Formation and Subdivision of Deformation Structures During Plastic Deformation

Science  12 May 2006:
Vol. 312, Issue 5775, pp. 889-892
DOI: 10.1126/science.1124141


During plastic deformation of metals and alloys, dislocations arrange in ordered patterns. How and when these self-organization processes take place have remained elusive, because in situ observations have not been feasible. We present an x-ray diffraction method that provided data on the dynamics of individual, deeply embedded dislocation structures. During tensile deformation of pure copper, dislocation-free regions were identified. They showed an unexpected intermittent dynamics, for example, appearing and disappearing with proceeding deformation and even displaying transient splitting behavior. Insight into these processes is relevant for an understanding of the strength and work-hardening of deformed materials.

Metals and alloys are typically poly-crystalline aggregates where each grain is characterized by the orientation of its atomic lattice. When deformed plastically, line defects (dislocations) are introduced into the lattice of each grain (1). These defects organize into dislocation boundaries separating (nearly) dislocation-free regions with almost perfect lattices, which we term subgrains. As an illustration, a transmission electron microscope (TEM) image is shown (Fig. 1). With increasing deformation, the flow stress increases, and the dislocation structure shrinks in length scale; the subgrains become progressively smaller, and the orientation difference between neighboring subgrains becomes larger (2).

Fig. 1.

TEM image of 99.99% pure Cu, deformed to a strain of 2% in tension. The dislocations (black line segments) organize into walls and dislocation-free regions.

Understanding the arrangement of dislocations is essential for science and industry, because their patterns determine many physical and mechanical properties, such as electrical resistivity of semiconductors or strength anisotropy and fatigue failure of (cubic) metals. Furthermore, dislocation patterns are of generic interest, because they are observed in a broad class of materials and over many different length scales, ranging from mm-sized structures in semiconductors (3) to structures with a size of 10 nm in severely deformed metals (4). Nevertheless, very central questions have not been settled. These include the following: How and when do ordered dislocation structures form? How is the shrinkage in length scale, and hence the subdivision of the deformation structure, accomplished?

Traditionally, deformed structures are characterized in two ways: by electron microscopy (EM) (58) and by line profile analysis of x-ray diffraction patterns (912). EM provides detailed maps of sections (Fig. 1), but the dynamics observed on such sections is not representative of the bulk because of artifacts such as dislocation migration toward the free surfaces and stress relaxation. Line profile analysis can, in principle, probe the bulk dynamics in polycrystals, but the results are averages over many subgrains and many grains, all with different orientations and neighboring relations.

We present results on the dynamics of individual, deeply embedded subgrains. The material was 99.99% pure Cu with an average grain size of 36 μm. Several 300 μm–thick specimens were studied during tensile deformation. To enable such measurements, we established a dedicated x-ray diffraction setup at the Advanced Photon Source synchrotron (Materials and Methods). A combination of x-ray optical elements generates a highly penetrating 52-keV x-ray beam, which at the same time exhibits high flux, narrow energy spread and divergence, and beam dimensions smaller than the average grain size. This beam impinges on the specimen and is used for in situ transmission studies. The resulting diffraction patterns are acquired concurrently by two area detectors for gathering information on the grain (detector A) and subgrain scale (detector B). Detector A is associated with an intermediate angular resolution of ∼0.02Å–1 (in reciprocal space units), whereas the resolution for detector B, due to a distance of 4 m to the sample, is ∼0.0005Å–1. The latter corresponds to an angular resolution of ∼0.004° and is an order of magnitude better than that attainable by EM. A sketch is shown in Fig. 2 along with examples of raw data.

Fig. 2.

(A) Sketch of the experimental setup. The real and reciprocal space coordinates (x, y, and z) and (qx, qy, and qz), respectively, are defined, together with the scattering angle of 2θ. The directions qy (the radial direction) and (qx and qz) are parallel and perpendicular, respectively, to the ideal reciprocal lattice vector for the reflection investigated (represented by G). They are related to the elastic strain and orientation distributions of the grain, respectively. X-ray diffraction patterns are acquired by using the two area detectors A and B, by rotating the sample around the x axis in small intervals. (B) Full diffraction pattern obtained with detector A at a strain of 3%. (C) Corresponding high-resolution image of the 400 reflection acquired by detector B. By stacking such high-resolution images, a 3D reciprocal space map of the reflection is obtained.

The experiment was initiated by characterization of the undeformed state using detector A. By applying experimental procedures described in the literature (1315), we identified the diffraction patterns from single grains, and we tested whether these selected grains are positioned near the center of the specimen. In the following, we present data for a grain with a volume of ∼8000 μm3 and with its [100] direction close to the tensile axis. The beam was set to illuminate a fixed subvolume of this grain by confining it with slits (Fig. 2A) to 14 μm by 14 μm.

The sample was strained to an elongation of 3% and then from 3% to 4.2% in steps of ∼0.04%. After each strain increment, a three-dimensional (3D) reciprocal space map of the 400 reflection originating from the grain sub-volume of interest was generated by acquiring a set of images with detector B while rotating around the x axis (Fig. 2A) in intervals of 0.004°. Furthermore, at a strain of 4.07%, the sample was translated with respect to the beam in steps of 2 μminboth x and y directions (Fig. 2A), and a reciprocal space map was made at each position. By a knife-edge algorithm similar to the ones presented in literature (13, 16), it is possible to determine the spatial position of features in the reciprocal space map from these scans.

All reciprocal space maps gathered show similar features, and no qualitative changes are observed in the investigated strain interval. A typical 3D map is shown in movie S1. A projection of this map, integrating along the diffraction vector, is shown in Fig. 3A. A striking feature of the high-resolution map is that the reflection comprises a set of individual peaks. Tests were performed to ensure that these peaks were not artifacts caused, for example, by the beam being partially coherent. The individual peaks were identified as diffraction spots arising from individual subgrains within the grain of choice. This claim is substantiated by three facts. First, the size of the associated diffracting entities, as deduced from the integrated intensities, were in the range of 1 to 3 μm, in good agreement with TEM results (Fig. 1). Second, the peaks were very sharp with typical full widths at half maximum of 0.001 to 0.003Å–1 in all directions of reciprocal space. This implies that none or, at most, very few dislocations were present in the subgrain, which again corresponds well with TEM results. Third, most of the peaks originated from one and only one position in the grain. (Those that did not were seen as composed of overlaying contributions from several subgrains.) Consequently, individual diffraction peaks observed in reciprocal space correspond in real space to dislocation-free regions separated spatially from each other by regions of increased dislocation density. The cloud of enhanced intensity between peaks is tentatively identified as arising from these disordered dislocation boundary regions, the broad walls in Fig. 1. On the basis of intensity ratios, the volume fraction of clearly identifiable subgrains as determined by x-rays is 30%. Considering that this is a lower limit, the value corresponds well with the volume fraction of 55% found by TEM (17).

Fig. 3.

(A) Projection onto (qx, qz) plane of an intensity map acquired at an elongation of 3.49%. The map is truncated in qz. The color scale is indicated above the map. (B) The intensity distribution of a few of the peaks appearing in the map projected onto qy (red, blue, and magenta). The corresponding profile for the entire mapped intensity is indicated in black. The observed width of the individual peaks is close to the experimental resolution as measured with a standard powder.

Examples of the intensity distribution parallel to the diffraction vector for selected peaks are provided in Fig. 3B. The centers of the peaks are clearly separated, indicating quite different elastic strains of the individual subgrains. The integrated profile in Fig. 3B (black curve) comprises contributions both from individual subgrains and from dislocation boundaries. According to the classical composite model (11), this curve is a superposition of two corresponding symmetric profiles, each broadened by the respective dislocation density. Forward stresses (enhancing the external load) in the dislocation walls and back stresses (opposing the load) in the dislocation-depleted regions lead to shifts in the radial positions and cause theasymmetryinthe blackcurve.In view of our results (movie S1 and Fig. 3B), this picture holds only for the contribution of the dislocation boundaries. The contribution of the subgrains cannot be described by a shifted profile broadened by a substantial dislocation density but rather as a number of individual sharp profiles stemming from an ensemble of dislocation-free subgrains each experiencing a different stress (with a back stress in average). This finding resolves two experimentally based objections against the composite model: (i) the lack of dislocations within subgrains (18) and (ii) the nearly constant internal stress within individual subgrains (19). The existence of nearly perfect subgrains in a deformation structure has a strong impact on existing work-hardening theories in particular for larger strains, because many of them—for example, (2024)—are built on versions of the composite model.

We created a video of the evolution of the reciprocal space map as a function of strain, and excerpts are shown in movie S2. As expected, the envelope of the reflection representing all parts of the diffracting subvolume broadens continuously with strain. Within the envelope, the evolution of nonoverlapping peaks can be traced. Because the integrated intensities and the radial and transverse positions of the peaks are linearly related to volume, elastic strain, and rotation of the corresponding subgrains, the subgrain dynamics and their reorientation can be monitoredindetail.

On the basis of an analysis of about 20 peaks, the peaks display unexpected intermittent dynamics. They appear and disappear again during straining. Some peaks, such as the one in Fig. 4A, even exist only for a short strain interval. From the decrease in average boundary distance observed by TEM (2), following an inverse square root law, we expected a decrease of the average subgrain volume, and consequently an increase in the number of subgrains, by 67% in the strain interval from 3% to 4.2%. The decreasing boundary spacing is traditionally attributed to mutual trapping of dislocations into new boundaries. Accordingly, a dislocation-free region without orientation spread may split into two regions with distinct orientations. No such event has been observed yet. The newly appearing peaks cannot be caused by reorientation of an existing subgrain, because the maximum orientation difference caused by dislocation slip during a strain increment of ∼0.04% is ∼0.05° and such reorientations can be excluded from inspection of the 3D reciprocal space maps. Rather, the distinct diffraction peaks emerge from the intensity-enhanced cloud, and the corresponding dislocation-free regions materialize temporarily as islands in a sea of dislocations. Changing their volume constantly, dislocation-free regions emerge and vanish fluctuatingly, reflecting the underlying stochastic dynamics of the entire dislocation ensemble (25). Such an intermittent behavior of the subgrains may explain two currently unsolved questions, namely how dislocation structures remain roughly equiaxed during hot-working (8) and how dislocation boundaries maintain a preferred orientation during, for example, cold-rolling (26).

Fig. 4.

Two examples of subgrain dynamics. Subfigures are excerpts from larger projections onto (qx, qz) plane of reciprocal space maps, with contour lines, axes, and color scale identical to those of Fig. 3A. These are shown left to right as a function of external strain (with strain increments relative to the first map). (A) An example of a peak that appeared, grew in intensity, and disappeared (the arrow indicates the peak at its maximum). (B) An example of a peak that split into two parts (as indicated by the arrow), which then recombined. The peak shifted simultaneously along qz, corresponding to a rotation of the subgrain by ∼0.05°.

Several peaks exhibited the behavior illustrated in Fig. 4B: The peak split into two subpeaks, which then recombined with further strain. Such a splitting of ∼0.002Å–1 or ∼0.02° can be caused by a single dislocation in the subgrain. Hence, despite the fact that the x-ray beam illuminates ∼6 × 106 dislocations at one time, we stipulate that such events can provide insight into the behavior of one or a few dislocations trapped inside subgrains.

A second sample was characterized during continuous deformation with a strain rate of 2.5 × 10–6 s–1 from 0% up to 3% elongation. The appearance of sharp peaks on top of a structureless cloud is characteristic of all strains greater than or equal to 0.4%. This observation strongly indicates that subgrain formation is initiated shortly after onset of plastic deformation. At 3% the strain was fixed, whereas the data collection continued. These data directly address a long-standing question of whether the dislocation patterns persisting after termination of the deformation are identical to the patterns existing during deformation. [The dislocation patterns present after terminating deformation can be preserved by neutron irradiation before unloading (27, 28) and hence observed by TEM after thinning.] Within observable error no changes were observed upon fixation. This is direct evidence that, at least in this case, the interruption of the test does not influence the patterning.

The method presented here can provide unique in situ information on the pattern-formation process, relevant for guiding and testing modeling efforts on the subgrain scale. In outlook, we have verified the applicability of resolving individual peaks corresponding to subgrains at much higher strains, namely 50% deformation. Furthermore, by scanning wires (16) or conical slits (13) positioned between the sample and the detector, the method can be extended to provide complete 3D spatial information. Such procedures could be added, at the expense of time, at selected strains.

Supporting Online Material

Materials and Methods

Fig. S1

Movies S1 and S2

References and Notes

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