International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870722003226

Mixed-phase heterogeneity

Krystyna Lawniczak-Jablonskaa*

aInstitute of Physics, Polish Academy of Sciences, Lotnikow 32/46, 02-668 Warsaw, Poland
Correspondence e-mail: jablo@ifpan.edu.pl

The common understanding of heterogeneity, or a heterogeneous material, is as an opposite to homogeneity, or a homogeneous material. A material can exhibit heterogeneity with respect to elemental, chemical or phase composition, colour, density or any other physical property. X-ray absorption spectroscopy (XAS) is an element-selective technique and a `fingerprint' of local atomic order; therefore, it has particular importance in studies of elemental, chemical and phase-composition heterogeneity in materials. Several examples of XAS investigations of heterogeneous materials are presented with an emphasis on the different methods used for data analysis.

Keywords: mixed-phase materials; heterogeneity.

1. Quantitative estimation of the number and the amount of different compounds

Applications of X-ray absorption spectroscopy (XAS) to disentangle the contributions of different compounds in heterogeneous materials are founded on the evidence that a given chemical compound has an X-ray absorption near-edge structure (XANES) with a characteristic shape. The XANES spectrum is closely related to the density of unoccupied states distribution allowed by electron-transition rules, and this density reflects the chemical bond. This fact opens a way to apply XANES to the quantitative estimation of the content of different compounds linked to a given element. Convincing examples of the sensitivity of XANES to chemical and particularly oxidation states have been reported in numerous publications. The K edges of metals are reported most frequently (Figs. 1[link]a and 1[link]b). One can notice not only changes in the fine structure of XANES spectra but also in the chemical shift indicating the difference in charge transfer in different compounds. Comparison of the spectrum of a studied sample with spectra of known compounds yields a qualitative assessment of the chemical and structural environment of an absorbing atom. Thus, one can obtain information about charge transfer from the edge position and anticipate possible compounds.

[Figure 1]

Figure 1

(a) Mn K-edge XANES of metallic manganese and different manganese oxides. (b) Mg K-edge XANES of magnesium in different compounds.

2. Least-squares linear combination fitting

The most popular method for quantitative analysis of the content of different chemical compounds of a given element is least-squares linear combination (LC). This method is simple and easy to implement. It is based on the least-squares algorithm to fit the sum of a given number of reference spectra to an experimental spectrum. The spectrum of the sample of interest is modelled by weighting the spectrum of each known reference and minimizing the difference between the modelled and measured spectra. The weighting factor is the fraction of each reference present in the sample. The IFEFFIT (Ravel & Newville, 2005[link]) or WinXAS (Ressler, 1997[link]) programs implement reliable algorithms for this kind of analysis. The LC performs well if the number of potential chemical and structural references is small and if the spectra of the chosen references exhibit unique features that allow differentiation between them. Moreover, the reference spectra should be of good quality and should be recorded under similar conditions. Furthermore, a proper energy-alignment procedure should be applied.

The LC method has been widely exploited in the investigation of geological samples, which are usually mixtures of many compounds. A method that is initially applied in studies of heterogeneous materials is X-ray diffraction (XRD). Nevertheless, not all compounds can be detected by XRD. The requirement for the presence of a crystal structure needs to be fulfilled. Moreover, the compounds should be present in a sufficient amount to give diffraction peaks in the diffraction pattern for the dominant phases.

A good example of such studies is presented in Klepka et al. (2010[link]). In this paper, the identification of the minority chemical compounds containing magnesium, manganese and chromium in the mineral ilmenite originating from geological deposits in Norway, Australia, China and India is presented. The most remarkable result was obtained for Norwegian ilmenite, where magnesium was found to be present as three chemical compounds [MgTiO3 at 58%, MgSiO3 at 30% and probably a mixture of MgO and Mg(OH) at 12%; Fig. 2[link]a]. Moreover, magnesium formed different chemical compounds in each of the investigated ilmenites. For example, in ilmenite from Australia only MgTiO3 was found (Fig. 2[link]b). The content of magnesium measured by conventional electron-probe microanalysis (EPMA) was at a level of 3–8%. The results of XANES analysis were then used in Rietveld refinement of the powder diffraction data. In the case of Norwegian ilmenite, the presence of MgTiO3 and MgSiO3 phases was confirmed. MgO/Mg(OH) does not form a crystalline phase. Therefore, XAS analysis can help in the identification of minority phases due to its elemental sensitivity. In addition, it was proved that the chemical binding of magnesium in ores of ilmenite depends on the climatic and geological conditions at the place of origin.

[Figure 2]

Figure 2

(a) Mg K-edge XANES of Norwegian ilmenite (line) and LC fitting result (line with circles). (b) Comparison of the Mg K-edge XANES spectra of Australian ilmenite with that of MgTiO3.

An interesting overview of the application of XAS over more than 20 years for the investigation of arsenic in solids including rocks, soils, sediments, synthetic compounds and numerous types of biota, as well as aqueous phases, is presented in Foster & Kim (2014[link]). This review discusses the different types of analysis that were performed on As XANES spectra and describes the benefits, drawbacks and limitations of each analytical approach. Analysis was performed to (i) identify the oxidation state and quantify the relative abundance if multiple oxidation states were present, (ii) select the appropriate number and type of reference compounds to be used in fitting and (iii) quantify the relative abundance of species in a multi-species sample.

Impressive studies of artworks exploiting the LC method and a modern detection system have been presented by Monico et al. (2015[link]). These authors report the nature and distribution of secondary chromium compounds from lead chromate-based pigments (chrome yellows) and quantitatively determine their abundance in two paint micro-samples taken from artworks by Vincent van Gogh. The findings formed the basis to construct a model of paint ageing. This study illustrates the potential of the XAS method for the analysis of cultural heritage.

Although the method works reliably when only a few reference spectra are fitted, it can be applied to more complicated cases, as reported in Piskorska et al. (2007[link]) for composite materials. BN composites are attractive to industry due to their combined high hardness and low friction, high resistance to corrosion and attractive thermal and electrical properties. Several so-called binding phases are formed during the technological process of composite preparation. Therefore, the edges of several elements (Ti, Si, C, N and B) should be examined. In such cases, from an analytical point of view, it is very useful to confirm the considered set of phases using other techniques. X-ray photoelectron spectroscopy (XPS) and/or XRD can be used as complementary methods (Piskorska et al., 2007[link]; Lawniczak-Jablonska et al., 2006[link]; Figs. 3[link]a and 3[link]b). Some discrepancies between the peak intensities in the fitted spectra and the composite spectrum of a BN sample with binding phase Ti3SiC2 (Fig. 3[link]a) may indicate that the formed phases are defective or of altered stoichiometry (in comparison with the reference samples TiB2 and BN used in fitting). Another reason may be a nonlinear background caused by the composite low conductivity since the spectrum was registered by the total electron-yield method. Nevertheless, both analyses confirmed that boron forms two phases, i.e. TiB2 and BN only. Moreover, Ti K-edge and N K-edge XANES studies validated the existence of these phases (Lawniczak-Jablonska et al., 2006[link]). Taking into account that each method (XAS and XPS) provides information at different depths, the difference in the amount of specified compound found by a given method may provide a suggestion of the spatial distribution of the phases in the composite material.

[Figure 3]

Figure 3

(a) B K-edge XANES of the composite formed by BN and the binding phase Ti3SiC2 (taken at a 1:3 volume ratio) after 3 min at a pressure of 109 Pa and a temperature of 1750°C. The reference compounds TiB2 and BN were used in the fit. (b) The XPS spectrum of the boron 1s level from the same sample showing two binding energies of boron characteristic of TiB2 and BN. Both analysis confirmed that boron is bound in two phases, TiB2 and BN.

3. Principal component analysis

An alternative method to check whether all considered references are necessary or sufficient to fit the spectrum under consideration is principal component analysis (PCA).

PCA has been used in chemical spectroscopy since the 1970s. The first application of this method to the analysis of XANES spectra was published in 1992 (Fay et al., 1992[link]). The application of PCA is helpful when one is not sure how many or which references are needed to decompose a studied sample, and when other methods cannot provide a sufficient set of compounds. If one chooses the wrong group of reference spectra, the analysis cannot be correct. PCA decomposes a set of data files mathematically into the minimum number of (principal) components needed to describe the data. These principal components are those which contain the signal and are mathematically sufficient to reconstruct the considered experimental spectrum from some linear combination of the principal components. The remaining components in the system are considered to be noise. Therefore, this procedure allows the determination of the number of components or reference spectra that are needed to describe the set of measurements within experimental error. The results obtained from PCA offer constraints that can then be applied in the traditional analysis. The best situation is when the principal components form a set of reference spectra, but this is not always the case. By adding or removing some of the reference spectra one can judge how important the considered reference compound is in analysis.

A comprehensive comparison of PCA results with least-squares LC for S K-edge spectra of humic acid samples is presented in Beauchemin et al. (2002[link]). It was shown that PCA provided a statistical basis for choosing the number of reference species to be included in the fitting model by indicating which references were statistically more likely to explain the spectra of the humic acid samples. The selected references and the scaling coefficients obtained by the PCA approach deviated by ≤6 mol% from the results obtained by performing LC fitting using a number of binary, ternary and quaternary combinations of sulfur references. Statistical ranking of the most likely reference spectra contributing to the unknown spectra enhanced LC fitting by reducing the analysis to a smaller set of reference spectra. The PCA approach is a valuable contribution to other spectral fitting techniques since it provides statistical criteria that improve insight into the data and leads to a more objective approach for fitting.

Technical developments in the equipment at XAS beamlines, particularly in the recording of spectra (Quick-EXAFS or energy-dispersive experiments), has opened the way to performing in situ and in operando XAS experiments. This approach now offers access to a deeper and more accurate temporal description of the chemical species involved in the processes by tracking the relationship that exists between the local order around the active elements. In such experiments, XAS data are collected as a function of time while changing a variable of the experimental conditions such as temperature, concentration or pressure. It has become routine to collect hundreds of spectra from a single sample. For analysis of such data sets, automated data-processing procedures such as least-squares fitting of data with LC of reference spectra and PCA are powerful tools, but have some limitations. Namely, LC requires reference spectra and PCA often provides mixed components that are difficult to interpret. In such a case, multivariate curve resolution with alternating least-squares fitting (MCR-ALS) can be used as a method to separate constituents from XAS data.

4. Multivariate curve resolution by the alternating least-squares (MCR-ALS) method

Tauler (Tauler, 1995[link]; de Juan & Tauler, 2003[link]) was the first to propose the MCR-ALS method for analysis of spectroscopic data from mixed-constituent materials. The basic assumption required for application of the MCR-ALS method is an inner linear structure of the data set, which is fulfilled in the case of XAS. MCR techniques do not require a priori information concerning the set of components except for an estimation of their number. For this approach, the PCA algorithm can be used. During PCA analysis, the experimental spectra are decomposed into orthogonal spectral components which do not have any direct physical meaning. Once the number of components has been estimated, ALS optimization is started using initial estimates, for example from PCA, with the MCR-ALS method imposing physically and chemically meaningful constraints, in contrast to mathematical or statistical constraints as in PCA. These meaningful constraints can be, for instance, non-negativity of XAS absorbance or component concentration, profiles without double peaks or the concentrations of all the components being equal to a constant value. An interesting example of application of the MCR-ALS method to monitoring the activation of a copper alumina catalyst is presented in Cassinelli et al. (2014[link]). The authors discussed the strategy for the use of the MCR-ALS method and the reliability of the results for time-resolved XAS data. The intermediate copper species determined by MCR-ALS is in a monovalent oxidation state and is characterized by a local order of two O atoms at a distance of 0.1915 nm. After isothermal treatment for 30 min, the intermediate species appears to be resistant to complete copper reduction. Copper speciation obtained using the MCR-ALS method (Fig. 4[link]b) leads to very different results to those obtained using LC and bulk CuO, Cu2O and metallic copper as reference spectra (Fig. 4[link]a). Moreover, the concentration profile determined by MCR-ALS was fully consistent with the reducibility of the catalyst measured using a temperature-reduction program.

[Figure 4]

Figure 4

The fraction of copper components during the activation of a copper alumina catalyst estimated by performing LC analysis (a) and MCR-ALS analysis (b) on the same experimental data set (taken from Cassinelli et al., 2014[link]).

Another interesting example of the application of MCR-ALS together with XRD to monitoring the crystallization of titania polymorphs in solution is presented by Kränzlin et al. (2014[link]). The authors define four crystallization stages, which allow dynamic changes of the crystalline and noncrystalline species to be followed and the information content to be optimized. From these studies they found that nucleation and growth are independent processes, despite the commonly accepted polymorphic crossover from anatase to rutile that is activated by the critical size of nanoparticles in solution. Moreover, 5.9 nm rutile nanoparticles were formed prior to the formation of 8.4 nm anatase nanoparticles. The authors conclude that the crystallization process is started by the formation of an intermediate noncrystalline phase and by time-dependent changes in the chemical environment. This conclusion was proved by applying the MCR-ALS method to determine the number of chemical species in solution from the XANES data. It was shown that the time-dependent changes in the chemical environment directly generate the activation energies involved in the nucleation of either rutile or anatase TiO2. Furthermore, these results show that thermodynamic equilibration does not determine the nucleation and growth of TiO2 crystals, since the rutile phase forms first and, initially, also with smaller sizes than the anatase phase. Therefore, the size-induced phase transformation is not an inherent material property of TiO2 but strongly depends on environmental variables.

5. Summary

The XAS technique is a method that is sensitive to element and local atomic order. Thus, it is a perfect tool to disentangle the contributions of different compounds in heterogeneous materials. Heterogeneous materials are very important in the technology of nanostructures, catalysts, colloids and composites. Minerals, soils and even the dyes used in art are examples of mixed-phase materials. Several analytical methods exist to perform XAS analysis of these materials, but each of them should be used together with all additional information. The least-squares linear combination method is simple and easy to implement. This method provides a reliable solution if the number of potential chemical and structural references is small and if the spectra of the chosen references exhibit unique features that allow differentiation between them. Principal component analysis is a trustworthy method to check whether all of the considered references are necessarily needed or are sufficient to fit the spectrum. In the case of the analysis of a large number of spectra from a time-evolved sample, multivariate curve resolution with alternating least-squares fitting is nowadays frequently applied and has great potential for the comprehensive analysis of large data sets.

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