A. Strain hardening and strain rate sensitivity of Au microspheres

It is common when characterizing the isothermal plastic deformation of bulk metal samples to express the equivalent flow stress as a power–law function of the equivalent strain and strain rate. Here we apply this approach to assess the dependence of ${\bar \sigma _{{\rm{VM}}}}$ upon ${\bar \varepsilon _{{\rm{VM}}}}$ and $\dot \varepsilon$ for the Au microspheres. Logarithmic plots of ${\bar \sigma _{{\rm{VM}}}}$ versus ${\bar \varepsilon _{{\rm{VM}}}}$ and $\dot \varepsilon$ yield slopes of magnitude n and m respectively. Figure 4(a) shows n plotted versus $\dot \varepsilon$ . The larger, 6.0 and 3.0 µm diameter, Au spheres displayed a strain hardening exponent of n = 0.40–0.50 and n showed clear strain-rate dependence. These values are similar in magnitude to those reported for bulk fcc metals such as Au. ^{Reference Kelly and Nicholson23} The smaller, 0.8 and 1.0 µm diameter, microspheres displayed lower strain hardening exponent of n = 0.27–0.33 and n was significantly less dependent upon strain-rate. Figure 4(b) shows the strain rate exponent m plotted against $\dot \varepsilon$ . The 6.0 µm diameter spheres displayed decreasing m, from m = 0.08 to 0.03, as ${\bar \varepsilon _{{\rm{VM}}}}$ increased from 0.05 to 0.20. A strain rate exponent of m = 0.03 is typical of bulk fcc metals. ^{Reference Dao, Lu, Shen and Suresh24} The smaller 0.8 and 1.0 µm diameter spheres displayed significantly higher m values ranging between m = 0.2 to 0.3 over the same strain range. Our observation that m is significantly increased when the diameter of the Au spheres is less than about 1 μm is similar to what was reported from uniaxial strain rate change tests performed on nanocrystalline Ni ^{Reference Wang, Hamza and Ma25} and uniaxial compression tests performed on Cu nanopillars. ^{Reference Jennings, Li and Greer26}

FIG. 4.

(a) Strain hardening exponent n, (b) Strain rate sensitivity m, and (c) Apparent activation volume V* plotted as a function of average centerline Von-Mises equivalent plastic strain ${\bar \varepsilon _{{\rm{VM}}}}$ , for all diameters of Au microspheres tested in this study.

The sample size dependence of the strain- and strain rate-sensitivity of the flow stress of Au microspheres described above point to a clear sample size dependence in the operative dislocation nucleation/glide mechanism. The deformation of small free-standing ductile metal volumes, such as our 0.8 µm diameter Au spheres, will likely be affected by the process of dislocation nucleation at the free surface and glide of these dislocations through the small sample with little chance of interaction with other dislocations. Atomistic simulations have predicted that dislocation nucleation from free surfaces has a very low associated characteristic activation volume of V* = 1–10b ³. ^{Reference Zhu, Li, Samanta, Leach and Gall27} This will result in a very low strain rate sensitivity since the parameters V* and m are related as ^{Reference Wang, Hamza and Ma28–Reference Zhu, Li, Ogata and Yip30}

where k is the Boltzmann constant and T is the absolute temperature.

Plastic deformation of the large, 3 or 6 μm diameter, Au spheres will likely involve more dislocation nucleation from inhomogenities within the interior of the sample and obstacle-limited glide of the dislocations through the sample. The characteristic activation volume for such a process is significantly higher, typically between V* = 100–1000b ³. ^{Reference Rodriguez31}

Substituting the values of m and ${\bar \sigma _{{\rm{VM}}}}$ from our study into Eq. (1) allows us to determine an apparent activation volume V* which is characteristic of the deformation process in the Au microspheres. Figure 4(c) indicates the calculated V* as a function of ${\bar \varepsilon _{{\rm{VM}}}}$ . In this figure V* is normalized with respect to b ³, where the Burgers vector of Au is taken as b = 0.289 nm. ^{Reference Nix and Lee32,Reference Bhakhri and Klassen33} For the smaller, 0.8 and 1.0 µm diameter, spheres V* remains at 4–6b ³ and is independent of strain while for the larger, 3.0 and 6.0 µm, diameter spheres V* increased from 12 to 42b ³ with increasing strain. The low V* values, 4–6b ³, for the smaller diameter spheres is similar to previously reported estimates for fcc metals by Mook and Gerberich, ^{Reference Mook, Lund, Leighton and Gerberich19} and is characteristic of dislocation surface nucleation events. These findings are also consistent with frequent discrete load jumps that were observed in the F–h curves of the 0.8 and 1.0 µm diameter spheres [Fig. 3(a)]. The larger, 3.0 and 6.0 µm diameter, spheres displayed smoother F–h curves [Fig. 3(b)] and displayed larger values of V* and its tendency to increase with increasing strain is typical of the operation of a dislocation–obstacle interaction limited deformation mechanism which is common in bulk ductile metals. ^{Reference Kocks, Argon and Ashby34}

B. Initiation of plastic deformation

The sample size- and strain rate-dependence of the initiation of plastic deformation within the Au microspheres was studied to provide a more detailed assessment of the operative rate-dependent dislocation nucleation process. The initial yield force F* at which the measured F–h response deviated from the ideal elastic response was recorded for microspheres of diameter from 0.8 to 6.0 μm that were compressed at several loading rates (Fig. 5 and Table I). The ideal elastic F–h response was expressed by the following Hertz equation describing the elastic contact of two spheres, one of radius R and the other of infinite radius ^{Reference Kogut and Etsion35}

where the reduced elastic modulus E _r is given as

FIG. 5.

Representative F–h curves of the Au microspheres compressed at three loading rates, (a) 0.8 and (b) 6.0 µm diameter Au microspheres. For each test the initial portion of the F–h curve is well matched by the F–h elastic response predicted by Eq. (2) (red lines). The load corresponding to the first deviation from the F–h elastic response indicates F* and is indicated by the black arrows. For the sake of clarity, the displacement data for each loading rate is offset by 10 nm.

Our calculations used isotropic elastic properties of E ₁ = 79 GPa and ν₁ = 0.42 for the Au microsphere and E ₂ = 1141 GPa and ν₂ = 0.07 ^{Reference Maharaj and Bhushan36} for the diamond flat-punch.

The average initial yield equivalent shear stress τ* was then calculated as

where A _f is the area of the {111} facet upon which the flat-punch contacted the Au sphere and the factor 0.5 represents the characteristic Schmid factor of a ductile fcc metal. ^{Reference Gall, Diao and Dunn20,Reference Heyer, Brinckmann, Pfetzing-Micklich and Eggeler37} Values of F*, A _f, and τ* are given in Table I. It should be noted that finite element elastic simulation indicates that the use of the Hertz equation to determine F* results in an uncertainty of about 10% due to the geometrical differences between the ideal flat surface/sphere point contact of the Hertz model and the flat surface/facetted-sphere contact of the microspheres of this study. This uncertainty is well within the sample-to-sample variability displayed by the data in Table I.

The average equivalent shear strain rate at the point of initial yielding was obtained from Eq. (2) as

where $\dot F = {\rm{d}}F{\rm{/d}}t$ is evaluated at F*.

Plots of τ*/G versus ${\dot \gamma ^ * }$ (Fig. 6) indicate that the shear stress required to initiate initial plastic deformation increases with increasing strain rate and decreasing sphere diameter. In these plots the elastic shear modulus of G _Au = 23.6 GPa was used. ^{Reference Kamimura, Edagawa and Takeuchi38} Included on the graph is the estimated theoretical shear strength which is normally estimated to be between G/30 and G/10 for fcc metals such as Au. ^{Reference Hull and Bacon39–Reference Bei, Shim, Pharr and George41} It is notable that τ*/G of the smaller, 0.8 and 1.0 µm diameter, spheres approaches the estimated theoretical strength limit for Au.

FIG. 6.

Normalized initiation of plastic deformation shear stress τ*/G versus strain rate of Au microspheres. Included in the plot are dashed lines indicating common estimates of the ideal theoretical shear strength (τ_ideal = G/30 to G/10) of a metal.

The strain-rate dependence of τ* suggests that the deformation mechanism controlling these events involves a time-dependent process. Previous molecular dynamic simulation of deforming Au micro-particles demonstrated that first dislocation nucleation events occur at periphery of the contact region between the facetted sphere and the compression platen. ^{Reference Mordehai, Lee, Backes, Srolovitz, Nix and Rabkin15} A plausible theory to explain the observed size dependence of τ* is that the platen/sphere contact area increases with increasing sphere diameter and thus the number of interfacial vacancies increases and the rate of diffusion of these vacancies controls the observed strain rate sensitivity of τ*. This theory is supported by the findings of Salehinia et al. ^{Reference Salehinia, Perez and Bahr42} which demonstrate that the presence of vacancies beneath the indenter during nano-indentation of Ni strongly affects F*.

The hypothesis that the strain rate dependence of τ* is linked to vacancy diffusion enhanced dislocation nucleation from the platen/sphere contact area can be tested by calculating the apparent activation energy of the process and comparing its value to that characteristic of vacancy diffusion. Application of the Orowan equation allows us to describe the shear strain rate, $\dot \gamma$ , as a function of the mobile dislocation density ρ_m, Burgers vector b, and the average dislocation glide velocity $\bar v$ as ^{Reference Orowan43} ;

The dislocation velocity can be expressed in terms of a stress- and temperature-activated atomic jump frequency as ^{Reference Bhakhri, Wang, Ur-rehman, Ciurea, Giuliani and Vandeperre44} :

where ν₀ is a characteristic athermal atomic jump frequency, τ_p is the athermal lattice resistance to dislocation glide, τ is the applied shear stress, and the remaining terms are previously defined. Combining Eqs. (6) and (7) with the assumptions that τV* ≫ kT and the apparent activation volume V* can be treated as constant, the following expression for τ can be arrived at

This relationship forms the basis of conventional thermal-rate-dependent analyses of mechanical flow stress of materials. ^{Reference Bhakhri, Wang, Ur-rehman, Ciurea, Giuliani and Vandeperre44–Reference Pirouz, Demenet and Hong46} In this study we use this equation to assess the ${\tau ^ * } - {\dot \gamma ^ * }$ response shown in Fig. 6. The linear dependence of the data when plotted on a semilogarithmic plot indicates a constant activation volume associated with the dislocation nucleation process occurring at τ*. Assuming that τ* occurs by the nucleation of a dislocation in an essentially dislocation free microsphere, ρ_m will be very small and the third term on the right side of Eq. (8) is much less than the second term. In such a case, the slope of the ${\tau ^ * } - {\dot \gamma ^ * }$ plot represents kT/V*. This analysis method was used to determine the activation volume V* associated with the initial dislocation nucleation process (Table I and Fig. 7).

FIG. 7.

The change of apparent activation volume, V* during the initiation of plastic deformation of different diameter Au microspheres.

The dependence of V* upon microsphere diameter is shown in Fig. 7. The calculated V* of 0.8 and 1.0 µm diameter Au spheres is about 0.4b ³ and 0.7b ³ respectively. These V* values are similar in magnitude to those obtained from a host of small specimen tests on systems where heterogeneous dislocation nucleation events from pre-existing defects such as vacancy clusters, impurity atoms, or free surfaces control the deformation process. ^{Reference Somekawa and Schuh29,Reference Wo, Zuo and Ngan47} The values of V* are, however, considerably smaller in magnitude than those calculated for the bulk deformation Au microspheres (described in the previous section).

The apparent activation energy of the initial dislocation nucleation process is expressed as Q* = τ_{0 K} V*. We calculated the athermal stress, τ_{0 K} of Au using room and high temperature nano-indentation hardness (<1 μm depth) measurement data obtained from literature. ^{Reference Smith and Zheng48,Reference Vieregge49} We used our experimentally obtained V* values along with the calculated value for the athermal stress of Au, τ_{0 K} = 256 MPa to calculate values of Q* (Table I). Our calculated Q* values increased from 0.02 to 0.16 eV with increasing microsphere diameter from 0.8 to 6.0 µm (Table I). The values of V* and Q* obtained from this study are of the same order of magnitude as those for atomic surface self-diffusion as reported by Liu et al. ^{Reference Liu, Cohen, Adams and Voter50} and are also consistent with the findings from a recent study by Chen et al. ^{Reference Chen, He, Shin, Richter and Gianola51,Reference Li52} on the kinetics of incipient plastic deformation in Pd whiskers.