Observing formation and evolution of dislocation cells during plastic deformation

Metals and alloys are typically polycrystalline aggregates. When deformed plastically, dislocations are introduced into the lattice of each grain^1,2. For materials with medium to high stacking fault energies the primary mechanism for facilitating the shape change is dislocation slip. The interplay between the plastic flow and minimisation of elastic energy implies that the dislocations organize into boundaries. Moreover the micro-structure often evolves to comprise two types of boundaries, on a coarser scale Geometrically Necessary Boundaries, GNBs, reflecting systematic variations in the plastic flow, and on a finer scale Incidental Dislocation Boundaries, IDBs, thought to represent statistically trapped dislocations³. The IDBs separate nearly dislocation-free regions called cells. With increasing deformation, the flow stress increases, and the entire hierarchical structure shrinks in length scale. Empirically, structural properties such as cell size and mis-orientation distributions have been shown to exhibit scaling with the applied field for plastic strains larger than 5–10%^4,5,6, when separating IDBs from GNBs.

However, despite extensive studies – motivated by the socioeconomic impact of metals-we are still unable to predict the type of micro-structure that forms as function of material and processing from first principles. While correlations have been established with the active slip systems^7,8,9, the mechanisms underlying the cell formation and cell sub-division are not known. This prohibits realizing the vision of “materials science in the computer” in this field. At the root of this predicament are two issues. Firstly, the complexity and computational effort in handling the large sets of dislocations involved has been prohibitive. Discrete^10,11,12 and Continuum¹³ Dislocation Dynamics models (DDD and CDD) can at best simulate representative volumes up to about 0.5 and 1% strain, respectively. For that reason until recently even basic patterning has been elusive¹⁴.

Secondly, it is challenging to visualise the micro-structural evolution experimentally in a representative way, as the multi length scale problem requires a combination of contrast to cells and local dislocation content over a large representative volume, in situ and within the bulk of a sample. Traditionally, dislocation structures are mapped by electron microscopy (EM)^{7,15,16,17,18,19}. EM provides very detailed maps, but is inherently limited in terms of representative volume by the use of thin foils or micro-pillars. Hence, the multiscale dynamics may not represent bulk conditions. For bulk studies x-ray diffraction based imaging has emerged as a powerful tool. Multi-grain X-ray imaging modalities such as 3DXRD and DCT can provide comprehensive information on the grain level, well suited for interfacing with crystal plasticity models^{20,21,22,23,24}. X-ray scanning nano-beam methods on the other hand have been demonstrated to visualise a few dislocations^25,26,27. For quantitative measurements of dislocation densities line profile analysis of x-ray diffraction patterns prevails^28,29,30,31, but results are averages over grains or over the illuminated volume, and typically only represent a fraction of the dislocations present.

To address this multiscale experimental problem we have established Dark Field X-ray Microscopy³², DFXM. With modalities similar to dark field TEM³³, this method enables large field-of-view visualisations of both dislocations^34,35,36,37 and cells^32,38 within the bulk. We here apply the technique to a comprehensive study of the structural evolution within Aluminum during in situ tensile deformation from an applied strain of \(\epsilon = 0\) to \(\epsilon = 0.046\). The inspected volume comprises \(\sim\) 40,000 dislocation cells, providing excellent statistics in this initial patterning range, where the structure is little known and the overlap with the strain regime available in simulations is most prominent.

The sample of choice is a single crystal of pure Aluminum, with [111] parallel to the tensile axis, see Fig. 1a. The mechanical device is mounted in the DFXM microscope such that diffraction in the vicinity of the [111] reflection is imaged for the illuminated layer. See Fig. S2 for more details of the loading device. By tilting goniometer angles \(\phi\) and \(\chi\), see Fig. 1a, contrast is provided to orientation and strain components. Three complementary modalities are illustrated in Fig. 1b–e. Weak beam contrast provides statistics over dilute dislocations ensembles at very low strains (\(\epsilon \le 0.005\)), orientation contrast provides statistics over cell properties, once these have formed, while the peak broadening \(\Delta q\) is a proxy for the total dislocation density, \(\rho\): \(\Delta q \sim \sqrt{\rho }\). By repeating the mapping for a set of \(z_l\) layers, a 3D map can be created. For definitions, algorithms and specifications, see sections 3.1 and 3.2.

Single crystals of the present orientation form parallel planar GNBs in addition to cells³⁹. The GNBs are clearly manifest in both EM and DFXM when inspecting planes which include the TA (see Fig. S20 and Zelenika et al.³⁸). In this work we report on the cell evolution primarily within a plane perpendicular to the TA. Here the GNBs are known to be less visible³⁹. This was confirmed by the absence of preferred directions or orientation correlations between cells in the present data, see Figs. S11 and S12. This indicates that a model that does not explicitly take the crystallography of the deformation process (slip planes) into account may be adequate.

For \(\epsilon < 0.01\), the sample exhibits a set of isolated dislocations and dislocation entanglements, as evidenced by Fig. 1b. In Section 3.6 these clusters are quantified in three complementary ways, which sample the elastic field associated with the dislocations differently. Consistently the three approaches reveal that the clusters are randomly positioned and distributed homogeneously within the Field-Of-View (FOV). Moreover their size distributions are consistent with log-normal distributions.

With increasing applied strain, the orientation spread and the peak broadening-both averaged over the entire volume inspected-grow approximately linear with the strain, see Fig. S5. Moreover, domains of approximately uniform intrinsic orientation distribution appear, see Supplementary Video 1 nd Fig. S21. We define cells as domains where the Kernel Average Misorientation, KAM, of all boundary voxels are above a threshold, \(\theta _{\text {KAM}}\). By means of morphological operations, the boundaries are made 1 pixel thin, and as a consequence a tessellation of the sample is obtained, see Fig. S9. Anticipating scaling we set \(\theta _{\text {KAM}}\) to be a linear function of \(\epsilon\). The resulting KAM filter superposed on the orientation map are shown as function of \(\epsilon\) as Supplementary Video 3 and Fig. S23. Snapshots are provided in Fig. 2. Inspection reveals that the tessellations are indeed similar, justifying the linear ansatz. This analysis also allows us to address a long-standing question: when are the cells formed? As shown in Fig. 3a the sample undergoes a transformation from no cells to site-saturation of cells in the range \(\epsilon = 0.008\)–0.023.

In a complementary approach we study the order in the structure on the micrometer length scale by deriving the autocorrelation function of the orientation maps. These are presented as function of \(\epsilon\) in Supplementary Fig. S26. The absence of any side peaks clarifies that there is no long range ordering of cells. Next in Fig. 3b for \(\epsilon = 0.046\), we compare the autocorrelation function with a hard sphere model of the cells with all parameters defined by the experimentally determined size distribution, see also Suppl. Information. The excellent match with experimental data informs that the cells “do not see each other”. We interpret this as evidence of the local and random nature of the cell formation process. From Fig. 3c it appear that this screening of the surrounding takes place as soon as the cells are fully formed, at \(\epsilon = 0.024\). Moreover, it is shown that results are consistent with with the available TEM results for \(\epsilon > 0.05\).

The cell map at \(\epsilon = 0.046\) comprises \(\approx 40,000\) cells within the 9 layers analysed-one of them shown in Fig. 2d. This large ensemble allows us to distinguish with certainty between functional forms for various distributions, thereby strongly constraining models of structural evolution. The cell size distribution is found to be consistent with a log-normal distribution, see Fig 4a, while statistical tests reject other commonly used functions, cf. Fig. S13. Likewise, the distribution of misorientation angle between neighboring cells is well described as a \(\chi\) function, Fig 4b. The corresponding peak broadening distribution, shown as Fig 4c, is a proxy for the local total strain. The distribution is bimodal, which we interpret as an elastic strain and a plastic strain component. The former is to a good approximation normal, within data uncertainty the latter may be normal or log-normal. As expected the plastic strain component is predominantly present in the rather broad cell walls, cf. Fig. 1e. Distributions of aspect ratio are provided in Supplementary Information.

Next, with the statistical tools presented, we describe the micro-structural evolution. Within the applied strain range from \(\epsilon = 0.013\)–0.046, where the material goes from predominantly formation (increasing cell sizes) to predominantly fragmentation (decreasing cell sizes), the distributions evolve in a consistent way, as summarised in Fig. 4d–f. Specifically, within statistical error the size distribution is log-normal throughout and exhibit scaling (\(\sigma\) is constant.) Likewise, the misorientation angle distribution grows linear with strain and exhibits scaling (within experimental error k is constant). Moreover, the bimodal model for \(\Delta q\) is valid throughout. The area fractions of cell interior and wall regions remain approximately constant, as do the profile of the elastic component associated with the cell interior. In contrast the wall distribution widens in a linear fashion with \(\epsilon – \epsilon _0\), with \(\epsilon _0 = 0.012\) being the onset of cell formation, cf. Fig. 4f.

The existence of a log normal cell size distribution throughout is consistent with both cell formation and division being multiplicative stochastic processes. This finding leads to suggest that the combined cell formation and division process can be described as a Markovian growth-fragmentation process⁴⁰. Used e.g. in population science⁴¹ and chemical engineering⁴², such processes are mathematically proven to give rise to log-normal distributions and scaling. Specifically, our interpretation is as follows: individual dislocation entanglements appear randomly in time and space. Similar to particle creation by diffusion they grow with a growth rate that is proportional to their size. Following impingement, the cell pattern exhibits scaling with \(\theta _{\textrm{KAM}} \sim \epsilon\). In the cross-over from formation to subdivision their area remains about constant, while new dislocations continue to build up the boundaries, the IDBs. The growth in dislocation density leads to a linear growth in the average misorientation across the IDBs. With the cells exhibiting no long range order, the IDB statistics can be modelled as a sum of stochastic processes, consistent with misorientations being associated with a chi distribution⁴³. Finally, during fragmentation the larger cells are more likely to divide. This is corroborated by a positive correlation between cell heterogeneity and cell size, cf. Fig. S19.

In the past, log-normal-distributed plastic strain⁴⁴, misorientation⁴⁵ and geometrically necessary dislocation densities⁴⁶ have been obtained experimentally with surface studies at the grain scale for polycrystalline metals deformed to strains exceeding \(\epsilon =0.1\) and across a range of materials and operative micro-mechanical mechanisms like slip and twinning. Log-normal distributions have also been predicted for the plastic strain component by crystal plasticity simulations⁴⁷ of grain ensembles, without any dislocations and for cell size and misorientation angle by assuming a cell splitting probability proportional to the cell surface area and misorientation evolution by rotational diffusion⁴⁸.

Uniquely, in this work we study the deformation mechanisms at the length scale and through the strain range where the dislocation patterning (the cell formation) takes place. The DFXM maps shown here represent a representative volume deeply embedded in a sample sufficiently large that the mechanical conditions represents bulk conditions. The voxelated maps presented can be input to crystal plasticity FEM or CDD models. Subsequently two 3D movies may be compared: one experimental and one with the simulations. As demonstrated in the past e.g. for grain growth such a comparison of hundreds of thousands of points in space-time will provide unprecedented opportunities for further optimisation and validation of models⁴⁹. For CDD such an approach also overcomes the limitation to the maximum strain range available in simulations today. The methodology presented applies to poly-crystals and all crystallographic space groups, limited mainly by the spatial resolution of 100 nm.

While this study is focused on the [111] tensile direction to study the formation of cells, other orientations with lower symmetry could exhibit a different cell morphology, as shown in previous studies^7,50. Extending future research to include different systems would be valuable to gain further understanding on these differences.

To study the individual formation and fragmentation events we are commissioning a new goniometer and a more mechanically stable loading device, making it possible to track the micro-structural features in space and time by DFXM. Moreover, this may be complemented by mapping the local (purely elastic) longitudinal strain component, informing of the local stress state⁵¹. In our view the entire methodology established in this way-DFXM analysis and interfacing to modelling – is key for understanding a range of other processes in metals, such as annealing (recrystallization) and ductile damage, as it enables the combination of in situ mapping of variations in the deformed structure and local stress state with large volume identification of nuclei and voids, respectively.