Determination Approach of Dislocation Density and Crystallite Size Using a Convolutional Multiple Whole Profile Software

Keywords:
line profile analysis,
whole profile fitting,
neutron diffraction,
780-MPa grade bainitic steel,
dislocation density,
crystallite size

2018 Volume 59 Issue 7 Pages 1135-1141

Details

Abstract

Neutron diffraction profile analysis using the whole profile fitting method is useful for obtaining microscopic information on metallic materials. To determine an appropriate fitting approach for obtaining reasonable and non-arbitrary results, we applied diffraction line profile analyses using the Convolutional Multiple Whole Profile (CMWP) method to diffraction patterns obtained using the Engineering Materials Diffractometer (TAKUMI, BL19) at the Materials and Life Science Facility (MLF) of the Japan Proton Accelerator Research Complex (J-PARC). The tensile specimens of 780 MPa grade bainitic steel were uniaxially stretched until the plastic strain reached a value of 0.05. We performed CMWP analyses on the obtained diffraction patterns during tensile test with various initial parameters of dislocation density and crystallite size. These parameters were optimized in the fitting procedures to minimize the weighted sums of squared residuals (WSSRs). Following this approach, we found that unsuitable initial parameter values resulted in unreasonable convergence. Therefore, initial fitting parameters should be chosen to ensure that the initial profiles are as broad as possible. Reasonable results were obtained following this suggestive approach even when the strain anisotropy parameter is set to arbitrary values.

Fig. 8 Converged dislocation density ρ and crystallite size *L*_{0} at various strain anisotropy parameter *q*. (a) Comparison among all the analyses results. (b) Comparison between before tensile tests without stress relaxation and with stress relaxation. (c) Effect of stress relaxation. In (b) and (c), pairs of (*q*, WSSR) are shown and underlined ones are of tensile tests with stress relaxation.

1. Introduction

Line profile analysis is useful for evaluating the microstructural characteristics of metals and the other crystalline materials. This theory has been well understood since the 1950s, for example, Williamson-Hall method^{1}^{)} and Warren-Averbach method,^{2}^{)} which provide dislocation density and crystallite size. However, it is problematic to apply these classical methods to some metals, such as steel and copper, which have elastic anisotropy. Furthermore, there is a relationship between scattering vector and dislocation direction and these factors are not considered in the analyses. To overcome this, Ungár proposed the Modified Williamson-Hall and Warren-Averbach method.^{3}^{)} These proposed techniques represent a turning point for using line profile analyses in materials science and engineering research. In this method, microstructural parameters are determined by independently fitting each diffraction peak. This method has been further developed to determine microstructural parameters more accurately using whole profile fitting procedures, which determine microstructural parameters by fitting multiple diffraction peaks simultaneously, and have attracted attention in recent years.^{4}^{–}^{8}^{)}

X-ray analyses are typically used in line profile analysis.^{9}^{–}^{11}^{)} However, only near-surface microstructural information of material is obtained from X-ray analyses due to the small penetration ability of X-rays into metals. Diffraction profile analyses using neutron beams, therefore, have recently emerged as a method to characterize material microstructures. This is because neutron diffraction profile analyses gives bulk microstructure information in the large gauge volume and the information correlates well with macroscopic mechanical behavior properties of materials.^{12}^{)} Thus, neutron diffraction profile analysis enables *in-situ* and non-destructive observation during material testing, such as uniaxial tensile tests.

The whole profile fitting method, which is one of the diffraction profile analyses, is compatible with time-of-flight (TOF) methods at pulsed neutron source since multiple diffraction peaks can be measured simultaneously. Recently, the Convolutional Multiple Whole Profile (CMWP) fitting method has become more widespread for line profile analysis.^{6}^{)} CMWP fitting is performed using the CMWP software developed by G. Ribárik and T. Ungár (available from a reference^{13}^{)}). In the CMWP fitting method, however, several free parameters related to microstructural characteristics must be optimized by fitting the whole diffraction pattern. This means that the convergent solution may vary depending on the analysis procedure because there are many combinations that can be determined by a number of fixed parameters, order of fixing parameters, values of initial parameters, etc. Therefore, the results of analyses may bring an unreasonable local solution and the users must decide whether the convergent solution is a local or an optimum solution. Although, line profile analysis experts in the materials science could appropriately determine the optimum solution, this may be difficult for less-experienced scientists and engineers. Therefore, a simple and useful procedure for determining the optimum solution must be established. To archive this, it is important to first investigate the method to obtain converged dislocation density and crystallite size because these parameters mainly affect the shape of diffraction profile, as investigated using conventional Williamson-Hall method. In determining dislocation density and crystallite size, it is also important to investigate the effect of strain anisotropy parameter because strain anisotropy parameter changes with plastic deformation.^{14}^{)} Therefore, we must investigate the effect of strain anisotropy parameter in the proposed procedure.

In this study, we propose a novel procedure to obtain reasonable and non-arbitrary results for neutron diffraction profiles analysis using the CMWP method. In the parameter space of dislocation density and crystallite size, we investigate the weighted sum of squared residuals (WSSRs) between the theoretical and measured profiles to obtain optimum convergence. To confirm the accuracy of the converged dislocation density and crystallite size, we compare the change of converged parameters during tensile testing and stress relaxation. We also investigate the effect of strain anisotropy parameter on the converged dislocation density and crystallite size.

2. Experimental Procedure

In the experiments, tensile specimens of commercial 780-MPa grade steel were prepared so that the longitudinal direction of specimen was parallel to the rolling direction. The steel has composition of 0.09C–0.7Si–1.4Mn–Bal.Fe and was received as hot-rolled steel. The dimensions of the specimens are shown in Fig. 1.

Fig. 1

Specimen dimensions.

Neutron diffraction measurements were conducted using the engineering materials diffractometer (TAKUMI, BL19) at the Materials and Life Science Facility (MLF), Japan Proton Accelerator Research Complex (J-PARC). The accelerated proton beam power was approximately 500 kW. The optical setup is shown in Fig. 2. The specimens were attached to a tensile test machine in the diffractometer. As the center of each specimen was irradiated by a pulsed neutron beam, the strain gauge was attached 5 mm away from the center (Fig. 1). The angle between the longitudinal direction of the specimen and the incident neutron beam was 45°. The two detectors were oriented at +90° and −90° relative to the incident beam direction. The diffraction profiles were acquired by the +90° detector to obtain the diffraction pattern changes in the longitudinal direction. The neutron diffraction measurements were conducted during the tensile tests, which are discussed in the following paragraph.

Fig. 2

Setup of the tensile testing machine and detectors.

Two types of tensile tests were conducted: (i) a tensile test with stress relaxation where crosshead movement was held for a given duration at a prescribed strain, and (ii) a tensile test without stress relaxation where the specimen was immediately unloaded when a prescribed strain was attained. Figure 3 shows stress variations with time during the tensile tests. As previously reported,^{15}^{)} stress relaxation is observed because the stress slightly decreases under the constant crosshead displacement. To obtain a diffraction profile which has a statistic that is adequate for performing line profile analyses, the strain of the samples must be kept constant for a certain period of time during the diffraction measurement. However, the microstructure is dynamic during tensile tests. Then the diffraction profiles obtained during the tensile tests exhibited no significant variations due to the lack of statistics. Therefore, we analyzed only four diffraction profiles, which were obtained before and after each tensile test. In the experiments, a load of 300 N was applied to each sample during the first 900 s and the neutron diffraction measurements were performed. This load, which was small enough to avoid any plastic deformation, was applied to remove the clearance in the tensile testing machine. A tensile strain was then applied up to a nominal strain value of 0.05 at a strain rate of 10^{−4} s^{−1}. After the nominal strain reached a value of 0.05, the applied load was immediately decreased to 300 N in the tensile test without stress relaxation. Neutron diffraction measurements were then performed for 900 s. In contrast, the crosshead was stopped for 7200 s when the nominal strain reached a value of 0.05 in the tensile test with stress relaxation. Then, the applied load was decreased to the initial load of 300 N, and neutron diffraction measurements were performed for 900 s.

Fig. 3

Variation of stress before, during, and after the tensile tests. Time period of diffraction profile analyses was depicted.

The neutron diffraction profiles were analyzed using the CMWP software.^{13}^{)} The diffraction peak indices of 110, 200, 211, 220, 310, and 222 were employed for the analyses. In the CMWP method, each measured diffraction pattern was fitted by a theoretical diffraction pattern calculated with five fitting parameters (strain anisotropy parameter *q*, variance of the lognormal crystallite size distribution σ_{LN}, effective outer cut-off radius of dislocation *R*_{e}*, dislocation density ρ, and crystallite size *L*_{0}) through performing iterative calculations. These five parameters correspond to the parameters *a*, *b*, *c*, *d*, and *e* in the CMWP software as follows:

\begin{equation} q = a \end{equation} | (1) |

\begin{equation} \sigma_{\text{LN}} = \frac{c}{\sqrt{2}} \end{equation} | (2) |

\begin{align} &\rho = \frac{2}{\pi(b_{\text{Burgers}}d)^{2}}\ \\& (\text{$b_{\text{Burgers}}$: Magnitude of the Burgers vector}) \end{align} | (3) |

\begin{equation} R_{\text{e}}^{*} = \cfrac{\exp\biggl(-\cfrac{1}{4}\biggr)}{2e} \end{equation} | (4) |

\begin{equation} L_{0} = \frac{2}{3}\exp\left(\frac{5}{4}c^{2} + b\right). \end{equation} | (5) |

In the CMWP analyses, firstly, the transition of fitting parameters in the first few iteration steps were investigated to approximately estimate the optimum convergence value. Then, the iteration number was set to three. This iteration number was ideal for evaluating transitions of fitting parameters, because analyses with less than two iteration steps were not sufficient for observing the transition of fitting parameters and the calculation time required for analyses with greater than four iteration steps were too long. The analysis conditions are summarized in Table 1. The transition of crystallite size and dislocation density at each iteration step was recorded. The WSSRs for each pair of dislocation density and crystallite size were recorded to evaluate the fitting quality. The fixed parameters *q*, σ_{LN}, and *R*_{e}* were set to values that would have realistic orders of magnitude. Initially, the *a* value was set to 1.96, which is the averaged value of *q* for edge (*q* = 1.29) and screw (*q* = 2.64) dislocations. The *q* values were computed by ANIZC.^{16}^{)} Subsequently, the parameter σ_{LN}, which is related to *c*, has an order of magnitude of 10^{−1} or 10^{0}, as depicted in reports using steels.^{17}^{,}^{18}^{)} Then, the parameter *c*, which is related to σ_{LN}, was set to a value of 0.7. Finally, the dislocation arrangement parameter *M** is expressed as:

\begin{equation} M^{*} = R_{\text{e}}{}^{*}\sqrt{\rho } \end{equation} | (6) |

Table 1 CMWP analysis conditions with three iterations.

Through the aforementioned analyses, we obtained the initial dislocation density ρ^{i} and crystallite size *L*_{0}^{i}, which enable iterative calculation to be stable. The CMWP analyses were then conducted iteratively until the fitting parameters converged. The analysis conditions are summarized in Table 2. As a control experiment, the CMWP analysis with an initial dislocation density ρ^{ii} and a crystallite size *L*_{0}^{ii}, which was not predicted to converge the iterative calculations, was conducted. Fixed parameters σ_{LN} and *R*_{e}* were set to the same value as the analyses described in Table 1. The strain anisotropy parameter *q* was set to values of 1.29, 1.96, and 2.64. The transition of fitting parameters and WSSRs were obtained from these analyses.

Table 2 CMWP analysis conditions. Iterative calculations were repeated until the fitting parameters converged.

3. Results and Discussion

3.1 Proposal of analysis approach to obtain a non-arbitrariness convergence
Figure 4 shows the transition of fitting parameters in each of three iterations of CMWP analysis that were performed with various initial values. The pairs of initial dislocation density and crystallite size are marked with round solid plot (symbol: ●). These fitting parameters were transitioned with iteration steps. Then pairs of fitting parameters at first iteration step are marked with round hollow plot (symbol: ○). Similarly, the symbols representing fitting parameters at each iteration step are defined in Fig. 4. Although the iteration step was set to a value of three, some analyses were repeated for four iterations unintentionally due to the performance of the CMWP software. The pairs of initial dislocation density and crystallite size are divided into three groups (Group A, B, and C) according to the characteristics of the transition fitting parameters. During the analysis with initial parameter values in Group C, the fitting parameters varied in the direction parallel to the horizontal (associated with crystallite size) and vertical axis (associated with dislocation density). The ranges of initial crystallite size 1.84 ≤ *L*_{0} ≤ 1846 nm and initial dislocation density 10^{12} ≤ ρ ≤ 10^{16} m^{−2} are realistic orders of magnitude that are common for steels.^{14}^{,}^{17}^{–}^{20}^{)} The fitting parameters of initial value in Group C transitioned within these ranges. Therefore, the analysis with initial parameters in Group C possibly results in stable iterative calculations and optimum convergence. In contrast, the fitting parameters did not vary in directions parallel to both horizontal and vertical axis in the analyses with initial parameter values in Group A. In the analyses with initial parameter values in Group B, the dislocation densities did not vary in iterative calculations, but the crystallite size varied. These results indicate that large initial dislocation density can lead to realistic solution, but small initial dislocation density can lead to unreasonable local solutions. Moreover, in Group C, even though the parameters varied in all initial dislocation densities, small initial crystallite size varied greater than large crystallite size. The tendency is clear in Groups A and B as the parameters varied toward horizontal axis direction in Group B, but did not vary in Group A. These results indicate that small initial crystallite size may lead to reasonable solution, but large initial crystallite size may lead to local solutions. In general line profile analyses, the peak broadening is used to measure dislocation density and crystallite size as a large dislocation density or a small crystallite size broadens the width of their diffraction profile. Therefore, it was suggested that initial fitting parameters should be selected to ensure that the initial profile is as broad as possible.

Fig. 4

Transition of fitting results using the CMWP method after tensile test without stress relaxation with three iterations. Initial dislocation densities were 10^{12}, 10^{13}, 10^{14}, 10^{15}, and 10^{16} m^{−2} and initial crystallite sizes were (a) 1.84, (b) 18.5, (c) 184, (d) 922, and (e) 1846 nm.

Figure 5 illustrates the analysis processes with two conditions: (i) an initial dislocation density of ρ^{i} = 10^{14} m^{−2} and a crystallite size of *L*_{0}^{i} = 1.84 nm, which are chosen according to the suggestive approach in Fig. 4, and (ii) an initial dislocation density of ρ^{ii} = 10^{12} m^{−2} and crystallite size of *L*_{0}^{ii} = 18.4 nm, which is in Group B (Fig. 4). Note that the dislocation density is not predicted to change during iterative calculations in condition (ii). Representatively, results of after tensile test without stress relaxation is shown because the other results demonstrated quantitatively similar results. In these analyses, the parameter *a* was set to a value of 1.96, and the iterative calculation was repeated until the iterative calculation converged. In the analysis with condition (ii), as expected from the analyses of Group B in Fig. 4, the dislocation density was constant throughout the iterative calculation. In contrast, in the analysis with condition (i), the dislocation density and crystallite size varied and converged to parameter values that were considered to be empirically reasonable. Less than 39 iterations were required for completing the four diffraction patterns. The WSSRs for all pairs of dislocation density and crystallite size obtained in the above process in Figs. 4 and 5 are shown in Fig. 6. The WSSRs with converged values in Fig. 5 were less than 0.85 and had the lowest values of all pairs of dislocation density and crystallite size. Therefore, this suggest that the converged parameter values should be the optimum. Figure 6 shows that the gradient of WSSR around Group A is relatively small, which explains the lack of consequent variation in both parameters. Since the gradient of WSSRs in the direction of the horizontal axis was steeper than in the direction of vertical axis around Group B, the only parameter of crystallite size varied around Group B. Figure 7 presents the measured and fitted profile obtained from the analyses of condition (i). Figure 7 shows that the theoretical and measured profiles agree well.

Fig. 5

Transition of fitting results obtained using the CMWP method after tensile test without stress relaxation with condition (i): initial dislocation density of 10^{14} m^{−2} and crystallite size of 1.84 nm and condition (ii): initial dislocation density of 10^{12} m^{−2} and crystallite size of 18.4 nm. The iterative calculations were repeated until the fitting parameters converged.

Fig. 6

WSSR distributions in the parameter space of dislocation density and crystallite size after tensile test without stress relaxation.

Fig. 7

Measured and CMWP fitted diffraction patterns after tensile test without stress relaxation (a) and the difference between experimental result and fitting one (b).

In the aforementioned discussion, the converged dislocation density and crystallite size were obtained, but strain anisotropy parameter was fixed to a value of *q* = 1.96. However, Umezaki *et al.* reported that the strain anisotropy parameter varies during plastic deformation.^{14}^{)} Therefore, we must consider whether the aforementioned approach for obtaining the optimum convergences can be applied for any strain anisotropy parameter. Furthermore, the effect of strain anisotropy parameter on the dislocation density and crystallite size should also be considered.

Figure 8 shows the change in convergence values with some strain anisotropy parameters where the error bars represent asymptotic standard errors and analyses conditions are shown in Table 2. In these analyses, the initial values of dislocation density and crystallite size were ρ^{i} = 10^{14} m^{−2} and *L*_{0}^{i} = 1.84 nm, respectively. These initial values were obtained from the suggested values determined from the analyses in Fig. 4. Figure 8 shows that the iterative calculations at any *q* value were performed stably and reached the convergence although the initial fitting parameters were decided from the investigation at *q* = 1.96 in Fig. 4. This suggests that the proposed approach to decide the initial fitting parameters at one specific *q* value can be applied to analyses at any strain anisotropy parameter.

Fig. 8

Converged dislocation density ρ and crystallite size *L*_{0} at various strain anisotropy parameter *q*. (a) Comparison among all the analyses results. (b) Comparison between before tensile tests without stress relaxation and with stress relaxation. (c) Effect of stress relaxation. In (b) and (c), pairs of (*q*, WSSR) are shown and underlined ones are of tensile tests with stress relaxation.

The WSSR was observed to be the lowest when the value of *q* was 1.29. However, this condition cannot be assured in case of dislocation of a specimen in an as-rolled condition as well as for a specimen that depicts a strain of 0.05. This is because *q* = 1.29 indicates that the only type of dislocation observed is the edge dislocation. One of possible ways for obtaining true values of ρ, *L*_{0}, and *q* is to fix *q* value obtained using other method, *e.g*. modified Willamson-Hall method.^{3}^{)}

Changes in the dislocation density and crystallite size were roughly obtained although there is an possibility that the analyses cannot produce true value of ρ, *L*_{0}, and *q*. In Fig. 8(a), comparing the results from before tensile tests (round plots: ○ and ●) and with those from after tensile tests (square plots: □ and ■), the increase in dislocation density and decrease in crystallite size with tensile tests were observed in each *q* value. This implies that general plastic deformation behavior can be holistically observed. In Fig. 8(b), the results from before tensile test without stress relaxation (hollow round plots: ○) and with stress relaxation (solid round plots: ●) were not noticeably different as the error bars at same *q* value generally overlapped. This result reflects the irregularity of the measured profile for each sample or measurement condition. In Fig. 8(c), results from after tensile test without stress relaxation (hollow square plots: □) and with stress relaxation (solid square plots: ■) shows the change in the parameters due to stress relaxation. By only considering the results at a constant *q* value, the crystallite sizes were not significantly different since the error bars almost overlapped. However, a decrease in dislocation density with stress relaxation was observed since the error bars did not overlap. The decrease in dislocation density during stress relaxation qualitatively agrees with previous TEM observations.^{21}^{,}^{22}^{)} However, the strain anisotropy parameter *q* may possibly change during stress relaxation. If this is the case, the results of Fig. 8(c) only show the possibility of a change in *q* or ρ with stress relaxation.

In this study, although we focused on the change in dislocation density and crystallite size, which strongly affect the shape of diffraction profile, the other parameters must be discussed to evaluate the change in microstructure in detail. Our proposed approach will be helpful further analyses that aim to obtain the initial dislocation density and crystallite size.

4. Conclusion

We performed neutron diffraction profiles analyses using CMWP method to develop a simple procedure for obtaining reasonable and non-arbitrary results. The software used to perform CMWP analysis was developed by G. Ribárik and T. Ungár. The following conclusions are drawn from using CMWP analyses with various initial parameter values.

- (1) Analyses results with several initial values show that small initial dislocation density and large initial crystallite size values may lead to unreasonable local solutions. To resolve this, fitting parameters should be selected so that the initial profile is as broad as possible. Initial values, which are chosen according to this approach, yields reasonable and non-arbitrary convergence values even when the strain anisotropy parameter is set to arbitrary values.
- (2) The analysis result according to the aforementioned method indicates that there is a change in microstructural parameters during plastic deformation and stress relaxation. In this study, we were only concerned with two significant parameters, namely dislocation density and crystallite size. It is a future objective to further investigate the effect of other parameters to evaluate change in microstructural parameters.

Acknowledgments

The neutron experiment was performed at the Materials and Life Science Experimental Facility of the J-PARC under a user program (Proposal No. 2015A0294). We deeply thank Dr. Harjo and Dr. Kawasaki at J-PARC for their experimental assistance. This work was partially supported by Photon and Quantum Basic Research Coordinated Development Program of MEXT, Japan, Research Group of the Iron and Steel Institute of Japan, and JSPS KAKENHI Grant No. 25289265. We would like to acknowledge the experimental assistance of Ms. Saori Iwata at Tokyo City University and Mr. Hayato Komine at Waseda University.

REFERENCES

- 1) G.K. Williamson and W.H. Hall: Acta Metall.
**1**(1953) 22–31. - 2) B.E. Warren and B.L. Averbach: J. Appl. Phys.
**21**(1950) 595–599. - 3) T. Ungár and A. Borbély: Appl. Phys. Lett.
**69**(1996) 3173–3175. - 4) G. Ribárik, T. Ungár and J. Gubicza: J. Appl. Cryst.
**34**(2001) 669–676. - 5) T. Ungár, J. Gubicza, G. Ribárik and A. Borbély: J. Appl. Cryst.
**34**(2001) 298–310. - 6) G. Ribárik: Ph.D thesis, Eötvös Lorand University, (2008).
- 7) P. Scardi and M. Leoni: Acta Crystallogr. Sec. A
**58**(2002) 190–200. - 8) Y.H. Dong and P. Scardi: J. Appl. Cryst.
**33**(2000) 184–189. - 9) M. Kumagai, K. Akita, M. Imafuku and S. Ohya: Adv. Mater. Res.
**996**(2014) 39–44. - 10) D. Akama, T. Tsuchiyama and S. Takaki: ISIJ Int.
**56**(2016) 1675–1680. - 11) S. Sato, K. Wagatsuma, M. Ishikuro, E. Kwon, H. Tashiro and S. Suzuki: ISIJ Int.
**53**(2013) 673–679. - 12) Y. Tomota, P. Lukas, S. Harjo, J.-H. Park, N. Tsuchida and D. Neov: Acta Mater.
**51**(2003) 819–830. - 13) “Convolutional Multiple Whole Profile fitting Main Page”. G. Ribárik and T. Ungár. http://csendes.elte.hu/cmwp, (accessed 2018-02-06).
- 14) S. Umezaki, Y. Murata, K. Nomura and K. Kubushiro: J. Japan Inst. Met. Mater.
**78**(2014) 218–224. - 15) K. Murasawa, H. Komine, Y. Otake, H. Sunaga, M. Takamura, Y. Ikeda and S. Suzuki: Key Eng. Mater.
**716**(2016) 948–953. - 16) A. Borbély, J. D.-Cernatescu, C. Ribarik and T. Ungar: J. Appl. Crystallogr.
**36**(2003) 160–162. - 17) T. Ungár, M. Victoria, P. Marmy, P. Hanák and G. Szenes: J. Nucl. Mater.
**276**(2000) 278–282. - 18) Y.Z. Chen, G. Csiszár, J. Cizek, S. Westerkamp, C. Borchers, T. Ungár, S. Goto, F. Liu and R. Kirchheim: Metall. Mater. Trans. A
**44**(2013) 3882–3889. - 19) Z.M. Shi, W. Gong, Y. Tomota, S. Harjo, J. Li, B. Chi and L. Pu: Mater. Charact.
**107**(2015) 29–32. - 20) D. Akama, T. Tsuchiyama and S. Takaki: ISIJ Int.
**56**(2016) 1675–1680. - 21) L. Xiao and J.L. Bai: Mater. Sci. Eng. A
**244**(1998) 250–256. - 22) M.S. Mohebbi, A. Akbarzadeh, Y.-O. Yoon and S.-K. Kim: Mech. Mater.
**89**(2015) 23–34.

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