Alloying effects on deformation induced microstructure evolution in copper

This section presents our findings on the interplay between alloying elements, plastic deformation, and microstructural evolution in polycrystalline Cu and Cu-1\(\%\)Pb. Initial microstructural characterization revealed right-skewed grain size distributions in both materials. However, for simplicity and quantitative comparison, we applied Gaussian statistics to the data. For pure Cu samples, the calculated mean grain size was 24 \(\mu\)m with a standard deviation of 30 \(\mu\)m. The Cu-1\(\%\)Pb samples had a calculated mean grain size of 21 \(\mu\)m with a standard deviation of 17 \(\mu\)m. The large standard deviation in pure Cu, exceeding its mean, reflects the significant variability in grain sizes. SEM analysis indicated homogeneously distributed lead particles within the copper matrix, with particle diameters ranging from 1 to 5 \(\mu\)m. However, the orientation of these particles could not be indexed through EBSD, TEM, or neutron diffraction techniques. Due to the small size of the Pb particles, which were not fully characterized by the experimental modalities used in this study, we did not monitor two-phase interactions, strain partitioning, or attributes like grain aspect ratio changes with deformation for both phases. These baseline characteristics set the stage for our subsequent detailed measurements and observations of the materials’ behavior under deformation.

The diffraction patterns obtained at varying strains for both Cu and Cu-1\(\%\)Pb specimens are shown in Figure 1. The diffraction profile asymmetry visible in Figure 1 corresponds to the characteristic peak shape asymmetry of time-of-flight neutron diffraction measurements. In the diffraction line profile analysis, this asymmetry is accounted for through the calibration/instrumental pattern. Although Cu and Pb have face-centered cubic (fcc) structures, their near-complete immiscibility in the solid state arises from a significant difference in atomic radii (37\(\%\))²². However, due to the small particle size and minimal Pb content in the compression specimens, only Cu peaks are discernible in the diffraction patterns. Nevertheless, a significant increase in the diffraction peak width is observed in both materials with increasing deformation levels. This broadening is primarily attributed to microstructural changes, such as defect generation, grain refinement, and strain accumulation, which are necessary to accommodate the externally imposed macroscopic load.

The texture evolution in Cu and Cu-1\(\%\)Pb up to a 40\(\%\) true strain is shown in Figure 2. Both materials were extracted from rolled plates and subjected to heat treatment after machining the compression specimens. Initially, Cu-1\(\%\)Pb exhibited a random texture, while the Cu specimen showed a weak remnant of the rolling texture. As a result, minor discrepancies are noticeable in the early deformation textures, which can be attributed to the slightly divergent starting textures between the two materials. Despite the initial texture differences, both materials demonstrate a consistent trend in the development of preferred orientation upon deformation, reaching a peak value of roughly 3.5 multiples of a random distribution (mrd) at 40\(\%\) strain. As expected during the compressive loading of fcc metals, a 110 fiber texture develops along the loading axis in both instances, as indicated by the increased intensity with increasing deformation.

Pole figures illustrating deformation texture evolution due to imposed compressive deformation for (a) Cu and (b) Cu-1\(\%\)Pb at different strains. The pole figures are plotted on a common scale ranging from 0.5 to 3.5 mrd. The center of the pole figure is parallel to the compression axis. With increasing strain, both materials show prominent increase in 110 peak intensity, indicating crystals with 110 plane normal orient along the loading direction as deformation progresses.

Figure 3(a) shows a Williamson-Hall plot derived from the full width at half maximum (FWHM) of the time-of-flight neutron diffraction data. The FWHM serves as an indicator of ‘microstrain’ or ‘intragranular strain’ resulting from variations in interatomic spacing associated with defects (such as dislocations or other heterogeneities), where K is defined as the reciprocal value of d, K=1/d. The instrumental resolution, determined from the annealed specimen at 0\(\%\) strain (assumed to be defect-free), consistently shows an increase in broadening with the scattering angle. The observed anisotropic line broadening at different strains reflects the average dislocation contrast factors in polycrystals, a signature behavior of peak widths of cold-worked metals. This non-linear broadening stems from the anisotropic contrast factor associated with each active crystallographic slip system in a specific crystal structure¹⁶. The average dislocation contrast factor of an hkl reflection in cubic polycrystals is expressed by Equation 1^23,24:

where q and \(\hbox {C}_{h00}\) are dislocation character parameter and average contrast factor for h00 reflections, respectively. Both depend on the single crystal elastic constants of the material and are determined numerically^24,25. The systematic increase in broadening with higher-order hkl reflections indicates an increase in dislocation density. As the broadening due to crystallite size remains constant with respect to hkl order when expressed in terms of diffraction vector length, the increase as a function of K is attributed solely to microstrain^26,27,28. Additionally, the inhomogeneous distribution of lead inclusions, initially segregated at grain boundaries in the Cu-1\(\%\)Pb compression specimens⁶, might also contribute to some extent of peak broadening.

(a) Peak width as a function of crystallographic planes and deformation for copper and copper with 1 wt.\(\%\) lead. Assuming that dislocations are the primary contributors to the observed broadening, the non-linear increase in width arises from the anisotropic elastic properties of the two materials. (b) Peak profiles of the (222) and (400) reflections for Cu (red) and Cu-1\(\%\)Pb (blue) after 40\(\%\) strain. Profiles are background-subtracted, centered, and normalized to both peak maxima and FWHM, with normalized intensities plotted on a logarithmic scale. Solid lines represent raw data, while dotted lines correspond to fits from line profile analysis. Although the FWHMs vary within a narrow range, the tail regions of the peaks from the two materials exhibit significant variations in broadening.

To isolate the effects of various microstructural factors on peak broadening, a dislocation line profile analysis (DLPA) was performed. Table 1 summarizes the microstructural information extracted from DLPA using the extended convolution multiple whole profile (eCMWP) software²⁶. The dislocation density measurements using DLPA are only reported for samples strained to 5\(\%\) and beyond, as the undeformed or slightly deformed samples do not exhibit detectable dislocation-related peak broadening, indicating dislocation densities below the neutron diffraction measurement threshold of approximately \(0.1*10^{14}\)\(\hbox {m}^{-2}\). This analysis yields several microstructure parameters, including the area-weighted mean coherently scattering crystallite size (sub-grain size, \(\chi\)), dislocation density (\(\rho\)), dislocation arrangement (\(M^{*}\)), and the parameter q which characterizes the dislocation contrast factors. From this, the fraction of edge and screw dislocations are estimated.

The dislocation density increases with deformation, reaching values of \(\sim 0.8 \times 10^{15} \hbox {m}^{-2}\) for Cu and \(\sim 1.4 \times 10^{15} \hbox {m}^{-2}\) for Cu-1\(\%\)Pb at 40\(\%\) compressive strain. Due to low dislocation density below 5\(\%\) strains, the DLPA was not performed. For the 5\(\%\) and 10\(\%\) strain data, refinements necessitated additional constraints to ensure convergence. Thus, the dislocation character parameter q was fixed at 1.7, suggesting equal proportions of edge and screw dislocations, while the dislocation parameter (\(M^{*}\)) was set to approximately 0.6 for both materials. At 5\(\%\) strain, both materials exhibited \(\rho\) values of around \(10^{13}\)\(\hbox {m}^{-2}\), which is at the detection limit of the neutron diffraction measurements.

The addition of an alloying element to Cu increased the dislocation content with deformation, nearly doubling the number of dislocations generated during deformation at the maximum strain achieved in the experiments. Concurrently, the crystallite size (\(\chi\)) – representative of the coherently scattering domain size – decreased with increasing deformation, slightly more so in Cu-1\(\%\)Pb at 40\(\%\) compressive strain. Typically, in metals, the coherently scattering crystallite size corresponds to the dislocation cell size within the grains delimited by high-angle boundaries.

A systematically lower \(M^{*}\) parameter, related to the dislocation arrangement²⁹, was observed in Cu-1\(\%\)Pb compared to Cu. This aligns with the observed difference in the tail region of the diffraction profiles for the two materials, as shown in Figure 3(b). The reduced \(M^{*}\) parameter in Cu-1\(\%\)Pb indicates a more ordered dislocation arrangement, i.e., having a higher dipole character, correlated with plastic deformation. As strain increases and the dipole character of the dislocation structure becomes more pronounced¹⁶, Burger’s vectors appear in pairs, cancelling out each other’s strain fields. This relationship is defined by \(M = R_{e}\sqrt{\rho }\), where \(R_{e}\) represents the outer cutoff radius beyond which dislocation strain fields are presumed non-interacting³⁰. In addition to the dislocation density (\(\rho\)), the arrangement of the dislocations also directly correlates with the stored elastic energy density within the material³¹. An arrangement with a pronounced dipole character, i.e., a lower value of \(R_{e}\) and \(M^{*}\), signifies a lower stored energy for the same density.

Given that the nature of a dislocation influences its interactions and annihilation^9,32, the ratio of edge to screw dislocations was computed as a function of strain for both materials after reaching a 15\(\%\) true strain, using the fitted dislocation character value (q). Previous research on fine-grained 99.9\(\%\) Cu compressed at low temperatures by Schafler et al.³² highlighted an uptick in the edge dislocation fraction with strain, culminating in an edge fraction approaching 75\(\%\) at a true strain of 22\(\%\) before plateauing. Generally, a similar trend favoring the increase in the net generation of edge dislocations with increasing strain is observed for both Cu and Cu-1\(\%\)Pb. However, the oxygen-free high thermal conductivity (OFHC) Cu, characterized by larger grains (averaging \(\sim\) 60 \(\mu\)m), showed a reversal in net edge versus screw dislocations generated past 30\(\%\) strain. This observed shift may primarily be attributed to noise, evident as pronounced fluctuations in the tail region of the diffraction line profiles, making it challenging for eCMWP to accurately decouple dislocation character and arrangement.

Compared to neutron diffraction measurements, which provide an indirect yet quantitative approach to microstructural characterization, electron microscopy techniques like Electron Backscatter Diffraction (EBSD) and Transmission Electron Microscopy (TEM) offer direct insights into local microstructures, dislocation cell structures, and arrangements³³. The synergy of these methods can enhance our understanding of microstructural evolution inferred from neutron diffraction^15,34.

Figure 4 shows the Kernel Average Misorientation (KAM) maps of deformed microstructures at varying strains for Cu and Cu-\(1\%\)Pb. The local grain morphology and orientation changes pre- and post-deformation were characterized using EBSD, and the KAM and the distance between a pixel and its nearest grain boundary were computed using the Dream.3D software³⁵. Cu displays KAM accumulation within the grain interior, a phenomenon absent in Cu-1\(\%\)Pb, particularly at lower strains. Moreover, Cu reveals a crystallographic orientation-dependent deformation accumulation within the grain interior, leading to pronounced orientation gradients in certain grains, while Cu-1\(\%\)Pb does not exhibit this crystallographic orientation dependency. Instead, elevated KAM values are concentrated near the grain boundary in Cu-1\(\%\)Pb. Figure 4(b) shows the probability distribution of KAM values across different strain levels for both materials. Although both materials show a rise in mean KAM values with deformation, Cu-1\(\%\)Pb consistently exhibits lower mean KAM values at lower strains. Interestingly, Cu-1\(\%\)Pb, which initially displayed high KAM values predominantly near the grain boundary at smaller strains, began to develop an orientation gradient within the grain interior at larger strains, resembling the deformation pattern seen in Cu at equivalent strain levels.

(a) KAM maps of deformed microstructures at different strain levels for Cu and Cu-\(1\%\)Pb. (b) Probability distribution of KAM values at different strain levels for the two materials.

EBSD and DLPA provide complementary but distinct microstructural information. EBSD, with a 0.5 \(\mu\)m resolution, cannot directly detect sub-grains below this size. The KAM values from EBSD reflect larger scale misorientations and inferred GND densities, which may not correlate with sub-grain sizes detected by DLPA. Cu-1\(\%\)Pb exhibits consistently smaller \(\chi\) values than Cu, especially at higher strain levels (past 5\(\%\) strain), indicating finer coherent scattering domains throughout deformation. This refinement is attributed to Pb particles impeding dislocation motion and promoting smaller subgrain formation. These observations are consistent with the higher dislocation density and more homogeneous strain distribution measured in Cu-1\(\%\)Pb. This observation supports the dual role of Pb particles as dislocation sources and obstacles, resulting in a more refined dislocation substructure in Cu-1\(\%\)Pb compared to pure Cu during plastic deformation.

Figure 5 shows a comparison of the dislocation structures in both Cu and Cu-1\(\%\)Pb at 5\(\%\) and 20\(\%\) strains. All bright-field TEM micrographs were captured under consistent two-beam conditions, with the \(\hbox {g}_{020}\) direction being strongly excited. After 5\(\%\) deformation, elongated, isolated dislocations are observed in both Cu and Cu-1\(\%\)Pb (Figures 5(a,c)). Upon deformation to 20\(\%\) strain, there is a significant increase in dislocation density (Figures 5(b,d)). The dislocation densities were estimated to be 0.97±0.18 \(\times 10^{14} m^{-2}\) and 3.75±0.75 \(\times 10^{14} m^{-2}\) for Cu, and 1.35±0.21 \(\times 10^{14} m^{-2}\) and 5.99±1.08 \(\times 10^{14} m^{-2}\) for Cu-1\(\%\)Pb, at 5\(\%\) and 20\(\%\) compressive strain, respectively. The elevated dislocation densities observed in Cu-1\(\%\)Pb, relative to Cu at identical strain levels, are consistent with the results from the neutron diffraction measurements.

Comparison of dislocation structures in Cu and Cu1\(\%\)Pb at different strain levels. Bright field (BF) TEM micrographs of Cu deformed to (a) 5\(\%\) and (b) 20\(\%\) strain show an increase in dislocation density with strain. In contrast, BF TEM micrographs of Cu1\(\%\)Pb at (c) 5\(\%\) and (d) 20\(\%\) strain indicate a higher dislocation density than that in Cu at equivalent strain levels. All TEM micrographs were captured under two-beam conditions with \(\hbox {g}_{020}\) strongly excited, as indicated by the orange arrows in the lower right corners of the micrographs.