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    Review of 'Statistical analysis of dislocation cells in uniaxially deformed copper single crystals'

    Statistical analysis of dislocation cells in uniaxially deformed copper single crystals
    Contribution on fundamentals of deformation and hardening of crystalline metals.
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    Statistical analysis of dislocation cells in uniaxially deformed copper single crystals

    The dislocation microstructure developing during plastic deformation strongly influences the stress-strain properties of crystalline materials. The novel method of high resolution electron backscatter diffraction (HR-EBSD) offers a new perspective to study dislocation patterning. In this work copper single crystals deformed in uniaxial compression were investigated by HR-EBSD, X-ray line profile analysis, and transmission electron microscopy (TEM). With these methods the maps of the internal stress, the Nye tensor, and the geometrically necessary dislocation (GND) density were determined at different load levels. In agreement with the composite model long-range internal stress was directly observed in the cell interiors. Moreover, it is found from the fractal analysis of the GND maps that the fractal dimension of the cell structure is decreasing with increasing average spatial dislocation density fluctuation. It is shown that the evolution of different types of dislocations can be successfully monitored with this scanning electron microscopy based technique.

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      Reviewed for PREreview  (https://doi.org/10.5281/zenodo.7017120



      The paper contributes to a substantial literature on fundamentals of deformation and hardening of crystalline metals. The brief literature review focuses towards characterisation and understanding of self-organised patterning in the dislocation arrangement and the possibility that this is fractal in nature. 

      The experiments centre on a series of Cu single crystals deformed to different maximum stresses and resulting plastic strains (from ~2% to ~11%). A variety of methods are used to characterise dislocation sub-structures and internal/residual elastic strains (and from these stresses) within the deformed crystals in the unloaded state. Methods include X-ray diffraction peak profile analysis, HR-EBSD mapping of stress and GND density distributions, and some TEM diffraction contrast imaging limited to one sample. The samples are inherently soft and difficult to work with and the clean data presented points towards careful experimentation. 

      A strength of the work is bringing multiple advanced techniques to bear on the same sample set. The majority of the conclusions confirm findings or ideas that are already in the literature but the use of GND density maps from HR-EBSD analysis to probe fractal nature of dislocation cells is new. The work is an interesting contribution to understanding of dislocation patterning during deformation.


      Reading the preprint provoked the following comments and questions some of which might be useful to the authors.


      Main Points

      • (page 9) Equations 6 & 7 imply that spatial gradients in both the lattice rotations and the elastic strains are used in calculating three components of the dislocation density tensor. It has typically been the case that only the lattice rotations have been considered for metallic samples as these are generally much larger. This allows six constrains on the dislocation density tensor rather than three if elastic strains are also included. The result of neglecting the contributions from strain is often referred to as the Nye tensor, while retaining it leads to the Kroner dislocation density tensor. A statement clarifying whether elastic strains are included here (Kroner analysis) would be helpful.
      • I found section 3 interesting and it provoked the following thoughts/questions: (i) exactly at the free surface s'i3 terms should go to zero which is used to justify forcing  s'33=0 to allow separation of normal strains in HR-EBSD. In reality HR-EBSD data comes from small but finite depth, while large elastic strains near dislocations are also highly localised. Does this free surface effect generate significant differences in probability distributions for s'13 and s'23 terms compared to say s'12 ? (ii) the same contrast factor is used for analysis of normal and shear stress probabilities – are there obvious differences in the resulting dislocation content estimates?, and (iii) given the complex dislocation arrangements (multiple line directions, multiple Burger’s vectors etc) what are the likely routes for generating better contrast factors for different stress components available from HR-EBSD?
      • I also found the fractal analysis interesting and should flag my very limit knowledge in this area which may generate naïve comments. There is substantial pre-processing of GND density data to generate binary maps discriminating cell walls and cell interiors. Does analysis of TEM diffraction contrast images present similar challenges or is the less quantitative nature of the data a blessing in this regard? A point I kept returning to was that any study of patterning of cell wall structures ought to have a good (quantitative) definition of what a cell wall is. The multi-step filtering described in section 4.3 indicates that this definition is not completely straight forward and may not be the same here as used for previous work based on TEM micrographs. 
      • Is it possible that the use of multiple length scales (ie sub-area sizes) in defining cell walls itself propagates into the subsequent finding of fractal properties, and/or influences the fractal dimension measured?
      • Adding some histograms into figures 6 and 7 would be helpful, and the number of different sub-area sizes used should be stated.
      • Section 4.1 describes box counting which is useful to those new to fractals (like me!), but appears to not be used in anger. In contrast the Hausdorff dimension is used in the results and discussion without being introduced/defined.
      • (page 16-17) The fractal behaviour must have bounds at high (sample, map size) and low (burger’s vector, pixel size) sizes – the size range corresponding to self-similar fractal scaling should be given.
      • As the relative dislocation density fluctuation increases there may also be an increase in the area fraction of cell wall (is this the case?). Is there a necessary link between cell wall area fraction (f) and the fractal dimension, and does this contribute to the relationship shown in figure 15 (right)?
      • What significance does fractal dimension of 1.8-1.9 for the cell wall structure have in terms of deformation/hardening behaviour. Does it offer ways to distinguish models? 
      • Fig 14 is a very interesting plot bringing together multiple measures of the dislocation content. I thought a little more discussion might be given (i) comparing the lower total dislocation density given by EBSD than XRD and the greater divergence at greater deformation, (ii) comparing GND densities as a fraction of the total dislocation content – the difference is much larger here for single crystals than was reported for polycrystalline Cu (Wilkinson et al Appl. Phys. Lett. 105, 181907 (2014); https://doi.org/10.1063/1.4901219) presumably as a result of strain gradients required to accommodate grain-grain differences in deformation. It seems inappropriate to de-select a data point from this figure on the grounds that XRD and EBSD measurements were carried out at different locations on the sample – this is presumably also the case for the data that is reported.
      • Fig 16 gives an insightful view of the GND data. The dipolar (+/-) nature of some walls compared to others with single sign is clearly shown. I had initially anticipated that two of these maps corresponding to Burgers vectors leading to zero Schmidt factor might be marked less intense than the others but that does not appear to be the case.  


      Minor Points

      • Figure 1 is a really nice depiction of the sample geometry and axes systems used. However, the rest of the document does not always refer back to this with complete consistency – for example EBSD stress maps are given in for sxy in fig 5, s22 in fig 13 but these a presumably referenced to the x’, y’, z’ axes shown in fig 1 (similarly Nye tensor components eg fig 13)
      • Crystallographic notation should be check throughout – [uvw] specific direction, <uvw> family of directions, (hkl) specific plane, {hkl} family of planes
      • C and sigma are used for multiple parameters – it would be better to use other symbols to avoid ambiguity especially for those newer to the field. (C: contrast factors & correlation integral, sigma: stress and relative dislocation density fluctuation)
      • Fig 2 – the line for hardening rate is not red as stated in the text bottom of page 3)
      • A reference for background removal from x-ray peak profiles would be useful for those new to the field (page 5), and exposure times for data acquisition should be given.
      • For HR-EBSD it would be good practice to specify the exposure time used, and the size and number of ROIs used for the cross correlation analysis (page 6-7).
      • In heading of section 3: momentum-> moment
      • Fig 8 shows Taylor hardening model seems to work ok – could give gradient of fit line cf Gb
      • I did not follow the statement on page 14 that the term (1−f)ρc/f/ρw is in the order of unity for high stress – could you add a sentence?


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