High-resolution neutron diffraction study of microstructural changes in nanocrystalline ball-milled niobium carbide NbC0.93

Materials Characterization 109 (2015) 173–180

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High-resolution neutron diffraction study of microstructural changes in nanocrystalline ball-milled niobium carbide NbC0.93 Anatoly M. Balagurov a, Ivan A. Bobrikov a, Gizo D. Bokuchava a, Roman N. Vasin a,⁎, Alexander I. Gusev b, Alexey S. Kurlov b, Matteo Leoni c a b c

Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia Institute of Solid State Chemistry of the Ural Branch of the Russian Academy of Sciences, Pervomaiskaya Str. 91 GSP, 620990 Ekaterinburg, Russia Department of Civil, Environmental and Mechanical Engineering, University of Trento, via Mesiano 77, 38123 Trento, Italy

a r t i c l e

i n f o

Article history: Received 7 August 2015 Received in revised form 25 September 2015 Accepted 27 September 2015 Available online xxxx Keywords: Neutron diffraction High-energy ball milling Carbides Nanocrystalline material Microstructure

a b s t r a c t High resolution neutron diffraction was applied for elucidating of the microstructural evolution of nanocrystalline niobium carbide NbC0.93 powders subjected to high-energy ball milling. The diffraction patterns were collected with the high resolution Fourier diffractometer HRFD by using the reverse time-of-flight (RTOF) mode of data acquisition. The traditional single diffraction line analysis, the Rietveld method and more advanced Whole Powder Pattern Modeling technique were applied for the data analysis. The comparison of these techniques was performed. It is established that short-time milling produces a non-uniform powder, in which two distinct fractions with differing microstructure can be identified. Part of the material is in fact milled efficiently, with a reduction in grain size, an increase in the quantity of defects, and a corresponding tendency to decarburize reaching a composition NbC0.80 after 15 h of milling. The rest of the powder is less efficiently processed and preserves its composition and lower defect content. Larger milling times should have homogenized the system by increasing the efficiently milled fraction, but the material is unable to reach a uniform and homogeneous state. It is definitely shown that RTOF neutron diffraction patterns can provide the very accurate data for microstructure analysis of nanocrystalline powders. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The quantification of microstructural features such as shape and size of the crystallites, their distribution, and the type/quantity of defects is a necessary step to understand the behavior of a nanosized powder in order to manufacture and utilize it. The analysis of fine-grained materials therefore requires an accurate assessment not just of the structure but also of the microstructure, using fast, reliable and unbiased techniques. Among the available techniques to obtain this information in a non-destructive way, diffraction certainly plays a leading role, especially considering the big progress that the technique made in the last decades [1,2]. The methods nowadays available for the retrieval of microstructural information from diffraction data may be divided in two classes: one includes those techniques based on the study of some integral characteristics of the diffraction peak profiles (e.g., the breadth), while the other considers the methods aimed at modeling the diffraction pattern as a whole. The Scherrer equation [3] and the Williamson–Hall plot [4] are probably the most common exponents of the first class, as the microstructure ⁎ Corresponding author. E-mail address: [email protected] (R.N. Vasin).

http://dx.doi.org/10.1016/j.matchar.2015.09.025 1044-5803/© 2015 Elsevier Inc. All rights reserved.

(in terms of an “average crystallite size” and a “microstrain”) is directly obtained from the integral breadths of individual diffraction peaks. The Rietveld method [5] and the Whole Powder Pattern Modeling (WPPM) [6], conversely, take the whole pattern into account trying to interpret it in terms of physical models, respectively, for structure and microstructure. So far, the WPPM is the only method based on accurate models for the microstructure and can directly include the parameters of the size distribution of crystallites into the refinement procedure [7]. Most of these techniques are routinely employed for the analysis of X-ray data; neutron diffraction is seldom considered. This is astonishing, as neutron TOF (time-of-flight) diffractometers operational at pulsed sources have a great potential for the characterization of the microstructure of fine-grained materials. Their resolution function, in fact, is almost independent of scattering vector Q within a fairly wide range. This is certainly quite different from the conventional neutron diffractometers using a monochromatic beam where the resolution function R(Q) is close to parabolic with a rapid rise in the both directions from the minimum value. In addition, the application of correlation Fourier technique on pulsed neutron source and the realization of the socalled reverse time-of-flight (RTOF) data acquisition method make it possible to achieve very high resolution comparable with that of X-ray instruments while maintaining the short source-to-detector distance and, consequently, the high brightness. The resolution of RTOF

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Fig. 1. Dependence of the specific surface area of the NbCy powders (triangles, left scale) and the average crystallite size (cirlces, right scale) on the milling time, as defined by BET method.

diffractometer mostly depends on the maximum rotation speed of the Fourier chopper, and can be tuned for specific purposes. Diffraction peaks obtained on such an instrument should be symmetric, without long ‘tails’ observed on diffraction spectra from conventional TOF instruments, installed at spallation neutron sources. This fact greatly simplifies precise peak profile analysis. In a previous short Letter [8] it was shown how microstructural information can be extracted from TOF data using the modified Williamson–Hall method. In particular, for cubic niobium carbide NbC0.93 it was found that a correct interpretation of the diffraction data is not possible without the account for the anisotropic peak broadening due to microstrain. But the result of the investigation [8] was ambiguous, as the observed diffraction line profiles could not be accurately reproduced within the framework of the applied simplistic model. We have carried out additional experiments on the same material and further analysis using more advanced line-profile analysis methods.

Fig. 3. Neutron diffraction patterns of the NbC-n powders, n = 0, 1, 5, 10, 15, measured with HRFD. As the milling time increases, the peak widths and the incoherent background increase as well. Inset shows the evolution of peak (420) profile with increasing milling time. Initial patterns are normalized by the maximum intensity value for (420) reflection.

This helped reaching an excellent agreement between experiment and model, which reveals important aspects of the microstructure of powdered nonstoichiometric niobium carbide, and allows understanding of its evolution during the milling process. Consequently, the focus of this paper is to demonstrate the application of advanced line profile analysis methods to the neutron diffraction data and to validate the possibility to use the high-resolution TOF neutron diffraction for the precise characterization of the microstructure of nanomaterials. 2. Samples and experiment Niobium carbide powder with nominal composition NbCy, y = 0.93, was subjected to high-energy ball milling. Each time 10 g of initial NbCy powder was milled at 500 rpm in a Retsch PM-200 mill using the total of

Fig. 2. a) SEM images of the initial coarse-grained powder of NbC0.93 niobium carbide at different magnifications. b) SEM images of NbC0.93 nanocrystalline powders produced by ball milling during 5 h and 15 h. Figures a (left) and b are reprinted from [11] with permission from Elsevier.

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Fig. 4. Diffraction peaks (422) and (420) of the NbC-5 sample and their fitting by the pseudo-Voigt function (left, χ2 = 5.31) and the sum of the Lorentzian and Gaussian functions with slightly different positions (right, χ2 = 1.71). The experimental points, calculated curve and component contributions (Lorentzian + Gaussian) are shown. The difference curve normalized by an error in a point is indicated below. Vertical ticks indicate the component positions. The Lorentzian width ~4 times larger than the Gaussian one and its position is shifted toward smaller d.

450 balls weighing 100 g with the addition of small amount of isopropanol, which afterwards was removed by drying the powder [9]. Four milling times were employed (cf. [8,10]) and the resulting samples were designated as NbC-n, with n = 0 (initial powder), 1, 5, 10, and 15 (numbers identify the milling time in hours). A preliminary X-ray diffraction analysis of the samples, conducted on a Shimadzu XRD7000 diffractometer, confirmed the high purity and homogeneity of the original composition (Δy ≈ 0.01) and identified traces of tungsten carbide +6 wt.% Co alloy (ball material) in the milled powders. The specific surface Ssp of the initial coarse-grained NbC0.93 powder and the same powder after milling was examined by the Brunauer– Emmet–Teller (BET) method on a Gemini VII 2390t Surface Area Analyzer [11]. The average particle size D can be estimated from the value of the specific surface Ssp as D = 6/ρSsp (where ρ = 7.76 g cm− 3 is the density of NbC0.93) assuming that all particles are spheres of the same size. It was observed that Ssp increases linearly with the milling time and D decreases hyperbolically (Fig. 1). According to SEM (Scanning Electron Microscopy) data, the average particle size of the initial and milled NbC0.93 powders is several times larger than the size of the coherently scattering domains determined from X-ray and neutron diffraction data. This is indicative of agglomeration of powder particles, which is confirmed by electron microscopy results (Fig. 2(a)). In milled powders the agglomeration is more pronounced (Fig. 2(b)). It is well seen that large agglomerates to 5 μm in size consist of smaller particles of size ~ 200 nm (duration of milling t = 5 h) or ~ 100 nm and smaller (duration of milling t = 15 h). The

size distribution of particles in the initial and milled NbC0.93 powders was additionally determined by the laser diffraction method using a HORIBA LA–950V2 Laser Scattering Particle Size Distribution Analyzer. Before measurement, the aqueous powder suspension was exposed to ultrasound for 15 min in a USB-1/100-TH Reltec ultrasonic bath to break large agglomerates and to provide uniform distribution of carbide particles in throughout the liquid. The electron microscopy data on the particle size of NbC0.93 powders correlate with the laser diffraction results in terms of quality and order of magnitude. Neutron diffraction patterns were measured on the high-resolution Fourier diffractometer (HRFD) [12] installed at the long-pulse IBR-2 reactor in Dubna. The instrument employs the correlation technique of data acquisition, which provides a very high resolution (Δ d/d ≈ 0.0013 at d = 2 Å) that is practically constant over a wide interval of dhkl spacing. This resolution should guarantee the possibility of determining the microstrain in the domains at the level of ε ≥ 5·10−4, and the average size of coherently scattering domains of Lcoh ≤ 350 nm. NIST standard reference materials SRM 676 (Al2O3), SRM 660b (La11B6) as well as a custom Na2Al2Ca3F14 (NAC) powder were used for the determination of the resolution function of the HRFD. The neutron diffraction patterns of the NbC-n powders are shown in Fig. 3. These patterns demonstrate a notable increase in peak broadening due to the mechanical work and an increase in the incoherent background arising probably from a noncrystalline component or diffuse effects. A broad peak appears in NbC-5 and NbC-15 patterns at d ≈ 1.8–1.9 Å, presumably associated to some short-range order.

Fig. 5. Diffraction patterns of the NbC-0 parent powder, as measured on HRFD (at the left) and HRPT (at the right, λ = 1.1545 Å) and processed by the Rietveld method. The experimental points, calculated pattern and difference curve are shown. Vertical ticks indicate the calculated diffraction peak positions. The broadening of diffraction lines at larger d-spacings is clearly seen for the HRPT pattern.

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Additional measurements of the neutron diffraction patterns of initial NbC0.93 powder, and a powder milled for 10 h (NbC-0 and NbC10) were obtained at the SINQ spallation source (PSI, Switzerland) on the constant-wavelength HRPT instrument [13] operated at λ = 1.1545 Å. The HRPT diffractometer has a resolution reaching Δd/ d ≈ 0.001 in the minimum of the resolution curve which is close to what can be achieved with HRFD. 3. Processing of neutron diffraction data Inset of Fig. 3 shows the evolution of the (420) peak measured on the HRFD instrument with the milling time. Apart from the gradual increase of peak widths and background level, some asymmetry is evident at smaller d, especially around 5 h of milling. An attempt to describe the peak profiles with a pseudo-Voigt function (linear combination of a Gaussian and a Lorentzian) shows some limitations already for the less milled NbC-1 specimen. The agreement between calculated and experimental data is markedly improved if the diffraction profiles are described by Gauss and Lorentz components with independent positions indicating the possible presence of two powder fractions in the samples. This is illustrated in Fig. 4, where a reduced range of d-spacing containing just two peaks is shown. The use of two separate components improves the χ2-factor by ~ 3 times but at the expenses of certain instability in the convergence of the minimization process for some of the observed peaks. A feature common to all profiles is however the large difference between the widths of the two separate components. This effect is more pronounced for the samples with greater milling time.

Fig. 6. The part of the NbC-5 diffraction pattern, as measured at HRFD and processed by the Rietveld method assuming the presence of two fractions. The experimental points, calculated fractions contributions (two Lorenzians with different widths), and their sum are shown. The difference curve normalized by an error in point is indicated below. Vertical ticks indicate positions of wide (top row) and narrow (bottom row) Lorenzians.

The average peak width refined by the Rietveld method regularly and strongly increases with increasing milling time. This value does not provide any direct numerical information about the microstructure, but is qualitatively compatible with the picture of decreasing domain size and increase in the quantity of defects, expected in the case of milling. 3.2. Traditional analysis of the widths of the NbC-n diffraction peaks

3.1. Analysis using the Rietveld method The determination of the structural parameters of NbC0.93 was carried out using the FullProf [14] (for HRPT data) and MRIA [15] (for HRFD data) software packages implementing the well-known Rietveld method. The material has a face-centered cubic symmetry (space group Fm3m) with cell parameter а ≈ 4.47 Å, and atoms in the positions: Nb (4a) (000), C (4b) (½ ½ ½). The unit cell parameter, occupancy of carbon positions and temperature factors of both atoms were refined from the diffraction data. Simple peak functions were used to describe the peak profile in FullProf (Gaussian, Lorentzian, or their combination). In MRIA program, in addition to these profiles, we used the available functionality to model the peak shape after the smoothed most intense peaks in the measured pattern. As an example, Fig. 5 shows the data processing for NbC-0 as measured with HRFD (after MRIA, model peak shape) and HRPT (after FullProf, pseudo-Voigt peak shape). The agreement is quite good and very similar results were extracted from the two datasets. In particular, the initial occupancy of carbon was refined as 0.919(7) from HRPT and 0.924(8) from HRFD data, which compares well with the expected stoichiometry. The Rietveld analysis of the milled samples confirmed the presence of two fractions with very different peak widths and slightly different cell parameters. For the fraction with broad peaks Lorentzian function provided the best approximation of peak profiles. For the fraction with narrow diffraction peaks both Lorentzian and Gaussian profiles were suitable with only minor difference in the fit quality. This is illustrated in Fig. 6, which shows the result for the NbC-5 diffraction pattern, measured at HRFD and processed by assuming the presence of two fractions, both with a Lorentzian peak profile. For all milled samples, the unit cell parameter of the fraction with narrower peaks is larger than that of the fraction with broader peak by ca. 0.01–0.02 Å. Also there is a small (~0.01 Å) gradual decrease in the cell parameter with the milling time (Table 1). This behavior was already observed based on the data processing considering only one powder fraction [8]. We denote the fraction with the smaller cell parameter and broader diffraction peaks as F1 and the fraction with larger cell parameter and narrow peaks as F2.

The classic Williamson–Hall method [4], involving the analysis of the functional dependence of the integral peak breadth (or full width at half maximum, FWHM) on the scattering angle (or time of flight in case of TOF diffraction), was used to analyze the widths of the diffraction peaks. To apply this method there should be several diffraction peaks measured in a fairly large range of d-spacing, and as a result, it is possible separately determine the strain and size contributions to the peak widths. To apply the Williamson–Hall method on the data obtained with the TOF diffractometer, it is natural to use d-spacing as a variable. It was shown earlier [16], that for HRFD, the peak breadth ΔdR dependence on d-spacing for standard powders without any size and microstrain effects can be written as: 2

2

ðΔdR Þ ¼ C 1 þ C 2 d ;

ð1Þ

where C1 and C2 are constants related to the characteristics (resolution function) of the diffractometer. For real crystals, the additional peak broadening connected with microstrain, ε, and finite domain size effects, L, can be expressed as: 2

2

2 4

ðΔdS Þ ≈ð2εdÞ þ ðk=LÞ d ;

ð2Þ

Table 1 Structure and microstructure parameters for NbC-n powders, as obtained after the fitting of individual reflections with two independent components (according to the data obtained on HRFD). The cell parameter a, the average crystallite size bL NV and the microstrain bε N are given for both fractions F1 and F2 (subscripts 1 and 2, respectively). Sample NbC-0 NbC-1 NbC-5 NbC-10 NbC-15

bLNV1, Å

bLNV2, Å

58 22 14 7

2550 105 63 45 32

a1 , Å

a2, Å

bεN1

bεN2

4.4612 4.4592 4.4566 4.4445

4.4682 4.4667 4.4662 4.4672 4.4666

5.2·10−3 7.3·10−3 8.1·10−3 8.4·10−3

3.3·10−4 7.3·10−4 8.5·10−4 1.3·10−3 4.1·10−3

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Fig. 7. The (Δd)2 over d2 dependences for the NbC-0 and NbC-1 powders and contribution of the HRFD resolution function to the peak widths (bottom line), as determined from diffraction patterns of standard powders NAC and Al2O3 (squares and circles). The (Δd)2-line for NAC and Al2O3 is linear (no size effect), for NbC-0 sample a size effect is noticeable (L ≈ 255 nm). For the NbC-1 powder two (Δd)2-lines are shown: through all points (solid line) and through points without large anisotropic effect (dashed line). In the second case, the size effect (L ≈ 150 nm) is noticeable.

if Gaussians are used as distribution functions (k is the Scherrer constant [17], close to unity and taking into account the shape of the crystallites). Combining these two relations, we obtain the total width as a function of d-spacing: 2

2

4

ðΔdÞ ¼ C 1 þ ðC 2 þ C 3 Þd þ C 4 d ;

ð3Þ

where С3 ≈ (2ε)2, C4 ≈ (k/L)2. In the absence of any size effect (large crystallites), the (Δd)2 on d2 dependence is linear. Again, it is possible to determine ε and L independently if data are obtained in sufficiently large dhkl interval. An isotropic shape is assumed if the same k is used for all reflections. We should note here the obvious fact that the L value is related to the size of the coherently scattering domains and is therefore usually smaller than the “optical” grain size, or the grain size determined from BET data, as grains can have an amorphous outer layer or can consist of several domains. The dependence of (Δd)2 on d2 for the NbC-0 sample is shown in Fig. 7, together with the contribution of the HRFD resolution function to the peak widths. This dependence is linear for the NAC and Al2O3 specimens, while for the NbC-0 it can be better described by a parabola with a positive C4 coefficient, which corresponds to L ≈ 255 nm (at k = 1). A much larger difference is observed for the C3 coefficient,

Fig. 8. The (Δd)2 on d2 dependences for the Lorentzian (diamonds, left scale) and Gaussian (squares, right scale) components of the total width of the diffraction peaks for the NbC-5 sample. The resolution function is also shown. The model curves are drawn through the experimental points corresponding to the values of the anisotropy factor Гhkl = 0, 0.16, 0.25 and 0.33 (right-to-left).

177

Fig. 9. Comparison of lognormal distributions of crystallite sizes for NbC-0, NbC-5 and NbC-15 samples, as obtained from HRFD diffraction data. For NbC-0 distribution parameters correspond to the bL NV = 223 nm. For NbC-5, two fractions F1 and F2 with different distribution parameters are seen, which corresponds to the diffraction crystallite sizes of 26 nm and 71 nm. For NbC-15, the contribution of the second fraction is negligible, while the diffraction crystallite size of 12 nm corresponds to the main fraction F1.

indicating the presence of microstrain already in the NbC-0 powder. Any attempt to describe the (Δd)2 vs. d2 dependencies for all milled powders according to Eq. (3) were unsuccessful. In our previous paper [8] it was shown that, in respect to NbC0.93, this effect could be attributed to the anisotropy of microstrain, i.e. to the dependence of peak broadening on particular set of Miller indices. Fig. 7 indicates the experimental data for NbC-1 and two variants of their description with a polynomial function (3). Formal description of all points led to negative value of C4, which is in contradiction with its physical meaning. Elimination of a couple of points with the highest anisotropic contribution from the fit leads to a classic parabolic relation with C4 coefficient corresponding to L ≈ 150 nm. It is known that the peak width anisotropy effects can be taken into account by introducing a dislocation-related contrast factor Γ = (h2k2 + h2l2 + k2l2)/(h2 + k2 + l2)2 [18,19] in Eq. (3). For the milled NbC-n powders where two powder fractions, F1 and F2 (introduced in Section 3.1), were present in the sample we described individual diffraction peaks by a combination of Lorentzian (for the fraction F1 with broad peaks) and Gaussian (for the fraction F2 with narrow peaks) components with independent positions. The so-obtained dependencies of the full width of Lorentzian and Gaussian parts of each diffraction peak were then modeled by means of Eq. (3), taking into account the anisotropy factor Γ. For the NbC-5 powder, the dependency of (Δd)2 on d2 is shown in

Fig. 10. Lognormal distribution of crystallite sizes for NbC-10, as calculated from HRFD and HRPT diffraction data.

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Table 2 Microstructure parameters for NbC-n powders, obtained by the WPPM method. The parameters of the lognormal distribution (μ and σ) and the dislocation density ρ are given. The values for two fractions — F1 with broad (top line) and F2 narrow (bottom line) peaks — are given for NbC-5 and NbC-10. The volume-weighted average crystallite size bL NV was calculated according to (5). Sample

Diffraction instrument

μ

σ

NbC-0 NbC-1

HRFD HRFD

NbC-5

HRFD

NbC-10

HRPT

NbC-10

HRFD

NbC-15

HRFD

5.58 0 0.88 3.05 1.28 1.10 0 0.22

0.18 1.16 0.87 0.65 0.72 0.72 1.06 0.85

(ρ ± 0.05), 10−16 m−2

bL NV ±5, nm

0.1 0.6 5.3

223 85 26 71 17 14 39 12

6.7 5.5 6.2

Fig. 8 for both components: the lines correspond to Γ = 0, 0.16, 0.25, and 0.33 (in particular, Γ = 0 and 0.33 for h00 and hhh diffraction lines, respectively). It is seen that most of the points lie exactly on the curves for these values of Γ. Moreover, a parabolic dependence of (Δd)2 on d2 for different groups of points is clearly visible. Taking these Γ values into account as done in [8], it is possible to evaluate the coefficients С3 ≈ (2ε)2 and C4 ≈ (1/L)2 for fractions F1 and F2 in all NbC-n powders (Table 1). 3.3. WPPM analysis of NbC-n diffraction data The distinguishing feature of the WPPM method is the generation of the diffraction peak profiles based on physical models of the microstructure and characteristics of the instrument. The microstructure model can involve shapes and sizes of the coherently scattering domains, dislocations, stacking and twin faults, etc. The possibility of handling a domain size distribution [7] is particularly handy for the analysis of a real specimen. Both the WPPM method and the PM2K software package implementing it [20] have been recently extended and updated to allow for a full microstructural analysis from neutron data [Leoni M, personal communication]. The software was employed for the analysis of the HRFD and HRPT measurements. In addition to the instrumental resolution function, we consider a lognormal distribution of domains:

g ðDÞ ¼

"  2 # 1 1 lnD  μ pffiffiffiffiffiffi exp  2 σ Dσ 2π

ð4Þ

where μ and σ are adjustable parameters. The presence of dislocations is also accounted for. The volume-weighted average domain size that should correspond to the L-value obtained by the Williamson–Hall

method in an ideal case can be calculated from the parameters of the distribution as [7]: bLNV ¼

  3 7 exp μ þ σ 2 : 4 2

ð5Þ

The lognormal size distributions refined using WPPM method for NbC-n powders are shown in Figs. 9 and 10. For the NbC-0 powder, the size distribution could be roughly determined only from the HRFD measurements. This specimen has quite narrow reflections, typical for a material with large domain size and low defect content. The resolution of the diffraction instrument, which for the HRPT is lower on the average than for the HRFD, is the limiting factor for the determination of the domain size. The refined lognormal size distribution of NbC-0 (μ = 5.58, σ = 0.18) gives the average domain size b L NV = 223 nm calculated according to (5). This estimation is close to the value of L = 255 nm, as determined by the Williamson–Hall method. We should note that this value, obtained in conditions where the observed profile is rather close to the instrumental one, should only serve as a guideline. Small errors in reproducing the instrumental profile can in fact lead to large variations of the estimated domain size value. As already observed for the analysis conducted with the other techniques, for all milled NbC-n powders with n N 1, the description of peak profiles using the WPPM method requires the introduction of at least two independent powder fractions, F1 and F2, with different cell parameters and size distributions. The refinement of the size distribution parameters is quite stable for the F1 fraction (large peak widths), but poses some problems for the other fraction with NbC-5 being an exception, as the volumes of both fractions are similar. For NbC-15 only the diffraction crystallite size for the main fraction F1 was evaluated (bL NV = 12 nm) since the amount of the second fraction F2 was too small. For NbC-10, the parameters of the lognormal distribution for the F1 fraction are μ = 1.28, σ = 0.72 (HRPT data), μ = 1.10, σ = 0.72 (HRFD data), which results in bL NV = 17 nm and 14 nm, respectively (Fig. 10). Both instruments give therefore essentially the same results. All results are presented in Table 2. The average contrast factor for the dislocations, necessary to determine the dislocation density (ρ), was calculated on the basis of the elastic constants of NbC0.865 [21], thus dislocation densities presented in Table 2 can be considered as an estimation. 4. Discussion and conclusions A comprehensive analysis of the neutron diffraction data for the NbC0.93 powders subjected to high-energy milling revealed a rather complex picture of their microstructure. Already at a preliminary stage, it became clear that the profiles should be modeled by

Fig. 11. Dependence of the lattice parameters (in Å) of F1 (crosses) and F2 (triangles) fractions (at the left) on the milling time and the volume ratio of these fractions in a sample (at the right). Statistical errors for the lattice parameters are close to the symbol size; lines are drawn for better visualization.

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Fig. 12. Dependence on the microstructural characteristics of F1 and F2 fractions on the milling time: size of the crystallites (at the left) and microstrain calculated using the mean value bГhkl N =0.2 (the average over all possible orientations, at the right). For the sizes of the crystallites, the values calculated according to the Williamson-Hall (triangles) and the WPPM (crosses) methods are shown. Lines are drawn for better visualization. The statistical errors are close to the symbol size.

considering the specimens as composed of two independent fractions of powder, F1 and F2. The best description is achieved by a combination of two profiles with independent position and breadth, in particular, a(F1) b a(F2) and Δd(F1) ≈ 4·Δd(F2). The F1 and F2 unit cell parameters differ by at least ~0.01 Å; as the milling time increases, the cell parameter a(F2) remains almost constant, whereas a(F1) decreases gradually (Fig. 11). The same figure shows the relative change of the F1 and F2 volume fractions with the milling time. To interpret the data, we should therefore assume that the milling process initially produces a non-uniform powder: a fraction of material is more severely processed, and the rest of the material remains in a less-milled state. The phenomenon is here enhanced by the possibility of extracting a certain amount of the carbon from the material. The chemical composition of the NbCy samples can in fact be estimated from the lattice parameter dependence on y [22,23]. A simple calculation suggests that the F1 fraction has compositions NbC0.87, NbC0.84, NbC0.82 and NbC0.80 in NbC-1, NbC-5, NbC-10 and NbC-15, respectively. The F2 fraction, on the contrary, shows an almost constant composition NbC0.95. The two fractions have quite different levels of microstrain (Fig. 12 and Table 2): the larger the microstrain, the larger the quantity of possible fast diffusion paths in the material that can favor the elimination of the carbon from the structure. It is clear that the F2 fraction is the one less efficiently milled: its volume decreases with the milling time, the domain size is on average larger (and there is a long tail of the size distribution toward large domain sizes) and the quantity of defects is lower. On the contrary, the F1 fraction starts to appear immediately during the milling, increases in quantity and shows a smaller size and a larger content of defects, due to the mechanical action of the mill and the expected large plastic deformation involved in the milling process (Fig. 12). The smaller size and higher quantity of defects can promote a more rapid decarburization of the F1 fraction, as confirmed by the cell parameter decrease. A similar variation in the composition of nonstoichiometric carbides NbCy and TaCy during high-energy milling was observed experimentally also in earlier works [24,25]. Quite astonishingly, for the average domain size the quantitative results of the WPPM analysis match well the outcome of a more rough Williamson–Hall method estimation (Fig. 12). This is a very rare case. The advantage of the WPPM is in the possibility of following the evolution of the size distribution as a function of milling time, and in the physical meaning of the results by accurate modeling of the whole pattern. Of course, the integral breadth of peaks can be estimated even without doing a fit of the profiles, missing, e.g., the presence of several fractions, stacking faults, or complex dislocation strain fields in the material. The same NbC0.93 material was considered as uniform in [8]. As a result of analysis, presented in [8], the overall trend with a tendency to decrease the domain size with increasing milling time, is respected, but the

presence of two powder fractions cannot be inferred as in principle it is not necessary to calculate integral breadth values. An important methodological point of this work is that the physical model-based WPPM method has been for the first time used to analyze neutron diffraction RTOF data. Both dislocation density values and average domain sizes obtained on HRPT diffractometer utilizing monochromatic neutrons and on HRFD that uses a RTOF scheme are comparable, as well as the shape of crystal size distributions (Fig. 10). This implies that RTOF neutron diffraction patterns can provide accurate data for microstructure analysis of powder materials. As a concluding remark, it can be seen that the overall picture is also compatible with the qualitative initial observation of an increase in the background with the milling time. The progressive transformation of the material from coarse-grained to nanocrystalline is accompanied by a reduction in the carbon content and therefore by an increase in the diffuse scattering due to point defects. The local elimination of carbon, in the end, produces disorder on the surface and inside of the domains that can lead to a progressive decrease of the crystallinity of the system. Further investigation will be necessary to confirm this hypothesis, but it is clear that a careful control of the milling conditions is necessary in order to obtain a nanosized nonstoichiometric carbide powder with a uniform size distribution and a desired composition. Acknowledgments The experimental data were obtained using the IBR-2 (JINR) and SINQ (PSI) neutron sources. The authors are grateful to V.Yu. Pomjakushin and D.V. Sheptyakov for their help with the experiment on HRPT. Parts of this work were financially supported by the Russian Foundation for Basic Research (RFBR) within the project No. 14-2904091_ofi_m and jointly by RFBR and the Moscow Region Government (project No. 14-42-03585_r_center_a). M.L. is grateful to the Italian government for support (FIRB project RBFR10CWDA). We appreciate useful comments from the anonymous reviewer who helped to improve this manuscript. References [1] B.E. Warren, X-ray Diffraction, Addison-Wesley, Reading (MS), 1969. [2] E.J. Mittemeijer, P. Scardi (Eds.), Diffraction Analysis of the Microstructure of Materials, Springer-Verlag, Berlin, 2004http://dx.doi.org/10.1007/978-3-662-06723-9. [3] P. Scherrer, Bestimmung der Grösse und der inneren Struktur von Kolloidteilchen mittels Röntgenstrahlen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918 (1918) 98–100. [4] G.K. Williamson, W.H. Hall, X-ray line broadening from filed aluminium and wolfram, Acta Metall. 1 (1953) 22–31, http://dx.doi.org/10.1016/0001-6160(53)90006-6. [5] R.A. Young (Ed.), The Rietveld Method, Oxford Univ Press, Oxford, 1993http://dx. doi.org/10.1002/crat.2170300412. [6] P. Scardi, M. Leoni, Whole powder pattern modelling, Acta Crystallogr A 58 (2002) 190–200, http://dx.doi.org/10.1107/S0108767301021298.

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