# Characterization of Dislocation Rearrangement in FCC Metals during Work Hardening Using X-ray Diffraction Line-Profile Analysis

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{e}, can also be analyzed using the modified method [10,11,12,13]. The strength of the interaction between dislocations can be expressed by defining the dislocation arrangement parameter, M, as follows: $M={R}_{e}\sqrt{\rho}$. When M > 1, the interaction between dislocations is weak. By contrast, M < 1 indicates a stronger interaction between dislocations.

_{e}decreases by forming dislocation dipoles, tangles, and cell structures. Therefore, the dislocation states and substructures can be deduced from the value of M, which comprise R

_{e}.

## 2. Materials and Methods

^{−4}s

^{−1}at room temperature.

_{1}radiation, which was monochromatic (Johansson monochromator). Generally, an instrumental broadening correction was made using the diffraction pattern of LaB

_{6}powder (NIST SRM 660 series) or fully recrystallized metals [14,15]. According to the certificate of SRM 660 series, the diffraction peaks of SRM 660 are slightly broadened due to the domain size of approximately 0.7 μm. While the full width at half maximum (FWHM) of the 220 reflection of a fully recrystallized copper was 0.0439° for Cu Kα

_{1}, the 311 reflection of the LaB

_{6}powder (SRM 660b), which appeared near the 220 reflection of copper, exhibited a larger FWHM of 0.0571°, as shown in Figure 1. To avoid the effect of the domain size on the line-profile broadening, the diffraction pattern of the fully recrystallized copper was used for the standard line profile.

^{15}m

^{−2}was simulated by varying the value of M from 2 to 0.2. As shown in Figure 3a, the FWHM decreased with decreasing M values, and the Gaussian-like peak shifted to a Lorentzian-like peak. Thus, M could be evaluated from the Gaussian width (W

_{G}) and the Lorentzian width (W

_{L}) by the Voigt function. Figure 3b shows the variations in FWHM and ${W}_{\mathrm{L}}/{W}_{\mathrm{G}}$ as a function of M. A relationship between M and ${W}_{\mathrm{L}}/{W}_{\mathrm{G}}$ was observed. It should be mentioned that M decreased with increasing plastic strain as a result of dislocation rearrangement. Consequently, ${W}_{\mathrm{L}}/{W}_{\mathrm{G}}$ increased with increasing the plastic strain. On the other hand, FWHM increased with increasing the plastic strain as a result of an increase in dislocation density.

## 3. Results and Discussion

#### 3.1. Variations in M Values during Tensile Deformation in Nickel and AISI 310S Stainless Steel

^{2}[18] and that of 310S is 86 mJ/m

^{2}, which was calculated through the following expression [19]:

^{2}) = −5.3 + 6.2 (%Ni) + 0.7 (%Cr) + 3.2 (%Mn) + 9.3 (%Mo)

#### 3.2. Effect of Grain Size on M Values

#### 3.3. Effect of SFE on the Variation in M Value

^{2}[24]) was tensile deformed, and the variation in the obtained M values was compared with those of Ni (SFE = 149 mJ/m

^{2}[18]) in Figure 8. The mean grain size of the Cu specimen was approximately 20 μm, which is half of that for the Ni specimen. However, the difference in grain size did not influence the dislocation rearrangement, as demonstrated in Section 3.2. Therefore, we may assume that the difference in M values between Ni and Cu specimens only represents the effect of the corresponding SFEs.

#### 3.4. Effects of Solute Elements on the Variation in M Values

## 4. Conclusions

^{2}), the cell walls of Ni (exhibiting a high SFE of 149 mJ/m

^{2}), were sharply formed at an approximate M value of 0.3. Thus, based on obtained M values and SFEs, the dislocation substructures were observed.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hayashi, M.; Ito, Y.; Takano, K.; Mori, H.; Matsunaga, H.; Onuki, Y.; Suzuki, S.; Sato, S. Effect of kinds of alloying elements in solid-solution copper alloys on dislocation evolution. J. Jpn. Inst. Copper
**2020**, 59, 70–75. [Google Scholar] - Nakagawa, K.; Takano, K.; Matsunaga, H.; Mori, H.; Kitahara, A.; Onuki, Y.; Suzuki, S.; Sato, S. Variation in dislocation-strengthening factors of Cu-Zn alloys with solid-solute zinc contents. J. Jpn. Inst. Copper
**2020**, 59, 59–63. [Google Scholar] - Li, P.; Li, S.X.; Wang, Z.G.; Zhang, Z.F. Fundamental factors on formation mechanism of dislocation arrangements in cyclically deformed fcc single crystals. Prog. Mater. Sci.
**2011**, 56, 328–377. [Google Scholar] [CrossRef] - Zhang, J.; Jiang, Y. An experimental study of the formation of typical dislocation patterns in polycrystalline copper under cyclic shear. Acta Mater.
**2007**, 55, 1831–1842. [Google Scholar] [CrossRef] - Kashyap, B.P.; Tangri, K. On the Hall-Petch relationship and substructural evolution in type 316L stainless steel. Acta Metall. Mater.
**1995**, 43, 3971–3981. [Google Scholar] [CrossRef] - Ungár, T.; Dragomir-Cernatescu, I.; Louër, D.; Audebrand, N. Dislocations and crystallite size distribution in nanocrystalline CeO
_{2}obtained from an ammonium cerium (IV)-nitrate solution. J. Phys. Chem. Solids**2001**, 62, 1935–1941. [Google Scholar] [CrossRef] - Ito, M.; Sato, S.; Ito, Y.; Mori, H.; Matsunaga, H.; Maki, K.; Suzuki, S. Effects of microstructural characteristics on stress relaxation resistance of solid-solution hardened copper alloys. J. Jpn. Inst. Copper
**2017**, 56, 45–50. [Google Scholar] - Sato, S.; Shobu, T.; Satoh, K.; Ogawa, H.; Wagatsuma, K.; Kumagai, M.; Imafuku, M.; Tashiro, H.; Suzuki, S. Distribution and anisotropy of dislocations in cold-drawn pearlitic steel wires analyzed using micro-beam X-ray diffraction. ISIJ Int.
**2015**, 55, 1432–1438. [Google Scholar] [CrossRef][Green Version] - Sato, S.; Wagatsuma, K.; Ishikuro, M.; Kwon, E.P.; Tashiro, H.; Suzuki, S. Precise characterization of dislocations and cementite in pearlitic steels at different drawing strains using X-ray diffraction. ISIJ Int.
**2013**, 53, 673–679. [Google Scholar] [CrossRef][Green Version] - Ungár, T.; Borbély, A. The effect of dislocation contrast on x-ray line broadening: A new approach to line profile analysis. Appl. Phys. Lett.
**1996**, 69, 3173–3175. [Google Scholar] [CrossRef] - Ungár, T.; Ott, S.; Sanders, P.G.; Borbély, A.; Weertman, J.R. Dislocations, grain size and planar faults in nanostructured copper determined by high resolution X-ray diffraction and a new procedure of peak profile analysis. Acta Mater.
**1998**, 46, 3693–3699. [Google Scholar] [CrossRef] - Ungár, T.; Dragomir, I.; Révész, Á.; Borbély, A. The contrast factors of dislocations in cubic crystals: The dislocation model of strain anisotropy in practice. J. Appl. Cryst.
**1999**, 32, 992–1002. [Google Scholar] [CrossRef][Green Version] - Ungár, T.; Tichy, G. The effect of dislocation contrast on x-ray line profiles in untextured polycrystals. Phys. Stat. Solidi A
**1999**, 171, 425–434. [Google Scholar] [CrossRef] - Stokes, A.R. A numeridal Fourier-analysis method for the correction of widths and shapes of lines on X-ray powder photographs. Proc. Phys. Soc.
**1948**, 61, 382–391. [Google Scholar] [CrossRef] - Dey, S.N.; Chatterjee, P.; Sen Gupta, S.P. Study of deformation stacking faults and dislocation microstructures in Cu–1Sn–Zn alloys. Acta Mater.
**2005**, 53, 4635–4642. [Google Scholar] [CrossRef] - Ribárik, G.; Gubicza, J.; Ungár, T. Correlation between strength and microstructure of ball-milled Al–Mg alloys determined by X-ray diffraction. Mater. Sci. Eng. A
**2004**, 387, 343–347. [Google Scholar] [CrossRef] - Balogh, L.; Ribárik, G.; Ungár, T. Stacking faults and twin boundaries in fcc crystals determined by x-ray diffraction profile analysis. J. Appl. Phys.
**2006**, 100, 023512. [Google Scholar] [CrossRef] - Narita, N.; Hatano, A.; Takamura, J.; Yoshida, M.; Sakamoto, H. Stacking Fault Energy of Pure Nickel Evaluated from the Twinning Stress of Nickel-Based Alloys. J. Jpn. Inst. Met. Mater.
**1978**, 42, 533–541. [Google Scholar] [CrossRef][Green Version] - Schramm, R.E.; Reed, R.P. Stacking fault energies of seven commercial austenitic stainless steels. Metall. Trans. A
**1975**, 6, 1345–1351. [Google Scholar] [CrossRef] - Narutani, T.; Takamura, J. Grain-size strengthening in terms of dislocation density measured by resistivity. Acta Metall. Mater.
**1991**, 39, 2037–2049. [Google Scholar] [CrossRef] - Umezaki, S.; Murata, Y.; Nomura, K.; Kubushiro, K. Quantitative analysis of dislocation density in an austenitic steel after plastic deformation. J. Jpn. Inst. Met. Mater.
**2014**, 78, 218–224. [Google Scholar] [CrossRef][Green Version] - Nakagawa, K.; Hayashi, M.; Takano-Satoh, K.; Matsunaga, H.; Mori, H.; Kitahara, A.; Onuki, Y.; Suzuki, S.; Sato, S. Effect of grain size of Cu-Zn alloys on dislocation-strengthening factors. J. Jpn. Inst. Met. Mater. To be published.
- Hall, E.O. The deformation and ageing of mild steel: III Discussion of results. Proc. Phys. Soc. Sec. B
**1951**, 64, 747–753. [Google Scholar] [CrossRef] - Zhao, Y.H.; Horita, Z.; Langdon, T.G.; Zhu, Y.T. Evolution of defect structures during cold rolling of ultrafine-grained Cu and Cu-Zn alloys: Influence of stacking fault energy. Mater. Sci. Eng. A
**2008**, 474, 342–347. [Google Scholar] [CrossRef] - Taylor, G. The mechanism of plastic deformation of crystals. Patr I-Theoretical. Proc. R. Soc.
**1934**, 145, 362–387. [Google Scholar] - Simm, T.H. Peak broadening anisotropy and the contrast factor in metal alloys. Crystals
**2018**, 8, 212. [Google Scholar] [CrossRef][Green Version] - Simm, T.H.; Withers, P.J.; Quinta da Fonseca, J. Peak broadening anisotropy in deformed face-centered and hexagonal closed-packed alloys. J. Appl. Cryst.
**2014**, 47, 1535–1551. [Google Scholar] [CrossRef] - Simm, T.H.; Withers, P.J.; Quinta da Fonseca, J. An evaluation of diffraction peak profile analysis (DPPA) methods to study plastically deformed metals. Mater. Design
**2016**, 111, 331–341. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Convolutional multiple whole profile (CMWP) fitting for XRD patterns of the Cu-30% Zn specimen at a true strain of 0.34.

**Figure 3.**(

**a**) Simulation of the 220 Cu reflection at a dislocation density of 2 × 10

^{15}m

^{−2}, with varying M values from 2 to 0.2; (

**b**) variations in full width at half maximum (FWHM) of the 220 reflection for Cu Kα

_{1}and ${W}_{\mathrm{L}}/{W}_{\mathrm{G}}$ as a function of M.

**Figure 4.**Inverse pole figure (IPF) maps before tensile deformation of the specimens: (

**a**) 310S and (

**b**) Ni.

**Figure 5.**True stress-strain curves for 310S and Ni. Specimens were deformed at the strains denoted by each curve marks to conduct XRD measurements.

**Figure 6.**Variations due to true strain for the 310S and Ni specimens in (

**a**) FWHMs and ${W}_{\mathrm{L}}/{W}_{\mathrm{G}}$ of 220 reflection; (

**b**) dislocation density and M.

**Figure 7.**TEM images of tensile-deformed 310S (

**a**–

**d**) and Ni (

**e**–

**g**) at true strains (ε

_{t}) of 0.05, 0.12, 0.18, 0.41, 0.05, 0.10, and 0.34, respectively.

**Figure 9.**IPF maps for (

**a**) Cu-30Zn_5, (

**b**) Cu-30Zn_20, and (

**c**) Cu-30Zn_60 specimens before tensile deformation.

**Figure 10.**True stress–strain curves for Cu-30Zn_5, Cu-30Zn_20, and Cu-30Zn_60 [22]. Specimens were deformed at the strains denoted by each curve marks to conduct XRD measurements.

**Figure 11.**(

**a**) Variations of dislocation density and M with respect to true strain [22]; (

**b**) M as a function of dislocation density.

**Figure 13.**IPF maps for (

**a**) Cu-2Mg, (

**b**) Cu-2Sn, (

**c**) Cu-2Si, (

**d**) Cu, and (

**e**) Cu-10Zn specimens before tensile deformation.

**Figure 15.**Influence of varying dislocation density on M values for (

**a**) Cu-2X and (

**b**) Cu-x Zn specimens (X = Mg, Sn, Si; x, in mass% = 0, 10, 30.).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Nakagawa, K.; Hayashi, M.; Takano-Satoh, K.; Matsunaga, H.; Mori, H.; Maki, K.; Onuki, Y.; Suzuki, S.; Sato, S.
Characterization of Dislocation Rearrangement in FCC Metals during Work Hardening Using X-ray Diffraction Line-Profile Analysis. *Quantum Beam Sci.* **2020**, *4*, 36.
https://doi.org/10.3390/qubs4040036

**AMA Style**

Nakagawa K, Hayashi M, Takano-Satoh K, Matsunaga H, Mori H, Maki K, Onuki Y, Suzuki S, Sato S.
Characterization of Dislocation Rearrangement in FCC Metals during Work Hardening Using X-ray Diffraction Line-Profile Analysis. *Quantum Beam Science*. 2020; 4(4):36.
https://doi.org/10.3390/qubs4040036

**Chicago/Turabian Style**

Nakagawa, Koutarou, Momoki Hayashi, Kozue Takano-Satoh, Hirotaka Matsunaga, Hiroyuki Mori, Kazunari Maki, Yusuke Onuki, Shigeru Suzuki, and Shigeo Sato.
2020. "Characterization of Dislocation Rearrangement in FCC Metals during Work Hardening Using X-ray Diffraction Line-Profile Analysis" *Quantum Beam Science* 4, no. 4: 36.
https://doi.org/10.3390/qubs4040036