independent pile-ups are formed from the vertices along where =2+. From equation (S10) and (S12) we find that The total compressive strain  is the sum of the elastic and pseudo-elastic strains, e=/E and p-e, respectively. Therefore, the slope of the stress-strain do not correspond to the elastic modulus E but to an effective value Eeff=/ that satisfies Let us consider the following values: E=397 GPa, µ=82 GPa, =0.291, and m=0.47. We recall that E depends on the shape of the nanoparticle, while the values of shear modulus and Poison's ratio are material properties. Therefore we consider the values as independent and we do not relate between them. The effective elastic modulus is then Eeff=42 GPa. This value is indeed consistent with a reduction in the measured slope in the stress-strain curves, as was found experimentally.
However, despite the reduced predicted slope, the calculated effective modulus is not in accordance with the MD simulations in Fig. S4. We attribute it to the accuracy of equation (S11) in small dimensions. In nanoparticle heights of a few tens of nanometers, only a few dislocations are nucleated into the pile-up and the continuous dislocation density function given in equation (S11) overestimates number of dislocations. For instance, equation (S12) suggests that in order to add a dislocation into the pile-up, the compressive stress should increase by If we consider the nanoparticle in the MD simulations, which is 20.8 nm high, we find that  = 0.46 GPa, while in the MD an increase of 1.9 GPa in compressive stress was needed to nucleate new dislocations into the pile-up.
To better estimate the pseudo-elastic deformation for small number of dislocation, we suggest that the expression for N given by equation (3S) describes correctly its dependence on material properties and external stress, but with different value for . For the compressive stress to increase at a rate of 1.9 GPa per dislocation nucleation in a 20.8 nm high nanoparticle,  should be equal to 0.89 (as opposed to 2+). With this value of  we find that the effective elastic modulus is Eeff=163 GPa, which is in excellent agreement with the effective slope during the pseudo-elastic deformation in the MD simulations.
Moreover, for a 350 nm high nanoparticles, the model predict that the stress increments between dislocation nucleation into the pile-up is =0.16 GPa. In the range between a strain of 1% and the strain burst, where we suggest that hydroxide layer contribution is negligible, the compressive stress increases by 2.  Fig. 6  This allows room for the next dislocation in the pile-up to glide forward and to become the front dislocation.