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Fractal Correlation Analysis of X-Ray Diffraction Patterns with Broad Background Carlos Santolalla, Jose Alvarez-Ramirez, José Antonio de los Reyes-Heredia, and Gerardo Chavez Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie303069y • Publication Date (Web): 27 May 2013 Downloaded from http://pubs.acs.org on June 4, 2013

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Industrial & Engineering Chemistry Research

(Corrected Manuscript No. ie-2012-03069y) Fractal Correlation Analysis of X-Ray Diffraction Patterns with Broad Background

C. Santolalla, G. Chavez-Esquivel, J.A. de los Reyes-Heredia and J. AlvarezRamirez*

Chemical Engineering Department Division de Ciencias Basicas e Ingenieria Universidad Autonoma Metropolitana-Iztapalapa Apartado Postal 55-534 Mexico D.F., 09340 Mexico

*

Corresponding author. Jose Alvarez-Ramirez. Email address: [email protected]. Fax/Phone: +52-5558044600

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Abstract X-ray diffraction (XRD) patterns with broad background are commonly found in powders where crystallization is incomplete or mixed with amorphous material. This is the case of alumina used for, e.g., heterogeneous catalysis purposes where certain degree of amorphicity is desired for obtaining prescribed material texture (e.g., porosity and area). This work uses detrended fluctuation analysis (DFA), a method intended for fractal analysis of noisy signals, to characterize XRD pattern with broad background. The idea is that XRD with broad background is not fully random, but contains information on regularity patterns expressed as correlations of the intensity signal. Sol-gel alumina fired at 500 o C and mixed aluminium/zirconium oxides fired at three different temperatures were used as examples for illustrating the applicability of the method. It is shown that the fractal DFA is able to locate angular regions associated ideal Powder Diffraction File-ICDD lines of diverse alumina phases. The results are discussed in terms of the corresponding Raman spectrometry analysis for contrasting the possible phases contained in the material. A crystallinity index is introduced in terms of a distance to randomness, so the regularity of a given phase can be quantified when the material is not fully crystalline.

Keywords: X-ray diffraction; Characterization; Fractal analysis.


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1. Introduction Solids with well-defined crystallinity exhibit specific x-ray diffraction (XRD) patterns with sharp, easily observed, intensity peaks. Both the positions, which correspond to lattice spacing, and the relative intensity of the lines are indicative of a particular phase and material, providing a pattern for comparison. In contrast to solids with crystallinity where the XRD pattern exhibits sharp peaks, material with incomplete crystallinity or blends of crystalline and amorphous phases can lead to broad background intensity signals. Many polymers and semiconductors where crystallinity is constrained to a limited part of the molecular structure are examples of materials that can lead to XRD patterns with broad background fluctuations. Incomplete annealing at relatively low temperatures of ceramic material can also produce amorphous structures where crystalline phases can be mixed with amorphous material within a complex geometric configuration displaying non-regular pattern. Broad background XRD patterns can be obtained for nanosized material due to the dispersión effect induced by the distribution of small particle size. As an example of cases with broad background fluctuations, Figures 1.a and 1.b show the XRD pattern corresponding to two different calcination times (0 and 6 hours) for alumina prepared by the sol-gel method 2− 4 and fired at 500 o C by 8 hours. The XRD pattern presents a broad background for which angular regions of regular intensity cannot be easily distinguished. For this calcination conditions, bayerite and boehmite groups are likely converted into η - and γ -alumina phases. A prominence is observed at about 40 degrees for the non-fired sample, which can be related to the presence of two precursors for

γ -alumina; namely, the bayerite and boehmite 5 . After 6 hours of the calcination process, one could expect the formation of crystalline γ -alumina, which should be reflected as 3 ACS Paragon Plus Environment


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intensity peaks coinciding with the Powder Diffraction File (PDF) pattern. However, the background intensity fluctuations are so large for a clear conclusion of the formation of γ alumina in the fired material. The broad background in the intensity signal could be caused by incomplete material calcination, which can yield a complex mixture of crystalline and amorphous material. As a consequence, the interpretation of XRD patterns like those in Figure 1 is not, in general, straightforward for non-specialized practitioners. This work proposes a fractal analysis approach for characterizing XRD patterns with broad background. The motivation arises from the observation that sharp intensity peaks, related to a particular structure or phase, are regular patterns reflecting the underlying regularity in the material configuration. In this way, a sharp intensity peak at a given angle is displayed because the x-ray scattered radiation is concentrated on the detector by regular geometries corresponding to crystalline structures. In contrast, radiation concentration is not met for amorphous material where regularity is not longer present. The idea behind our proposal is that the broad background pattern is not fully random, but contains intrinsic correlations that can be related to the commonly displayed fractal patterns in the material structure 6, 7 . In this way, the approach consists in detecting regularities reflected as correlations in the intensity signal. In turn, the presence of correlations at a given angular region can be compared with ideal PDF cards to conclude the presence or not of suspected material phases. The detrended fluctuation analysis (DFA) analysis implemented within a rolling window framework is used in this work to obtain a correlation pattern corresponding to specific intensity signals 8 . Sol-gel alumina fired at 500 o C and mixed aluminum/zirconium oxides calcinated at three different temperatures were used for illustrating the applicability of the method. An index of crystallinity defined


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as a distance to randomness is introduced. For alumina, the fractal analysis results suggest a gradual transition between phases with calcination time. On the other hand, the fractal properties of the broad background fluctuations for alumina/zirconium oxides indicated a crystallinity degree depending on firing temperature and composition. The results are discussed in terms of the corresponding Raman spectrometry analysis for contrasting the possible phases contained in the material.

2. Materials and Methods 2.1. Materials High-purity aluminum tri-sec-butoxide (CH12H27AlO3, Aldrich, 99%) and zirconium propoxide (Zr[O(CH2)2CH3]4 , Aldrich, 70%) were used for all sample preparations.

2.2. Alumina Preparation In buthanol, 25.73 mL of aluminum tri-sec-butoxide was dissolved. To this solution, 1 g of oxalic acid (Baker, 99%), the hydrolysis catalyst, was added to obtain pH 5. Thereafter, 15 mL of water were slowly dropped into the solution, which was refluxed and stirred for 3 hours at 80 o C until forming a gel. Subsequently, samples were dried at 70 o C and annealed in air at 500 o C for 8 hours.

2.3 Mixed Aluminum/Zirconium Oxides Preparation Two groups of samples were prepared with Al/Zr-atomic ratio of 2 and 3 were prepared. The corresponding fraction mol of ZrO2 is 0.3 and 0.5 for AZ3 and AZ5, respectively. For simplicity, the first and the second sample groups are named AZ3 and AZ5, respectively. 5 ACS Paragon Plus Environment


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The mixed oxides were prepared according to the following procedure: The precursors aluminum tri-sec-butoxide and zirconium propoxide were dissolved with isopropilic alcohol (99 %, Baker) at 5 o C. Hydrolysis was induced by gently pouring nitric acid (67.5%, Baxter) dissolved with deionized water while maintaining the solution under constant stirring. The precipitate was heated at 100 o C for 24 h and then divided into three parts. Finally, the samples were calcinated at 500, 700 and 900 o C for 12 h.

2.4. XRD Analysis The calcination process was monitored with X-ray diffraction (XRD) analysis by sampling every hour. The XRD patterns were measured at room temperature with a Siemens D-5000 diffractometer having Cu Kα radiation and a secondary beam graphite monochromator. Intensities were measured in the 5-70 degrees 2θ range with a 0.02 degrees step size and a measuring time of 1.0 second per point.

2.5. Raman Spectroscopic Analysis Raman spectra were acquired on a T64000 Jobin-Yvon-Horiba triple monochromator with 1800 grooves 1/mm holographic gratings. A Spectra-Physics Stabilite-2016 argon ion laser was used for supplying 514.5 nm exciting radiation. A power of about 100 mW was employed. The slit width was 1/cm. The wavenumber accuracy of the recorded spectra was

± 1 1/cm. The powdered samples were pressed into wafers and rotated at high speed for reducing the damage of the heat effect on the sample.


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2.6. Detrended Fluctuation Analysis Auto-correlations are commonly found in time series from diverse fields from natural, social and economical systems. The auto-correlation function C (s ) is traditionally used as an index for quantifying the strength of persistence in time. In this form, if the elements of a time-series x k are not correlated, C (s ) should be zero for all time-scales s, meaning that

future information is not related to past information. In general, correlations exist only up to a certain number of days s* , implying that the correlation function vanish for time scale s* . For long-term correlations, C (s ) decays by a power-law C ( s ) = xk xk + s ≈ s −γ . For large values of s , a direct calculation of C (s ) can be hindered by the level of noise and by nonstationarities in the data. If the time-series is stationary, one can use standard spectral analysis techniques and calculate the power spectrum E ( f ) of the time-series as a function of the frequency f . For long-term correlated data, one has that E ( f ) ≈ f β , where

β = 1 − γ . Other traditional approaches use variogram-based methods, which are commonly used to estimate the scaling features of correlated time series. However, if the time-series is non-stationary, the detrended fluctuation analysis (DFA) offers an alternative for calculating the correlation exponent 8 . In this approach, the time-series xk , k = 1,..., N , is first integrated

Yk = ∑ (x j − xk ), k = 1,..., N k

j =1

where

xk =

1 N

N

∑x

j

j =1



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is the time-series mean. After dividing Yk into N s = [N / s ] not-overlapping segment of equal length s , a piecewise polynomial trend Ys ,k is estimated within each segment and the detrended series is calculated as Y~k = Yk − Ys , k . The degree of the polynomial can be varied

in order to eliminate linear, quadratic or higher order trends of the integrated time-series. Here we used linear polynomials for detrending. The fluctuation function is computed as

 1 F ( s ) =   sN s

1/ 2

 Y~k2  ∑ j =1  sN s

Each integrated time-series is self-similar if the fluctuation function F (s ) scale as a powerlaw with the segment size s (i.e., the number of strides in a segment of observation or the time scale). Typically, F (s ) will increase with segment size s . A linear relationship on a double log graph indicates that F ( s ) ≈ s α , where the scaling exponent α (also called the self-similarity parameter) is determined by calculating the slope of the line relating log F (s ) to log s. For a process where the value at one step is completely uncorrelated with any previous values (e.g., white noise), one has α = 0.5 . In contrast, long-range, persistent correlations are present if α > 0.5 . A value α < 0.5 signifies antipersistent correlations (a large stride interval is more likely to be followed by a small one and vice versa over different time scales). The scaling exponent a is related to the correlation and spectral exponents by α = 1 − γ / 2 = (1 + β ) / 2 . The DFA analysis provides quantification, given in terms of a scaling exponent, of the correlations contained in a sequence. The major problem of the DFA method is its sensitive to short-term memory. Therefore, in order to correct for short-range dependence we have applied the DFA analysis to blocks of shuffled data; i.e., one picks a random permutation of the data series within blocks of predetermined size (in general, small size 8 ACS Paragon Plus Environment

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blocks) and applies the DFA analysis to this shuffled data. This is justified due to the Lo's critique 9 that fractal scaling exponent estimators is sensitive to the presence of short range dependence and the effect of random permutations in these small blocks is necessary to destroy any particular structure of autocorrelation within these blocks. In this work, we used blocks of size 15 for performing shuffling of sequences. Figure 2 shows that after removing the background noise by Fourier filters, the resulting XRD patterns do not contain useful information on the presence or not of the different alumina phases. This suggests that the background noise can contain important information about the material phases, so the intrinsic structure of such broad background should be explored. The methodology uses the DFA analysis for detecting angular regions where the noisy XRD pattern I (2θ ) contains certain regularity in the sense that the signal exhibits departures from random behavior. The underlying idea is that intrinsic correlations of I (2θ ) reflect a certain degree of ordered patterns within the complex material geometry. In this way, a sharp peak in the intensity pattern I (2θ ) that reflects a highly organized material structured corresponds to high correlations. On the other hand, highly disordered geometries correspond to random behavior of the XRD intensity fluctuations, which in turn are quantified by a scaling exponent close to the value of 0.5 . Based on these ideas, the proposed methodology for analyzing XRD patterns with broad background consists in the transformation of the intensity signal I (2θ ) into a scaling exponent pattern α (2θ ) via the fractal DFA. That is, the following transformation is performed by the fractal analysis: DFA

I (2θ ) → α (2θ )



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The above transformation is achieved by implementing the DFA analysis described above within a rolling window framework. In this way, the analysis for characterizing the presence of material phases is performed in the correlations pattern α (2θ ) rather than in the original intensity signal I (2θ ) . The DFA analysis can be seen as a nonlinear operator that filters the intensity signal to remove high-frequency components into a transformed signal domain. By recalling that

α = 0.5 corresponds to independent, non-correlated signals and that high values of the scaling exponent reflect stronger correlations (i.e., stronger geometric order), a fractal crystallinity index is introduced as follows: C (2θ ) = α (2θ ) − 0.5 In this way, C (2θ ) = 0 will correspond to fully amorphous material without regularities in its geometric structure. The larger the value of C (2θ ) , the stronger the orderliness of the molecular lattice. Highest values of C (2θ ) should correspond to fully crystalline structures inducing sharp intensity peaks.

3. Results and Discussion 3.1. Monitoring the Calcination of Alumina Before presenting and discussing the XRD results and the corresponding DFA analysis, a brief description of the main process involving alumina calcination are in order. At least seven crystalline phases have been characterized for alumina; namely, η, γ, χ, δ, κ, θ and α (see references 10-12). Boehmite is the main precursors for alumina phases. The boehmite has a crystalline structure made up of layers, formed by sharing corner octahedra, parallel


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to the (100) plane. The octahedra are nearly regular and have four of their corners occupied by oxygen atoms and the other two by hydroxyls. The α-alumina is the only phase that can be obtained after thermal treatment above 1000

o

C. The other alumina phases appear

always mixed, because the transition temperature range between them is not so sharp. In this way, the transformation between alumina phases strongly depends on the precursors and the thermal treatment used in their stabilization. The preparation procedure described above involves the following transformation path for the different alumina phases 5 : a) Boehmite gel aged at 40 o C leads to bayerite (Al2O3 ⋅ 3H2O). b) Bayerite aged at 80 o C produces crystalline boehmite (Al2O3 ⋅ H2O). c) After calcination at temperature 500 o C, crystalline boehmite is transformed into γ -alumina (Al2O3 ⋅ n H2O, 0 < n < 0.6 ). The presence of γ * -boehmite, a phase having the same crystalline structure than boehmite, has been also reported. During the transformation of boehmite into γ -alumina, the hexagonal oxygen arrangement is converted into the cubic oxygen arrangement. Calcination at higher o

temperatures, from 600 to 900 C, leads to blends of γ -alumina and δ -alumina phases (Al2O3 ⋅ nH2O, n low). The above transformations are rarely perfect, leading in most cases to complex mixtures of amorphous and crystalline alumina phases. For instance, the steps involving boehmite gel aged at 40 o C and subsequently aged at 70 o C can produce a material containing both bayerite and crystalline boehmite. As a consequence, the XRD analysis should produce dispersed intensity patterns similar to those in Figure 1 where peaks correspondent to theoretical ICDD cards are not easily verified with direct inspection. Figure 3.a shows TEM images of the raw (i.e., non-calcinated) alumina where flakelike formation typical for boehmite can be observed. Figures 3.b and 3.c presents TEM 11 ACS Paragon Plus Environment


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images of calcinated alumina after 2 and 5 hours of calcination time. The presence of crystalline structures coexisting with amorphous material can be observed, indicating the gradual formation of γ -alumina. The crystalline lamellae exhibit a cubic formation, which is attributed to the presence of γ -alumina 13 . The high-resolution image yields a d spacing of about 4.55 A, which matches that of (111) and (400) crystal planes of γ -alumina. On the other hand, Figure 4.a shows the Raman spectra for the non-calcinated material and for alumina after different calcination times. The observed Raman peak positions in crystalline boehmite are at about 570, 1040 and 1460 cm −1 . During the calcination process, the γ alumina is formed at the expenses of the crystalline boehmite. However, it has been reported 13 that Raman peaks are hardly detected for the γ -phase. By normalizing the intensity patterns with respect to the intensity for boehmite at 200 cm −1 , it can be noted that the Raman peaks observed for boehmite exhibit an important decrement during the first two hours of calcination time. This feature is interpreted as an indicative of the formation of the

γ -alumina. Figure 4.b shows the evolution of the three Raman peaks. The intensity of the peaks located at 1040 and 1460 cm −1 exhibit a nearly monotonous decrement, with 35% and 55% decrements after 8 hours of calcination times. On the other hand, the peak located at 570 cm −1 exhibits a more interesting behavior, with an important decrement during the first three hours. After this, the peak intensity presents a recovery to subsequently decrease monotonously after the sixth hour of calcination time. The origin of this behavior is not clear for us, although it could be suggesting the formation of γ * -boehmite, as it has been reported early.


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The Raman spectra in Figure 4 suggested the γ -alumina formation at the expenses of the crystalline material (i.e., boehmite). In principle, the transformation process should be reflected in the XRD analysis of the calcinated alumina. However, Figure 1 shows XRD patterns with complex broad background. At this point, the detrended fluctuation analysis will be used for characterizing the XRD fluctuations for finding a correspondence with the evolution of the alumina calcination process. In this way, the XRD experimental data consisted of 3301 observations. For balancing locality and accuracy of the correlation computations (i.e., scaling exponent) computations, the rolling window size was selected as 300 observations. Smaller window size leads to numerical instabilities of the scaling exponent by means of the slope of log( F ( s )) versus log(s ) . On the other hand, larger window size can lead to excessive smoothing of the scaling exponent pattern, which can induce important uncertainties in the location of angular regions with high correlations. For the non-calcinated (i.e., raw) material, Figure 5 illustrates the effects of the window size in the scaling exponent computations. It is noted that, although the scaling exponent pattern is qualitatively similar, the corresponding to higher rolling window size exhibits less prominent variations as a consequence of an averaging effect. In this way, scaling exponent computations were performed over a window of 300 observations. Figure 6 shows the computations of the cristallinity index C (2θ ) = α (2θ ) − 0.5 for 0.0 and 6.0 hours of calcination time. It is noted that the pattern is quite noisy, induced by the broad background of the XRD pattern. However, a low-pass Fourier filtering for removing high-frequency scaling exponent fluctuations reveals the presence of important prominences about certain angular regions. Although the XRD patterns shown in Figure 1 are very similar, the cristallinity index patterns for the two calcination times have different



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behavior, with high values of the cristallinity index concentrated at different angular regions. This indicates that, as assumed in the methodology section, the broad background fluctuations are not fully random. In fact, Figure 6 shows that the intensity noise contains serial correlations reflecting features of the material intrinsic crystalline structure. The above results suggest that the crystallinity index, defined from the fractal scaling exponent analysis, can be used for exploring the evolution of the different material phases as the calcination time is increased. Figure 7 summarizes the DFA results by showing some agreements between the locations of fractal crystallinity index peaks with the location of some PDF lines for different alumina phases. By exhibiting the PDF lines and fractal crystallinity index coincidences, we are showing that high scaling exponent values could be related to the presence of crystalline alumina phases, which were masked by the effects of broad background in the XRD pattern. The following features can be commented: a) The crystallinity index C (2θ ) does not display sharp peaks as in the XRD intensity pattern I (2θ ) . Instead, the crystallinity index obtained from DFA is distributed about a maximum

values. The DFA analysis provides a mean for processing the broad background intensity fluctuations for revealing the presence of high correlation regions that could be attributed to one or more material phases. For instance, the angular region in the vicinity of the 20 degrees region after 2 hours calcination time indicates that a crystallinity pattern conformed by bayerite and complex oxide groups. Similarly, the 28 and 40 degrees vicinities can be related to complex crystallinity conformed by bayerite and boehmite. c) The DFA analysis can reveal the existence of angular regions associated with non-trivial correlations that are not exhibited by the original XRD pattern. In fact, the DFA suggested the existence of a non-random pattern in the 55 degrees vicinity, with crystallinity index similar to that for the


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55 degrees pattern. In contrast to the original intensity pattern I (2θ ) , the fractal crystallinity index suggests that the 50 degrees region can be related to the boehmite phase. c) For the raw material, the highest fractal crystallinity index is located in the 28 degrees region, in line with previous results indicating that the sol-gel preparation yields a high content of crystalline boehmite 13 . For 6 hours of calcination time, a high crystallinity index is observed, indicating that γ -alumina is formed with at the expenses of transformation from bayerite and boehmite phases 2,11,12 . d) The presence of γ * -boehmite at about 10 degrees can be indicated, as a relatively small intensity peak is observed in such angular region 2 . It should be remarked that the results in Figure 7 do not imply that only these phases are contained in the calcinated alumina. Rather, such results indicate that a fractal analysis of the broad background XRD signal can provide important insights in the presence of certain phases that are expected according to the underlying preparation method.

5.2. Calcination of Mixed Aluminum/Zirconium Oxides As a second example, the AZ3 and AZ5 mixed aluminum/zirconium oxides (Al2O3/ZrO2) described in Subsection 2.3 will be considered. In general, mixed oxides conform complex metastable phases where individual oxides provide a complex contribution to the crystalline structure of the calcinated material. In contrast to relatively pure material, like the alumina case considered above, the characterization of mixed oxides via XRD patterns and Raman spectroscopy is more challenging. Figure 8 exhibits the XRD patterns for the calcinated mixed oxides at thee different temperatures. For AZ3 and AZ5, an intensity peak is located at about 30 degrees, while peaks at about 45 and 68 degrees are obtained depending of the 15 ACS Paragon Plus Environment


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calcinations temperature. According to the phase diagram 15 , for temperatures between 500 and 1150

o

C, the calcinated mixed oxides are composed by monoclinic and zirconium

oxide and γ -alumina. In this way, the lines for monoclinic ZrO2 (Card 24-1165) and γ alumina (Card 10-0425) are also depicted as vertical lines in Figure 8. The intensity band between 28 and 40 degrees presents important variations, coinciding with lines of both ZrO2 and γ -alumina. On the other hand, Figure 9 presents the Raman spectra for the AZ3

and AZ5 formulations for the three calcinations temperatures. For all cases, two intensity bands can be observed at about 580 and 990 cm −1 . These bands can reflect the presence of

γ -alumina, corresponding to the first two intensity bands in Figure 4.a. In contrast to the case of pure alumina, the Raman bands do not exhibit a monotonous change with respect to the calcinations temperature, which can be reflecting the formation of metastable phases when the aluminum oxide interacts with the zirconia oxide. Figure 10 presents the variations of the crystallinity index with respect to the diffraction angle. For AZ3, the mixed oxide containing 0.7 mol fraction of γ -alumina, the larger variations are observed for the region between 28 and 40 degrees. The crystallinity index for relatively high diffraction angles is small, suggesting that the alumina has the most important role in the crystallinity formation during calcinations. In contrast, the crystallinity index for the AZ5 formulation with 50% mol fraction exhibits a more complex behavior. In fact, the crystallinity index changes in an important way for diffraction angles between 45 and 55 degrees. To gain insights on the crystallinity changes, as derived from the fractal detrended fluctuation analysis, the crystallinity index for the 28-40 and 45-55 diffraction angles is estimated and presented in Figure 11. Interestingly, for both regions the cystallinity index is not monotonous with respect to the calcinations temperature. In fact, 16 ACS Paragon Plus Environment

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according to the results in Figure 11, the crystallinity exhibits the smaller value for 700 o C. It can be suggested that crystallinity is higher for 500

o

C, for which the γ -alumina is

formed while the monoclinic phase of the zirconoium oxide is stabilized for higher temperatures. In this way, temperatures of about 700

o

C correspond to the formation of

metastable mixed oxide phases that are relatively amorphous as compared to individual oxides.

4. Conclusions This work showed that the broad background XRD patterns from powder analysis can be systematically characterized for extracting information on material crystallinity. The idea is to locate angular regions where XRD intensity fluctuations contain certain degree of regularity reflected as internal correlations, which are estimated by the fractal DFA method. Correlations are quantified in terms of an scaling index, the so-called scaling scaling exponent, which reflects the regularity of geometric patterns contained in the complex, possibly fractal, material structure. The analysis approach was illustrated with two typical materials (alumina and mixed oxides), showing that the method can provide information on the angular location of regularity peaks associated to prominent lines (PDF cards) of alumina phases. A crystallinity index was introduced in terms of a distance to randomness, so the regularity of a given phase can be quantified when the material is non-crystalline.



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References (1) Warren, B.E. X-ray Diffraction. Dover Publications, New York, 1990. (2) Wang, J.A.; Bokhimi, X.; Morales, A.; Novaro, O.; Lopez, T.; Gomez, R. Aluminum local environment and defects in the crystallization structure of sol-gel alumina catalyst. J. Phys. Chem. B 1999, 103, 299-303. (3) Fullerton, D.J.; Westwood, A.V.K.; Brydson, R.; Twigg, M.V.; Jones, J.M. Deactivation and regeneration of Pt/γ-alumina and Pt/ceria–alumina catalysts for methane combustion in the presence of H2S. Catalysis Today 2003, 81, 659-671. (4) Perdigon-Melon, J.A.; Gervasini, A.; Auroux, A. Study of the influence of the In2O3 loading on γ-alumina for the development of de-NOx catalysts. Journal of Catalysis.

2005, 234, 421-430. (5) Gates, B.C.; Katzer, J.R.; Schuit, G.C.A. Chemistry of Catalytic Processes. McGrawHill, New York, 1979. (6) Liu, Sh.; Liu, X.; Xu, L.; Qian, Y.; Ma, X. Controlled synthesis and characterization of nickel phosphide nanocrystal. Journal of Crystal Growth 2007, 304, 430-434. (7) Amin, J.S.; Ayatollahi, Sh.; Alamdari, A. Fractal characteristics of an asphaltene deposited heterogeneous surface. Applied Surface Science 2009, 256, 67-75. (8)] Peng, C.-K.; Buldyrev, S.V., Havlin, S.; Simons, M.; Stanley, H.E.; Goldberger, A.L. Mosaic organization of DNA nucleotides. Phys. Rev. E. 1994, 49, 1685-1689. (9) Lo, A.W. Long-term memory in stock market prices. Econometrica. 1991, 59, 12791313.


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(10) Liu., P.; Skogamo, J. Space-group determination and structure model for κ-Al2O3 by convergent-beam electron diffraction (CBED). Acta Crystallographica B 1991, 47, 425-433. (11) Liu, Q.; Wang, A.; Wang, X.; Zhang, T. Mesoporous γ-alumina synthesized by hydrocarboxylic acid as structure-directing agent. Microporous Mesoporous Materials 2006, 92, 10-21.

(12) Liu, Q.; Wang, A.; Wang, X.; Gao, P.; Wang, X.; Zhang, T. Synthesis, characterization and catalytic applications of mesoporous γ-alumina from boehmite sol, Microporous and Mesoporous Materials 2008, 111, 323-333. (13) Roy, A.; Sood, A.J. Phonons and fractons in sol-gel alumina: Raman study. Pramana – J. Physics 1995, 44, 201-209. (14) Zhang, X.; Honkanen, M.; Levänen, E.; Mäntylä, T. Transition alumina nanoparticles and nanorods from boehmite nanoflakes. Journal of Crystal Growth 2008, 310, 36743679. (15) Jayaram, V.; Levi, C.G.; Whitney, T.; Mehrabian, R. Characterization of Al2O3-ZrO2 powders produced by electrohydrodynamic atomization, Materials Science and Engineering: A, 1990, 124, 65-81.



100

Intensity (a.u.)

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Angle, 2θ (degrees)

Figure 1. (a) X-ray intensity pattern I (2θ ) for non-fired material prepared with the solgel method. Despite the broad background and the absence of sharp intensity peaks, a prominence at about 40 degrees suggests the presence of the alumina precursors bayerite and boehmite. (b) X-ray intensity pattern I (2θ ) for alumina after 6 hours calcination at 500 o C. After this calcination time, the presence of γ -alumina could be expected. However, the XRD pattern is so noisy for drawing conclusions on the strcture of the fired material. 20 ACS Paragon Plus Environment

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Intensity (a.u.)

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Boehmite PDF Lines Bayerite PDF Lines

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Intensity (a.u.)

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γ-alumina PDF Lines

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Figure 2. Same XRD patterns as in Figure 1 after applying a low-pass Fourier filter for removing the broad background. It is noted that the noise filtering does not provide useful insights for concluding about the presence or not of the different alumina phases.



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(a)

(b)

(c) Figure 3. TEM images for (a) non-calcinated material and for calcinated alumina after (b) two and (c) five hours of calcination times. The presence of crystalline formations in the calcinated material can be observed. 22 ACS Paragon Plus Environment

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Figure 4. (a) Raman spectra for raw and calcinated alumina. (b) Evolution of the Raman peaks with respect to the calcination time.


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(a) 400 observations

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Figure 5. Scaling exponent pattern for two different values of the rolling window size. Although the results are qualitatively similar, high rolling window size yield less prominent scaling exponent peaks as a consequence of an averaging effect over a high number of observations. In contrast, very low window size can lead to numerical instabilities in DFA computations.


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Crystallinity Index, C(2θ )

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Crystallinity Index, C(2θ )

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Figure 6. (a) Crystallinity index pattern for alumina calcinated at 500 o C after 0.0 hours (raw material) and 6.0 hours calcination times. Despite the similarity of the XRD in Figure 2, the crystallinity index patterns show important differences, indicating that the background noise contains information on the structure of the alumina. For better appreciation, the crystallinity index pattern was low-pass Fourier filtered for removing high-frequency components.



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(d) 8 hours

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Figure 7. Crystallinity index as a function of calcination time. (a) 0.0 hours. Three peaks are located at about 28, 43 and 53 degrees, which can be related to the presence of bayerite (pink lines) and boehmite (red lines) phases. (b) A gradual deformation of the peaks in (a) are observed after 2 hours of calcination time. (c) After 4 hours , some correlations peaks are found that can be related to the formation of γ -alumina (blue lines). (d) It is apparent that the transformation from bayerite and boehmite to

γ -alumina is almost complete after 8 hours. 26 ACS Paragon Plus Environment

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(a) AZ3

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Diffraction Angle (2θ)

Figure 8. XRD patterns for (a) AZ3 and (b) AZ5 mixed oxides calcinated at three different temperatures. In both cases, broad background fluctuations are observed. The PDF intensity lines for γ -alumina and the most prominent (cubic, tetragonal and monoclinic) ZrO2 lines are exhibited. The correspondence of some intensity peaks with PDF lines is evident for some XRD patterns.



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Figure 9. Raman spectra for (a) AZ3 and (b) AZ5 mixed oxides. The Raman peaks located at about 580 and 996 cm −1 can be observed.


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Crystallinity Index, C(2θ

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Diffraction Angle (2θ)

Figure 10. Crystallinity index pattern for (a) AZ3 and (b) AZ5 mixed oxides. Important variations are observed for the diffraction angle ranges 28-40 and 45-55 degrees.




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(b) AZ3 AZ5

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Figure 11. Crystallinity index pattern for AZ2 and AZ3 mixed oxides. The upper panel corresponds to the 28-40 degrees region, while the lower panel to the 45-55 degrees region.


Fractal Correlation Analysis of X-ray Diffraction ... - ACS Publications

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