*3.3. X-ray Diffraction Analysis of the Powders*

Several XRD patterns gathered at various stages of MA are shown in Figure 5a. As can be seen at the early stage of MA (5 h), the microstructure was purely bcc, and the main Fe-bcc phase was accompanied by several reflections of Mn-bcc. After 5 h of milling, no significant change in microstructure occurred as compared to the initial powder mixture (Figure 5c), besides broadening of reflections, and Mn peaks were still present. Eventually, after 30 h of MA, Mn reflections vanished, suggesting that it was finally dissolved in the Fe matrix. At this point, the Mn-induced phase transformation process was obvious, as the bcc peak intensities dropped drastically and fcc peaks became dominant (Figure 5a,c). After 60 h of MA, the bcc reflections were nearly elusive, with the exception of the most prominent [110] Fe-bcc peak. Prolonged milling caused a further increase in the intensity of the Fe-fcc phase. The Fe-bcc [110] peak trace, however, was still detectable even after 150 h of MA (Figure 5d).

**Figure 5.** Microstructural evolution of Fe-18Cr-18Mn alloy: (**a**) overall comparison of XRD patterns after a certain MA time, (**b**) insight showing the vanishing of Fe-bcc peaks, (**c**) comparison of initial and after 5 h of MA patterns with emphasis on Mn-bcc reflections, (**d**) changes of the shape and intensity of most prominent Fe-bcc [110] and Fe-fcc [111] reflections during the MA course.

XRD patterns were fitted using the pseudo-Voigt function and the extracted data (peak 2*θ* position and FWHM) were processed in accordance with the MWH procedure, using the calculated contrast factor values for Fe-bcc and Fe-fcc phases, and regarding the instrumental broadening correction and proper background definition. The phase percentage was estimated by the ratio of integrated areas of peak profiles of Fe-bcc and Fe-fcc phases, respectively, as presented in Figure 6.

**Figure 6.** Phase percentage of Fe-fcc.

Extracting information from the experimental peak profiles, where multiple phases exist simultaneously, outlines the major problem concerning the traditional LPA methods—overlapping among peak profiles (especially within the region of Fe-bcc [110] and Fe-fcc [111] peaks). The most common solution to this difficulty is pattern decomposition—separating the individual contributions from different phases and subtracting the instrumental and background effects.

The main results of the MWH analysis are collated in Figure 7. The experimental values of *q*, which imply the edge/screw characters of dislocation, were evaluated using the methodology provided by Ungár et al. [18] (Figure 7a). The FWHM values were plotted according to the MWH relation (Figure 7b), where the intersection at *KC*1/2 = 0 gave the average crystallite size *d*. It is clear that the data points follow the linear trend in almost the exact manner and, therefore, were fitted using the linear function. It should be underlined that, due to the progressive bcc→fcc phase transformation over the MA time, the reliable determination of FWHM of Fe-bcc reflections was rather unobtainable after 30 h of MA, caused by the strong broadening and low intensity. Although after 30 h of MA, there were still three apparent Fe-bcc reflections present (Figure 5a), they were not enough to calculate *d* accurately using the MWH method.

As presented in Figure 7c, the domain size was reduced from ~20 nm to around 10 nm during the first 60 h of MA and saturates at this level until the end of milling. The slope of the MWH linear approximation is also plotted in Figure 7c, which can be related to the dislocation arrangement parameter *M* and the square root of dislocation density *ρ*. However, the *M* can be determined thoroughly only with the Fourier methods, such as WPPM. Consequently, the determination of *ρ* from the results obtained by MWH alone is improper and should be avoided [38]. Nevertheless, the inspection of the MWH slope provides some insight into the strain accumulation in the material, and it can be noticed that it mimics the trend of *d* in the opposite manner (Figure 7c). Finally, the dislocation character trend was plotted in the form of the edge dislocation fraction *fe* (Figure 7d). In general, the dislocation structure is in favor of screw dislocations over the whole MA process, which might be caused by the rotational nature of the ball milling process, promoting the formation of twists in crystals [39]. The *fe* value fluctuates in the 0.2–0.4 region and never exceeds the half edge–half screw line. This supports the other authors' conclusions that edge dislocations are unstable in fine Fe domains [24,40].

**Figure 7.** The principal result of the MWH analysis: (**a**) determination of experimental *q* values, (**b**) FWHM values plotted according to the MWH procedure, (**c**) evolution of crystallite size and MWH slope, (**d**) fraction of edge dislocations during MA.

The diffraction data were furtherly analyzed by the WPPM approach and the quality of fit can be perceived in Figure 8. Plots present the experimental data compared with model WPPM diffractograms, further decomposed into single analyzed phases (Mn-bcc, Fe-bcc, and Fe-fcc). A flat, nearly featureless residual line (the difference between experimental and calculated patterns) proves good quality of modeling. The graphical analysis is essential to determine the quality of fit and to ensure that the model is chemically plausible. The quality of WPPM fit, such as in other least-squares minimization methods, can also be defined by inspecting the discrepancy factors. In this study, the most straightforward discrepancy index, the weighted profile *R*-factor, *Rwp*, did not exceed 3.46%, which indicates a robust quality of fit, particularly considering the multiplicity of phases and overlapping peaks. Another marker, goodness of fit (GoF, related to the ratio between actual and expected *Rwp*) had a maximum of 1.52 (pattern after 5 h of MA) and a mean value of 1.21 (where unity indicates the flawless fit).

Unlike the MWH method, the pattern is not decomposed into individual line profiles, but the whole diffractogram is modeled by optimizing the value of several physical parameters. Figure 8a presents the pattern obtained after 5 h of MA, with a detailed view of multiple, minor Mn-bcc reflections, successfully refined using WPPM. Figure 8b–d magnify the most interesting, from the modeling viewpoint, 2*θ* region, where prominent Fe-bcc [110] and Fe-fcc [111] peaks extensively overlap. Additionally, inserts show the fit quality of the remaining reflections.

**Figure 8.** WPPM-refined experimental patterns gathered after: (**a**) 5 h of MA, (**b**) 30 h of MA, (**c**) 60 h of MA, (**d**) 150 h of MA. Legend: hollow dot—experimental data, red line—WPPM model, orange line—Mn-bcc phase, blue line—Fe-bcc phase, green line—Fe-fcc phase, black line—residual.

Basic results derived from WPPM refinement, in terms of size and strain evolution during MA, are revealed in Figure 9. It is a known fact that MA prompts the volume expansion of the unit cell, predominantly as a result of severe plastic deformation and persistent dissolution of alloying additives and various contaminants originating from the milling equipment into the matrix [28]. As depicted in Figure 9a, the lattice parameter *a f cc* of the Fe-fcc phase increases constantly up to 90 h of MA and then exhibits a sharp rise and sets around this value.

The domain size *d* decreases with the MA time, following a similar trend as using the MWH method, and so does the lognormal standard deviation (Figure 9b,c). However, the *d* values obtained with the WPPM method are roughly half of that calculated by MWH. Both methods confirm that grain refinement is effective only during the beginning of MA (up to 30 h), with little reduction achieved afterward. It is worth mentioning that real materials have distributions of domain sizes, so the trends of the size distributions should also be followed, as demonstrated in Figure 10, and the corresponding values of the standard deviation of lognormal distribution are plotted in Figure 9c. It is clear that mechanical treatment causes the shift of the domain size distribution curve to lower values, the standard deviation being reduced accordingly, especially in the early stage of MA (Figure 10a). Eventually, a microstructure consisting of very fine (*d* = 6 nm) and narrowly distributed (*σ* = 3 nm) crystalline domains is established. Similar crystallite sizes were previously reported for other ball-milled fcc alloys (e.g., [41,42]).

**Figure 9.** Basic microstructural data derived from WPPM refinement in terms of the progression of following the parameters during MA: (**a**) lattice parameter, (**b**) lognormal domain size, (**c**) lognormal standard deviation, (**d**) dislocation density. Error bars correspond to the estimated standard deviation of the WPPM fit.

**Figure 10.** Lognormal domain size distributions of crystalline domains: (**a**) up to 30 h of MA, (**b**) after 30 h of MA, in terms of probability density function (PDF).

It is also clear that ball milling generates a high density of dislocations due to severe plastic deformation of ground material, which can be followed in Figure 9d. In general, the accumulation of defects in the Fe-fcc phase follows a similar trend to lattice expansion, with the same sharp rise after 120 h of MA. This is foreseeable, given the fact that the

lattice inflation is caused not only by the dissolution of alloying additives but is directly caused by an increase in the dislocation density [43] and is further promoted by extrinsic dislocations pilling up at the grain boundaries [44]. Rawers and Cook suggested that the strain on the grain boundary could extend into the nanograin itself, expanding the lattice [45]. In this study, from the moment of appearing (30 h) up to 90 h of MA, the *ρ* of Fe-fcc phase oscillates around 2 × 1016/m2, then experiences a sharp rise and sets around 3.5 × 1016/m2, the effective outer cutoff radius *Re* being 6.24 nm. The final value of *Re* is around that of the mean crystallite size *d* (6.0 nm), and corresponds to an average of half the dislocation per crystalline domain (for *d* = 6.0 nm: *d*/ *πd*3/6 ∼ 5.3 × 1016/m2), which is plausible. This is in line with HRTEM observations of ball-milled FeMo alloy by Rebuffi et al. [24], which concluded that dislocations are rather unlikely to be present in every single crystalline domain. As the accumulation of defects in powders, subjected to serious mechanical treatment, is the principal cause of microhardness growth during MA (Figure 4), *ρ* and *HV* can be related, which is demonstrated in Figure 11.

**Figure 11.** Relationship between dislocation density *ρ* and powder microhardness *HV*. Dashed line is the linear fit of the data.

Regarding the mutual dependence of *ρ* and *Re* parameters in the Krivoglaz–Wilkens dislocation theory, Wilkens introduced the dimensionless parameter *M* = *Reρ*−<sup>1</sup> = *Re*/*ld* called the Wilkens parameter. As can be noticed, *M* is also directly related to the average dislocation distance *ld*; therefore, it is used to determine the arrangements of dislocation arrangements in the strain fields [30]. The value of *M* much larger than unity (*M* 1) indicates that the dislocations are very weakly correlated and randomly arranged (weak dipole character). In contrast, *M* ≤ 1 signifies a distinctly correlated dislocation arrangement, accompanied by an intense screening of strain fields (*Re* < *ld*) (strong dipole character). In the present case, the decreasing value of *M* of the fcc phase to the value of ~1 with milling time (Figure 12) supports the hypothesis of gradual transformation from a dislocation cell structure with long-range strain fields into the nanocrystalline microstructure with a high density of dislocations, systematically distributed in grain boundaries.

**Figure 12.** Trend of the Wilkens parameter *M* in the function of milling time.

The WPPM modeling quality was good (mean GoF = 1.21) and hassle-free in almost all cases, except the problems encountered during refining the pattern obtained after 30 h of MA. In that case, during refinement of the Fe-bcc phase, *Re* constantly reached unrealistically small values (<1 nm), the lower limit of the continuum approach [46]. As a consequence, *ρ* rose to incredibly high levels a few times 1017/m2, accompanied with very low *M* (~0.35). When the dislocation density in material becomes sufficiently high (1016/m2), the strain field parameters (*ρ*, *Re*, *M*) can become uncertain in the sense that *Re* constantly approaches 0; this is a known intrinsic problem with the Krivoglaz–Wilkens theory caused by the strong reciprocal correlation of *Re* and *ρ* [47]. The simplest way to overcome this discrepancy is by imposing the arbitrary value of *Re* and keeping it fixed during refinement. Here, we assumed that *Re* was extended to the size of the entire crystalline domain (*Re* = *d*, ~6 nm), and with this constraint, *ρ* = 2.92 × 1016/m2 and *M* = 0.85 was obtained, which seemed perfectly reasonable.
