electronic reprint Nanocrystalline domain size

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Applied Crystallography ISSN 0021-8898

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Nanocrystalline domain size distributions from powder diffraction data Matteo Leoni and Paolo Scardi

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J. Appl. Cryst. (2004). 37, 629–634

Leoni and Scardi



Nanocrystalline domain size

research papers Journal of

Applied Crystallography

Nanocrystalline domain size distributions from powder diffraction data

ISSN 0021-8898

Matteo Leoni* and Paolo Scardi Received 9 April 2004 Accepted 3 June 2004

# 2004 International Union of Crystallography Printed in Great Britain ± all rights reserved

University of Trento, Faculty of Engineering, Department of Materials Engineering and Industrial Technologies, via Mesiano 77, 38050 Trento, Italy. Correspondence e-mail: [email protected]

The need for an a priori domain size distribution is one of the main limitations of existing line pro®le analysis methodologies. A numerical modi®cation of the whole-powder-pattern modelling algorithm is proposed, to allow the re®nement of a general domain size distribution from powder diffraction data. The shape of domains has to be inferred for the specimen under study. The algorithm is robust enough to unveil ®ne details in the re®ned distribution, as witnessed by the results of tests performed both on simulated and on real patterns of nanocrystalline ceria.

1. Introduction New methods for the analysis of X-ray diffraction (XRD) line pro®les are the object of several current research studies, aimed in particular at providing information on nano-sized materials otherwise accessible only (and not always) by transmission electron microscopy (TEM). Traditional line pro®le analysis (LPA) techniques, based on the processing of line pro®le data (mostly peak width and shape factor) obtained after ®tting individual re¯ections in the pattern, are being superseded by whole-pattern techniques, allowing a direct re®nement of physical model parameters on the experimental data. This migration is accompanied by the adoption of a Fourier formalism and/or by the elimination of a priori ®xed analytical pro®le functions. The latter is a particularly important point; the (arbitrary) bell-shaped function introduced for describing experimental line pro®les is often incompatible with the physical nature of the broadening sources present in the specimen under study, and it can therefore bias the results of LPA. This is, for instance, the case for the Voigt (Langford, 1978), the pseudoVoigt (Wertheim et al., 1974) with its various modi®cations and the Pearson VII (Hall et al., 1977) curves, which cannot reproduce the line pro®le arising from a lognormal distribution of spherical crystallites (cf. e.g. Scardi & Leoni, 2001), a distribution often met in nanocrystalline systems. As the diffraction pro®le results from a convolution of various sample- and instrument-related contributions, a Fourier approach is the natural framework for developing fast and ¯exible LPA methods (Klug & Alexander, 1974; Warren, 1990). This philosophy is put into practice by whole-powder-pattern modelling (WPPM), a recently devised, completely analytical, procedure for the modelling of size and strain broadening effects in XRD patterns (Scardi & Leoni, 2002, 2004). J. Appl. Cryst. (2004). 37, 629±634

WPPM adopts a limited number of assumptions on the microstructure of the material under study; the line pro®le produced by a crystallite size distribution and by various kinds of defects (dislocations, planar faults, grain surface relaxation effects, anti-phase boundaries, concentration gradients, point defects etc.) are convoluted together with the instrumental component, and the pattern is then directly synthesized through (fast) Fourier transformation. WPPM has been used successfully for describing the evolution of metal powders subjected to intensive ball milling (Scardi & Leoni, 2002) and for the investigation of cerium oxide nanocrystals produced by crystallization from an amorphous gel at increasing temperatures (Leoni & Scardi, 2004; Leoni et al., 2004). In all those cases, a lognormal distribution of spherical crystallites was chosen to describe the size-related broadening in the analysed specimen. Even if the spherical shape is a reasonable assumption (con®rmed by TEM), and there are indications that the size distribution is not far from lognormal, the adoption of a given size distribution is an undesired constraint on the modelling. Real size distributions can depart signi®cantly from a priori assigned ones, especially if a multimodal character is present. The present paper shows how to extract a general grain size distribution directly from powder diffraction data, i.e. without making any a priori hypothesis on the analytical shape of the distribution. The proposed method is illustrated by modelling simulated patterns of cerium oxide nanocrystals with mono- and multimodal distributions. The reliability is then assessed by analysing experimental data collected on ceria calcinated at increasing temperatures; the result is compared with a grain size distribution obtained by TEM and with analogous results produced by imposing an analytical shape on the distribution.

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DOI: 10.1107/S0021889804013366

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research papers 2. Theoretical background and proposed method Formulae for the description of the diffraction pro®les of a polydisperse powder showing a simple grain size distribution and the presence of various kinds of defects are readily available in the literature. Suitably framed within a selfconsistent convolutive algorithm, these formulae form the basis of the whole-powder-pattern modelling approach. In WPPM, the intensity, I…d †, of an hkl diffraction pro®le is described through its Fourier transform, C…L†, as [cf. equation (3.1) of Scardi & Leoni (2004), but note a typographic error therein (missing `exp')] R …1† I…d † ˆ k…d † C…L† exp…2iLdhkl † dL; where d is a length in reciprocal space (equal to dhkl in the Bragg condition), L is the Fourier variable, and k…d † includes all terms contributing to the diffracted intensity and not related to the line broadening sources (e.g. Lorentz±polarization factor, square modulus of the structure factor etc.). For the equation to be complete, the Fourier coef®cients, C…L†, must account for all sources of broadening affecting the pro®le under study. The advantage of the Fourier approach is immediate; the effect of the various broadening sources on a single re¯ection is a convolution product in real space (Warren, 1990), which transforms into a product of the corresponding Fourier transforms in Fourier space. The various terms composing C…L† are here grouped in two main classes, the ®rst accounting for size effects [AS …L†], related to the ®nite extension of the coherently diffracting domains, and the second [ANS …L†] accounting for all other (non-size) sources of broadening (e.g. lattice micro distortions, point, line and plane defects, etc.). For a list of ANS …L† terms see, for example, Scardi & Leoni (2001, 2002) and Leoni & Scardi (2004). Because of the convolution product, C…L† = AS …L† ANS …L†. It has already been demonstrated that grain shape and grain size distribution effects can be somehow decoupled in the size term. In particular, the size coef®cients can be written as [cf. equations (6) and (22) of Scardi & Leoni (2001)] 1 R1 R Ac …L; D† w…D† dD w…D† dD; …2† AS …L† ˆ L K c …hkl †

0

where Ac is the Fourier transform for the particular grain shape considered, K c …hkl † is a constant related to the shape under consideration [values for simple crystal shapes are reported in Table I of Scardi & Leoni (2001)] and w(D) is a weight function, related to the grain size distribution, g(D), through the relationship w…D† = g…D† Vc …D†, where Vc …D† is the volume of a crystallite of diameter D.1 Weighting the distribution on the volume of the diffracting crystallites is a 1

Expressions for various grain shapes are given, for example, by Patterson (1939), Wilson (1962), Langford & LoueÈr (1982), Vargas et al. (1983) and Scardi & Leoni (2001). Even though analytical Fourier transforms for simple crystal shapes and the corresponding diffraction pro®le have been known for decades, the introduction of a suitable grain size distribution, only envisaged in past literature, is a recent achievement (Langford et al., 2000; Scardi & Leoni, 2001).

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consequence of the physical nature of XRD as a volumesensitive probe. So far, the WPPM approach has considered analytical AS …L† values for the most common grain shapes and size distributions. The fully analytical approach allows a signi®cant increase in calculation speed, providing a re®nement in a reasonable time, although at the expense of ¯exibility. In fact, if no prior information is available on the specimen, the choice of a certain distribution may be arbitrary. For nanocrystalline powders, a lognormal distribution of spheres is commonly employed, especially for ®nely dispersed catalysts and powder produced via sol±gel (Krill & Birringer, 1998; Langford et al., 2000; Leoni et al., 2004). This choice has already been validated by several authors and is also partly supported by maximum-entropy calculations (Armstrong et al., 2004). Inspection of (2) reveals the possibility of extracting a size distribution directly from the diffraction data. Let us assume a certain shape for the crystallites in the powder (i.e. let us assume Ac and Vc are known); the integral in (2) can be replaced by a ®nite sum considering a step-wise grain size distribution (a histogram) instead of a continuous curve, and the sampled distribution values (column heights in the histogram) can then be treated as parameters to be re®ned. In this way the a priori (®xed-shape) analytical distribution [typically a two-parameter (mean and variance) function] is replaced by a generic unconstrained distribution. The parameters of the distribution can then be re®ned together with the other parameters of the model within the usual WPPM frame, the Rietveld method or any non-linear least-squares minimization approach, in general. Since the whole diffraction pattern is used in WPPM re®nements, the number of data points should be, in any case, suf®ciently large with respect to the number of re®ned parameters.

3. Refinement of simulated data To test the algorithm, the general grain size distribution was extracted from diffraction patterns synthesized using given analytical grain size distributions. Fig. 1(c) (dots) shows the simulated diffraction pattern of ceria (space group Fm3m, lattice parameter a0 = 0.541134 nm as in ICDD PC-PDF card #34-0394) calculated assuming both instrumental and size broadening. Instrumental broadening is chosen to be equal to that of a high-resolution Rigaku PMG/ VH instrument; the instrumental contribution is minimal with respect to the other sample-related broadening. Size broadening is assumed to be caused by a lognormal distribution, g(D), of spherical domains.2 The distribution parameters, i.e. = 1.45 (corresponding to a mean diameter hDi = 4.61 nm) and ! = 0.4 (cf. Fig. 1a, line), were chosen to mimic the behaviour of a representative nano-sized material. The 2 The lognormal distribution is de®ned as [Aitchison & Brown, 1969; Scardi & Leoni, 2002 equation (16a)] g(D) = ‰D ! …2†1=2 Šÿ1 exp‰ÿ…ln D ÿ †2 =…2!2 †Š, where is the lognormal mean and ! is a width parameter.

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J. Appl. Cryst. (2004). 37, 629±634

research papers corresponding weight function, w(D), i.e. the distribution actually seen by XRD, is shown in Fig. 1(b) (line). The result of modelling is shown in the same ®gure, superimposed on the starting data. The re®ned pattern, in particular, is shown in Fig. 1(c) (line); the residual is ¯at, demonstrating the good performance of the algorithm. The corresponding distribution and weight functions are shown in the histograms in Figs. 1(a) and 1(b). In this case, a very good agreement with the prior distribution is again evident. A weighted re®nement was actually carried out, in order to be more representative of a real case of study. Gaussian statistics were assumed, as the intensity of each data point was suf®ciently far from zero. The re®nement strategy involved multiple stages, reducing at each stage the range of diameters explored and increasing the sampling (i.e. the number of points) of the distribution until a suf®ciently detailed result was found. To avoid any bias, a `top-hat' function (uniform distribution) was chosen as the starting distribution; a large sampling step was selected, in order to branch quickly the region where most of the information is contained. Starting from this rough guess, a ®ner sampling of the distribution function was then made in the branched region and the single frequencies were further optimized. A remark must be made on the calculation speed: because of the discrete nature of the distribution to be re®ned, only sums and products of simple polynomials, i.e. fast operations,

Figure 1

Distributions and patterns of ceria. (a) The chosen distribution and (b) the corresponding weight function, w(D); the continuous line represents the distribution adopted in the simulation and the histogram is the result of the re®nement. (c) The simulated pattern of ceria (dots), ®tting result (line) and residual (difference between data and model, shown below). J. Appl. Cryst. (2004). 37, 629±634

are involved in the calculation of the Fourier transform. When the number of sampling points is not too high (e.g. below 500), the calculation speed is even higher than in the analytical WPPM, as complex error and gamma functions (entering the Fourier size term for lognormal and gamma distributions of grains) are no longer required. Calculation of the whole pattern of Fig. 1(a) for a 250-point distribution takes less than 50 ms on an AMD Athlon XP 2200+ machine. To evaluate the performance in a more complex case, three bimodal size distributions were considered, each of them in turn obtained as a weighted sum of two gamma distributions3 (1 = 5 nm, 1 = 20 and 2 = 12 nm, 2 = 20, respectively). Gamma distributions were chosen so as to depart from the `usual lognormal behaviour'. Figs. 2(a)±2(c) (lines) show the grain size distributions, g(D), while the corresponding weight functions, w(D), are shown in Figs. 2(d )±2(f ). The latter make quite evident the enhanced weight of large-size crystallites with respect to the small-size region. The corresponding simulated patterns obtained from analytical distributions are shown in Figs. 2(g)±2(i) (dots); to obtain these ®gures, the patterns corresponding to the two component distributions were independently generated and (suitably scaled) summed. As for the monomodal case of Fig. 1, the re®nement started from a uniform distribution, zero background and arbitrary microstructural parameters. The ®tting results are shown in Figs. 2(g)±2(i) as continuous lines. The match is good, as demonstrated by the ¯at residual. The same is also valid for the distributions, shown as histograms in Figs. 2(a)±2(c) (size distribution) and 2(d )±2(f ) (weight function). From Fig. 2, we can conclude that, used on simulated data, the algorithm seems able to reconstruct ®ne details of the distribution, even in cases where the volume weight is unfavourable [e.g. distributions shown in Figs. 2(b) and 2(c), where the fraction of small grains has little effect on the pattern]. To prove further the validity of the approach, the same patterns were analysed using WPPM and assuming the grains to be spherical and distributed according to a monomodal lognormal or gamma distribution. The results are shown in Figs. 2(a)±2(c) (size distribution) and 2(d )±2(f ) (weighted size distribution) as dashed (lognormal) and dotted (gamma) lines, respectively. Here the effect of volume weighting is more evident; the re®ned distribution is far from the original one unless a substantial fraction of large grains is present. The mean grain diameter of a forcedly assumed monomodal distribution seems somehow related to the mean grain diameters, 1 and 2, of the two size distributions composing the actual g(D). In particular, it is close to their volumeweighted average. Particularly interesting is the case shown in Fig. 2(a), where the grain size distribution seems completely missed when WPPM adopts the a priori selected analytical expression. The 3 The gamma distribution is de®ned as [Arley & Buch, 1950; Scardi & Leoni, 2002 equation (16b)] gg(D) =R‰=ÿ…†Š…D=†ÿ1 exp…ÿD=†, where ÿ…x† is 1 the gamma function, ÿ…x† = 0 yxÿ1 exp…ÿy† dy:

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research papers The corresponding weight function, w(D), and extracted size distribution, g(D), are shown in Figs. 3(e)± 3(h) and 3(i)±3( l), respectively. Some fuzziness is present, possibly related to the noise (quite evident in the pattern), increasing with the mean grain diameter. The distributions are in good agreement with the previous result of WPPM (Figs. 3e±3l, continuous lines), justifying again the use of a lognormal size distribution when dealing with nanocrystalline powders produced by this route. It should be noted that in the present algorithm no correlation is imposed between the various sampling points of the distribution, i.e. the re®nement routine treats them as Figure 2 Modelling result of simulated cerium oxide data. Distributions and corresponding patterns are shown for three completely independent. test cases. (a)±(c) Original distribution (continuous line), result of the present algorithm (histogram) and WPPM The agreement with the results assuming a lognormal (dashed line) and a gamma (dotted line) grain size distribution. (d )±( f ) Weight TEM distribution is also very functions w(D) `seen' by X-ray diffraction, referring to the distributions shown in (a)±(c) weighted by the volume good [cf. the histograms in of the corresponding crystallites. (g)±(i) Simulation (dots) and ®tting result (line) corresponding to the distributions shown in (a)±(c). The residual (difference between simulation and model) is also shown below the Fig. 3( j)], con®rming the pattern (in this case, very close to zero). possibility of obtaining a fairly accurate grain size distribution in a completely non-destructive way through the disagreement is less pronounced if the volume weighting is use of WPPM. considered (Fig. 2d), which is a reasonable conclusion The structural and microstructural parameters agree with considering that the diffraction pro®le is related to the weight previous results within the estimated standard deviations and, function more than to the distribution alone. quite surprisingly, the goodness-of-®t (GoF) indicator (McCusker et al., 1999) is slightly lower in the numerical than 4. Refinement of experimental data: nanocrystalline in the analytical case, again demonstrating the good perforceria mance of the numerical procedure. As a concluding remark, it is important to underline that the To test the algorithm on real data, four nanocrystalline ceria proposed procedure is intrinsically stable if (i) high-quality patterns, recently reported by Leoni et al. (2004), were chosen. (low-noise) data are available and if (ii) the distribution is not The four patterns are relative to nanocrystalline ceria powders too broad. To recognize the criticality of (i) and (ii), we can produced by sol±gel and calcinated at 573, 673, 773 and 873 K, look at the simulated distributions in Fig. 2. Noisy data can respectively (further details on the synthesis and data colleclead to a poor estimation of the lower end of the distribution tion can be found in the cited reference). (correlation with the background), whereas broad (or multiDiffraction data previously analysed by WPPM by assuming modal) distributions are intrinsically a challenge for LPA, as a lognormal distribution have been reanalysed using the the diffraction pattern is in¯uenced by the weight function present procedure, choosing a spherical shape for the crysmore than by the distribution. As a consequence, the contritallites and starting the re®nement from a uniform size bution of small grain sizes can be easily missed or confused distribution (i.e. assuming that no a priori information on the with the background. In any case, the re®ned distribution must distribution was available). The spherical shape is fully comply with the condition that the (size) Fourier cosine consistent with the TEM observations (Leoni et al., 2004). The transform must decrease steadily in order to have a typical resulting patterns are shown in Figs. 3(a)±3(d ). The residual is peak shape in the diffraction pattern. This condition intrinsiregular and ¯at in all four cases, closely resembling the result cally prevents large oscillations in the re®ned size distribution, obtained with the fully analytical WPPM (cf. Leoni et al., 2004; as far as the quality of the data is appropriate. Leoni & Scardi, 2004).

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J. Appl. Cryst. (2004). 37, 629±634

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Figure 3

Modelling result for nanocrystalline ceria powders calcinated for 1 h at (a) 573, (b) 673, (c) 773 and (d ) 873 K. Raw data (dots) and the result of the re®nement (continuous line) are shown, together with the residual curve (difference between data and model). (e)±(h) and (i)±(l ) show, respectively, the corresponding weight functions, w(D), and size distributions, g(D). The full histogram is the result of the present algorithm, whereas the continuous line is the traditional WPPM result. In ( j), TEM data are also shown by a hashed histogram.

5. Conclusions

References

A numerical modi®cation of the WPPM algorithm is proposed, allowing a general grain size distribution to be re®ned from powder diffraction data. Being framed in the WPPM context, the size distribution can be re®ned together with other structural and microstructural parameters of the specimen under study, accounting for the presence of other line broadening sources. When tested on simulated patterns, the proposed algorithm accurately reproduces the original size distribution down to the ®nest details. The algorithm was also used to analyse the experimental patterns of nanocrystalline ceria calcinated at various temperatures, with results comparable with those of the WPPM with an assigned lognormal distribution and in good agreement with the TEM observation. Besides validating the proposed approach, these results also con®rm the appropriateness of using lognormal distribution functions for the description of nanocrystalline powders produced by sol±gel.

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The authors thank Dr J. I. Langford for critical reading of the manuscript. J. Appl. Cryst. (2004). 37, 629±634

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