2023 Volume 64 Issue 8 Pages 1952-1958
In order to understand the influence of dislocation substructures on the size-dependent strength (smaller is stronger) of micron-sized metals, we have fabricated single-crystal cylindrical micropillars with various diameters approximately ranging from 1 to 10 µm, which were prepared on the surface of the fully annealed sample and the subsequently cold-rolled samples of high-purity aluminum (Al). The annealed micropillars exhibited a size dependence of the resolved shear stress required for slip. The shear stress (τi) normalized by shear modulus (G) and the specimen diameter (d) normalized by Burgers vector (b) followed the correlation of τi/G = 0.33(d/b)−0.63. The size-dependent strength was reduced by cold-rolling, resulting in lower power-law exponents (0.26∼0.31) for the correlation in the cold-rolled specimens. The fine dislocation substructures introduced by the cold-rolling could be associated with the reduced size-dependent strength, which can be rationalized using the stochastic model of the dislocation source length in an assumption of homogenously distributed dislocations existing in the experimental micropillars. The inhomogeneous dislocation substructure with various dislocation cell sizes would contribute to a variation in the measured strength depending on the location, likely due to the probability of exiting the dislocation cell walls (local variation in dislocation density) in the micropillars fabricated on the cold-rolled Al samples.
Fig. 9 (a) Experimental resolved shear stress required for initial slip (τi) as a function of the micropillar diameter (d), for the 15% cold-rolled micropillars, compared with the calculated values using different dislocation densities (ρ). (b) Schematic illustrations showing single-crystal micropillars with different diameters (d) fabricated on the grains with the dislocation substructure.
Micromechanical testing (e.g., micropillar compression test, micro-tensile test, and micro-bending test) has opened an opportunity to explore the mechanical response of materials at the scale of micron or submicron meters. In particular, the micropillar compression test1,2) is a promising technique to fundamentally investigate the mechanical response of single crystals. Extensive studies have revealed a trend of “smaller is stronger” for cylindrical single-crystal micropillars of pure metals3–10) and intermetallic compounds.11–13) It can be generally described with the following power-law relationship (with an exponent of m and a constant of A) between the resolved shear stress (τ) and specimen size (d: diameter of cylindrical specimens):14)
| \begin{equation} \frac{\tau}{G} = A\left(\frac{d}{b}\right)^{-m} \end{equation} | (1) |
In an aspect of the application of micropillar compression tests to commercial Al and its alloys, it is necessary to understand the size-dependent strength of micro-sized specimens. The commercial-purity Al and wrought Al alloy series are often formed into rod or sheet products by plastic-forming processes (e.g., extrusion, forging, and rolling). Such industrial processes would introduce high-density dislocations into the Al alloy products. It is therefore required to fundamentally understand the effect of initial dislocations on the size-dependent strength of micro-sized Al crystals. In addition, the micropillar compression test enables measuring the compression response at a wide range of initial strain rates to address the strain-rate sensitivity of flow stress for single crystals in conventional metal products.15–20) The role of dislocation substructure in the measured strength is necessary to understand the reliability of the strain-rate sensitive strength measured by micropillar compression tests.
It is well-known that the annihilation and rearrangement of dislocations often occur in the plastically deformed pure Al at ambient temperature and appear more pronounced in higher purity levels.21) The static or dynamic restoration process could lead to the development of dislocation substructures (e.g., dislocation cells21)), resulting in the local variation in dislocation density inside the plastically deformed pure Al. The previous analyses were carried out assuming single crystals containing homogenously distributed dislocations,5) whereby the effect of heterogeneity of dislocation density (which is associated with the dislocation substructure) on the size-dependent strength of micro-sized pure Al still remains unclear. In the present study, we used a fully annealed high-purity Al (99.99% purity) and those subsequently cold-rolled in different reductions. The compression response of cylindrical micropillars with various diameters approximately ranging from 1 to 10 µm was investigated to address the effect of cold-rolling reduction on the size-dependent strength of high-purity Al single-crystals. Dislocation substructures were observed in the experimental samples. Based on these results, the effect of dislocation substructures on size-dependence strength in high-purity Al was discussed in terms of the stochastic model of the dislocation source length.22–24)
High-purity Al sample with a purity level of 99.99% was used in the present study. The experimental high-purity Al was annealed at 300°C for 3.6 ks to obtain a fully recrystallized microstructure. The fully annealed sample was cold-rolled to different reductions of 15 and 30%. These samples were mechanically polished and electropolished using a solution of perchloric acid and ethyl alcohol. Their microstructures were examined with a field-emission type scanning electron microscope (FE-SE: JEOL JSM-7001FA). The crystallographic orientations were identified by electron back-scattered diffraction (EBSD) analysis. To observe the dislocation substructures in the fully annealed and cold-rolled samples, thin disc samples were prepared by twin-jet electropolishing using a solution with perchloric acid and ethyl alcohol and then observed using a transmission electron microscope (TEM: JEOL JEM-2100 Plus) operating at 200 kV. The hardness (HV) was measured using a Vickers indenter with a constant load of 9.8 N at ambient temperature. More than 7 indentation tests were performed on randomly selected areas of each sample.
Cylindrical micropillars with various diameters (1∼10 µm) were fabricated on the electropolished surface of the high-purity samples by the focused ion beam (FIB: JEOL JEM-932) system operated at 30 kV. The aspect ratio (height/diameter of cylindrical micropillars) approximately ranged between 3 and 6. The loading axis directions (measured by prior EBSD analysis) were set parallel to the high-index orientations favorable for only a single slip system activated in uniaxial compression. Compression tests were carried out in a load-controlling mode using a SHIMADZU DUH-211S nanoindenter equipped with a 20 µm flat-punch diamond tip at ambient temperature. For setting an initial strain rate of all specimens to approximately 10−3/s, various loading rates, ranging from 1 × 10−3 to 10 × 10−3 mN/s, were used. The macroscopic slip traces of deformed micropillars were observed by FE-SEM to determine operative slip direction and plane.
Figure 1 presents the EBSD orientation distribution maps for the annealed sample and cold-rolled samples, together with a SEM image showing the fabricated micropillars on the sample surface. The microstructural observation revealed a fully recrystallized microstructure consisting of coarse equiaxed grains (Fig. 1(a)). The gradient orientations were often detected within the equiaxed grains (Figs. 1(b), (c)), suggesting the presence of dislocation substructures developed by the prior cold-rolling. In these maps, the arrowheads indicate the identified grains with a particular orientation for preparing the micropillars (as shown in Fig. 1(d)). The annealed sample exhibited an average Vickers hardness value of 15 HV. The 15% and 30% cold-rolled samples exhibited relatively higher hardness values of 18 and 22 HV, respectively.
EBSD orientation maps of (a) annealed sample, (b) 15% cold-rolled sample, and (c) 30% cold-rolled sample. The arrowheads indicate the grains where the micropillars were fabricated. (d) Representative SEM image showing the micropillars fabricated on the sample surface.
Figure 2 displays bright-field TEM images showing dislocations in the annealed sample and cold-rolled samples. All images were taken with the same reciprocal lattice vector of g = 002. The annealed sample (Fig. 2(a)) showed a featureless morphology with a few dislocations, whereby a number of dislocations were observed in the 30% cold-rolled sample (Fig. 2(b)). Most of the dislocations appeared tangled, resulting in the formation of a dislocation substructure with a mean dislocation cell size of approximately 1 µm. An inhomogeneous dislocation substructure was found in the 15% cold-rolled sample (Figs. 2(c), (d)). A number of dislocations appeared to locally form cell walls (as indicated by arrowheads in Fig. 2(c)), resulting in a relatively coarsened cellular structure (average cell size of approximately 3 µm in local regions). Meanwhile, few dislocations were observed in the different regions (Fig. 2(d)). Considering the observed areas by TEM, it was assumed that the dislocation cell size would approximately range from 3 µm to above 6 µm in the 15% cold-rolled sample. The dislocation density (ρ) was quantified to 3.7 × 1012 m−2 in the annealed sample,8) whereas the ρ values were not quantified in the cold-rolled samples due to the difficulty of determining individual dislocation lines in the dislocation substructures (Figs. 2(b), (c)).
Bright-field TEM images showing dislocations in the (a) annealed sample, (b) 30% cold-rolled sample, and (c), (d) 15% cold-rolled sample.
Figure 3 presents the nominal stress-strain curves of the single-crystal micropillars with different diameters (d), which were fabricated on the annealed sample7,8) and cold-rolled samples. Each compression direction of single crystals is shown in a standard stereographic triangle of inverse pole figure and the stress for the slip initiation (yield stress or initial strain burst) was indicated by arrowheads in these figures. All flow curves were characterized by an elastic loading and a following plastic deformation containing intermittent strain bursts24) and negligible strain hardening. In the annealed sample (Fig. 3(a)), the transition behavior from elastic to plastic regions appeared to depend on the specimen diameters (d). Notably, Young’s modulus appears varied in the measured stress–strain curves, whereas the different elastic slopes would be due to the varied specimen geometries,25) which can be rationalized by using Sneddon’s solution of a flat punch indenting into an isotropic elastic half-space.26) In the flow curve of the small micropillar (d = 1.2 µm), a relatively large strain burst appeared at a high-stress level (about 160 MPa), whereas a trend was found that the larger micropillars showed smaller strain bursts at a lower stress level and exhibited a continuous yielding. The flow curves clearly indicate that a higher stress level could be required for the onset of plastic deformation in smaller single-crystal micropillars, which is indicative of the size-dependent strength of pure Al.7,8) The size dependence of strength appeared reduced in the cold-rolled specimens and less pronounced in the heavily rolled sample (higher reduction in cold-rolling). The 15% cold-rolled specimens showed comparatively continuous yielding behaviors with fewer strain bursts (Fig. 3(b)). In small micropillars (d ∼ 1 µm), the 15% cold-rolled specimens exhibited a lower stress level than the annealed specimens and the size dependence of the strength appeared less pronounced. These tendencies appeared more significant in the 30% cold-rolled specimens (Fig. 3(c)). The flow curves showed a continuous yielding, which was independent of the specimen size. These results demonstrated that the cold rolling process reduced the size-dependent strength of single-crystal high-purity Al micropillars.
Nominal stress-strain curves of the single-crystal micropillars with various diameters (d) fabricated on the surface of (a) annealed sample,8) (b) 15% cold-rolled sample, and (c) 30% cold-rolled sample.
Figure 4 displays the SEM images showing the compressed micropillars with a fixed diameter of approximately 4 µm. The SEM observations revealed the macroscopic slip traces on the surfaces of cylindrical micropillars, which was indicative of the activation of a single slip system. Such a slip trace was observed in all the deformed micropillars with different diameters. To identify the activated slip system under compression, the geometries of the compressed micropillars were compared with the pole figures obtained from the prior EBSD analyses. A representative result (30% cold-rolled specimen) is summarized in Fig. 5. The tilted view of the compressed micropillar (Fig. 5(a)) provides an angle between the compression direction and the activated slip plane. The top view (Fig. 5(c)) presents the displacement direction by slip. The comparison of these SEM images (Figs. 5(a), (c)) with the 111 and 011 pole figures (Figs. 6(b), (d)) addressed the dominantly activated slip system. The identified slip system corresponded well to one with the highest Schmid factor in all single-crystal specimens.
Representative SEM images showing single-crystal micropillars (a), (c), (e) before and (b), (d), (f) after the compression test: (a), (b) annealed sample, (c), (d) 15% cold-rolled sample, and (e), (f) 30% cold-rolled sample.
(a) 45° tilted view and (c) top view of compressed single-crystal micropillars fabricated on the sample surface of 30% cold-rolled sample and corresponding (b) 111 and (d) 011 pole figures for the determination of an activated slip system. Large circle symbols correspond to the direction normal to the activated slip plane in (b) 111 pole figure and the activated slip direction in (d) 011 pole figure.
The determined Schmid factor of the activated slip system (Fig. 5) can provide a calculation of the resolved shear stresses for slip initiation (τi) of high-purity Al single-crystals. The variations in the τI value of the annealed and cold-rolled specimens were summarized as a function of the specimen diameter (d). These results are shown in Fig. 6. It was somewhat difficult to determine the stress for the slip initiation (detecting the first discontinuous-displacement point from the flow curves) because the slope of the elastic regions deviated, which is presumably due to a slight misalignment of the flat punch contacting the top surface of micropillars.27) The stress of the comparably large strain burst or deflection points from the initial elastic region in the experimental curve (indicated by arrowheads in Fig. 3) was used as the stress for the slip initiation (macroscopically). The τi value was designated as the critical resolved shear stress (CRSS, τCRSS) in the present study. In addition, 0.2% proof stress was used as τi value in the continuous stress-strain curves. Both annealed and cold-rolled single-crystal specimens showed the size dependence of τi value (Fig. 6(a)). The τi value appeared to increase monotonously with decreasing d in the annealed high-purity Al (99.99%: 4N-Al) specimen. The reported τi values of pure Al with different purity levels (99.9% and 99.999%: 3N-Al and 5N-Al)28,29) corresponded well to the observed trend of the 4N-Al specimen (Fig. 6(a)). However, a lower slope was found in the cold-rolled specimens. A trend was observed that the 30% cold-rolled specimens exhibited higher τi values than the 15% cold-rolled specimens, in particular, in higher d (larger specimen size). Such a trend of the varied slopes was found in the resolved shear stress at a higher strain of 5% (τ5%), as presented in Fig. 6(b).
The present micropillar compression tests revealed the variations in the size-dependence of the critical resolved shear stress (used as the τi value) in high-purity Al single-crystals by changing the cold-rolling reduction (Fig. 6(a)). To quantify the change in the size-dependent strength, the experimentally measured τi values normalized by G were plotted as a function of d normalized by b, according to eq. (1).14) Figure 7(a) shows a variation in τi/G depending on d/b for the annealed specimens of high-purity Al (4N-Al),7,8) together with the previous results of the fully annealed 3N-Al and 5N-Al specimens.28,29) The approximate line fitted by the measured τi/G values showed the power-law relation with m = 0.657,8) (corresponding to that of pure fcc metals, 0.6613)) according to eq. (1). Although the 15% cold-rolled specimens exhibited scattered τi values in the specimen diameter (d) range of approximately 3∼8 µm (Fig. 7(b)), a lower m value (0.26) was measured on the fitted experimental data of the 15% cold-rolled specimen. The 30% cold-rolled specimen exhibited a similar m value (0.31). These results clearly indicate the reduced size-dependent strength in high-purity Al single-crystals by the prior cold-rolling. Notably, the 30% cold-rolled specimens exhibited a higher A value than the 15% cold-rolled ones, which was indicative of strengthening by fine dislocation substructures introduced by the higher-reduction cold-rolling (Fig. 2).
Variations in the resolved shear stress required for slip initiation (τi) scaled by shear modulus (G) as a function of the micropillar diameter (d) scaled by Burgers vector (b): (a) the fully annealed pure Al specimens and (b) the cold-rolled pure Al specimens.
The regression fit to the experimental data according to eq. (1) demonstrated that the size-dependent strength was less pronounced in high-purity Al single-crystals by cold-rolling (Fig. 8). The introduced dislocations and the substructure development (Fig. 2) could contribute to the variation in size-dependent strength by the cold-rolling. In general, the mechanism of size-dependent strength for the single crystals of metals was rationalized by truncation of single-arm dislocation sources10,22–24) or exhaustion/starvation of dislocation sources.25,30,31) Assuming the pre-existing dislocations were homogenously distributed within the experimental single crystals, the stochastic model of the dislocation source length (proposed by Parthasarathy et al.22)) could be applied to describe the size dependence of strength. This model can give a description of the onset of plastic deformation from a truncated weakest single-arm dislocation source defined by an average length ($\bar{\lambda }_{\textit{max}}$). In this model, the critical resolved shear stress (τCRSS) can be expressed using the friction stress (τ0) and the dislocation density (ρtot) as follows:
| \begin{equation} \tau_{\textit{CRSS}} = \frac{\alpha Gb}{\bar{\lambda}_{\textit{max}}} + \tau_{0} + 0.5G\sqrt{\rho_{\textit{tot}}} \end{equation} | (2) |
| \begin{align} \bar{\lambda}_{\textit{max}} & = \int\nolimits_{0}^{R}\lambda_{\text{max}}p(\lambda_{\text{max}})d\lambda_{\text{max}}\\ & = \int\nolimits_{0}^{R}\left[1 - \frac{\pi(R - \lambda_{\text{max}})(l - \lambda_{\text{max}})}{\pi Rl}\right]^{n - 1} \\ & \quad \times \left\{\frac{\pi[(R - \lambda_{\text{max}}) + (l - \lambda_{\text{max}})]}{\pi Rl}\right\}n\lambda_{\text{max}}d\lambda_{\text{max}} \end{align} | (3) |
| \begin{equation} n = \textit{Integer}\left[\frac{\rho_{\textit{tot}}}{12} \cdot \frac{\pi(d/2)^{2}h}{L_{\textit{seg}}}\right] \end{equation} | (4) |
(a) Calculated critical resolved shear stress (τCRSS) scaled by shear modulus (G) as a function of the micropillar diameter (d) scaled by the Burgers vector (b) for single-crystal pure Al by using the stochastic model of the dislocation source length.22) (b) Experimental resolved shear stress required for slip initiation (τi), for the annealed pure Al specimens and the 30% cold-rolled specimen, compared with calculated τCRSS using different dislocation densities (ρ).
On the basis of the stochastic model of the dislocation source length, the τCRSS values were calculated in cylindrical single-crystals of pure Al with various diameters (d). The τCRSS value of bulk single-crystals (corresponding to τ0 in eq. (2)) was applied to 1 MPa.32) The h value is an approximate average value (4d) of the cylindrical micropillars studied. The calculated τCRSS values scaled by G were plotted as a function of d scaled by b. The results are summarized in Fig. 8(a). In this figure, the τCRSS values were plotted in case of n (the number of pins) > 1. In the single crystals with various dislocation densities (ρ) ranging from 1011 m−2 to 1014 m−2, a general trend was found that the τCRSS value decreases with increasing d. The trend represents a smaller slope (size-dependent strength) and a higher saturated τCRSS value of the single crystal having a higher ρ, which is indicative of the reduced size-dependent τCRSS by the high-density dislocations in the single crystals. The fitting curves (τCRSS/G vs. d/b according to the stochastic model) to the experimental τi values of the annealed and 30% cold-rolled micropillars (Fig. 7) were presented in Fig. 8(b). The τi values of the annealed micropillars corresponded well to the calculated τCRSS/G curves for single crystals with ρ = 4 × 1012 m−2. The ρ value was in good agreement with the experimentally measured ρ value (3.7 × 1012 m−2) in the annealed sample.8) The τi values of the 30% cold-rolled specimens fitted well to the calculated τCRSS/G curve in case of ρ = 5 × 1013 m−2. The ρ value was not experimentally quantified in the cold-rolled sample (Fig. 2(c)), whereas it would be a reasonable value for the cold-rolled high-purity Al.21) The aforementioned results represent that the initial dislocations introduced by cold-rolling could dominantly contribute to the reduced size-dependent strength of the cold-rolled micropillars of high-purity Al.
The 15% cold-rolled micropillars exhibited scattered τi values in the diameters (d), approximately ranging from 3 to 8 µm (Fig. 7(b)), whereby it was difficult to fit the experimental τi values by the calculated τCRSS values under a certain ρ value. This suggests that dislocation densities would vary depending on the location in the 15% cold-rolled sample. Herein, two calculated τCRSS curves under different ρ values of 4 × 1012 and 5 × 1013 m−2 (in the annealed and 30% cold-rolled samples, respectively) were compared with the experimental τi values of the 15% cold-rolled specimens. The result is shown in Fig. 9(a). It was found that the scattered τi values were within a range of τCRSS values calculated under varied ρ values ranging from 4 × 1012 to 5 × 1013 m−2, indicating the micropillars with different dislocation densities fabricated on the surface of the 15% cold-rolled sample containing the inhomogeneous dislocation substructure (Fig. 2(c), (d)). It is noteworthy that a range of d (approximately 3∼8 µm) exhibiting the scattered τi values would correspond well to an approximate cell size in the dislocation substructure (Fig. 2(c), (d)). It can be therefore considered that the local variation in dislocation densities could contribute to a significant variation in the measured τi value of the 15% cold-rolled micropillars. The comparison of the experimental micropillars with the dislocation cells is schematically illustrated in Fig. 9(b). In the micropillars with a diameter (d) range of 3∼8 µm (equivalent to the dislocation cells), the local dislocation density would change in the micropillars depending on the location of the inhomogeneous dislocation substructure. When micropillars are prepared on dislocation cell walls in the 15% cold-rolled sample, such tangled dislocations would play a significant role in dislocation multiplication, resulting in reduced τi (low strength). In contrast, micropillars would often be prepared inside the dislocation cells (local regions with low dislocation densities), whereby such micropillars could exhibit high τi (high strength). The larger micropillars (d > 8 µm) could include the dislocation substructure (several dislocation cells) with a relatively high dislocation density (Fig. 9(b)). In case of smaller d than 3 µm, the effective ρ value might change according to the probability of existing the dislocation cell walls in the fabricated micropillars. The measured τi values appeared to follow the trend of τCRSS values calculated under a high ρ value of 5 × 1013 m−2, whereas the details still remain unclear. It would be required to precisely analyze the dislocation distribution inside the dislocation cells developed in the cold-rolled high-purity Al.
(a) Experimental resolved shear stress required for initial slip (τi) as a function of the micropillar diameter (d), for the 15% cold-rolled micropillars, compared with the calculated values using different dislocation densities (ρ). (b) Schematic illustrations showing single-crystal micropillars with different diameters (d) fabricated on the grains with the dislocation substructure.
In the present study, we investigated the size-dependent strength of single-crystal cylindrical micropillars with various diameters approximately ranging from 1 to 10 µm, which were prepared on the surface of the fully annealed sample and the subsequently cold-rolled samples of high-purity Al (99.99% purity). The key results were shown as follow.
The support from The Japan Institute of Metals and Materials was gratefully acknowledged.
