Chapter 21Quantitative Phase Analysis: MethodDevelopments

Luca Lutterotti

Abstract During the years several methods have been developed to identifyand quantify the phases in our samples. The newly developed methods respondto precise needs like increasing accuracy, lowering detection limits, automatizeand speed-up the process or overcome errors and limitations. Most of the lastdevelopments are based on the use of large set of data and full pattern analyses likethe Rietveld method. The progresses have been stimulated by the need to analyzenew and complex materials with the help of advanced hardware to collect quicklyand reliably our data. Refinement of old and new methods will be presented for thequantification of phases as well as some examples. Particular cases will be treatedfor layered materials and thin films, bulk amorphous, textured samples and claymaterials. The last frontier appears to be the combination of the diffraction withother techniques to improve the final analysis.

21.1 Introduction

In the quantitative analysis of our samples we may identify two principal casesthat it is worth to consider. The first case is when the sample is a powder or wasa bulk reduced to a powder; this is quite a common case. There is also a secondimportant case when the sample is a bulk not reducible to a homogeneous powder orit has a special structure/distribution of phases for which it is necessary to analyzeit in the original form. In the first case we can just utilize the usual methods for

L. Lutterotti (�)Dipartimento di Ingegneria dei Materiali e Tecnologie Industriali, Universita degli Studi diTrento, via Mesiano 77, 38123 Trento, Italye-mail: luca.lutterotti@ing.unitn.it

U. Kolb et al. (eds.), Uniting Electron Crystallography and Powder Diffraction,NATO Science for Peace and Security Series B: Physics and Biophysics,DOI 10.1007/978-94-007-5580-2 21,© Springer ScienceCBusiness Media Dordrecht 2012

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234 L. Lutterotti

quantitative analysis, provided we take every precaution to fulfill the requisites fora good analysis: homogeneous sample, a sufficient number of small grains andrandomly oriented.

When the sample is a bulk or a sample is not fulfilling all these requisites, weneed to employ special methodologies and measurement techniques to perform thequantitative analysis. In this chapter we will cover this second case showing someof the methods developed to quantify the phases amount and distribution throughdiffraction and the help of other techniques when necessary.

21.2 Layered Materials and Thin Films

There are several cases in which the material is structured in more layers. Each layermay have different phases and a phase may be present in more than one layer. If thesample is analyzed the traditional way, by assuming an homogeneous volume, thismay lead to two problems:

• The phase fraction we obtain does not really represent the composition of thesample and their quantification depends on the beam penetration.

• The absorption factor in the formula for the quantitative analysis is no morea constant value and this also will result in increasing the inaccuracy of theanalysis.

In such cases it is necessary to use a model for the layered material to take intoaccount thicknesses of the layers, absorption factors and phases distribution. Theabsorption factor depends on the phase content and thickness in each layer and thiscause a circular dependency in the quantification method. The Rietveld method as amulti parametric fitting technique can solve the problem by including a suitablemodel. This has been done and demonstrated for the analysis of the oxidationof some super-alloys [1]. Following that work we can compare an analysis madeassuming a homogeneous sample and subsequently adopting the correct model. Inthis example an Inconel 738 has been oxidized at high temperature (for details weremind to the original work) and the phases have been identified and quantified usingan X-ray diffraction pattern collected in Bragg-Brentano geometry, copper radiation.Table 21.1 reports the phase quantification by the Rietveld method assuming ahomogeneous sample and the layered model.

It is immediately clear how much informative is the correct model. Determininga phase fraction for the substrate using the homogeneous model does not makesense. A first indication that the sample has a layered structure is given by the Bfactors (a overall B factor for each phase has been used). Looking at the third rowof Table 21.1 phases on top of the layers show much higher intensity at low anglerespect to high angle and this is corrected in the Rietveld by a higher B factor.Instead for phases down in the sequence the reverse is observed. The correct layeredmodel has been constructed by trial and error, observing the B factor value. This hasbeen compared and validated using EDXS elemental maps in a cross section.

21 Quantitative Phase Analysis: Method Developments 235

Table 21.1 Rietveld phases quantification for the oxidized Inconel 738 assuming a homogeneoussample and a layered model [1]

Rietveld Alloyhomogeneous Cr2O3 TiO2 CrNbO4 Al2O3 matrix Ni3Al

Volume fraction 0.49 0.011 0.353 0.055 0.086 0.005B factor 0.297 �1.77 5.72 4.76 �4.75 �3Rietveld layered

modelVolume fraction Layer thickness

(�m)Layer # Cr2O3 TiO2 CrNbO4 Al2O3 Alloy

matrixNi3Al

1 – – 1 – – – 0.0892 0.952 0.048 – – – – 2.23 – – – 0.973 0.026 0.001 0.364 – 0.11 – – 0.845 0.045 55 – – – – 0.95 0.05 1B factor �0.19 0.73 �0.45 1.98 0.51 0.6

To correctly compute the absorption factor in a layered model we point the readerat the excellent formulation in [2] (see Eq. 23) that include also the case of a generalorientation of the sample surface respect to the incident and diffracted beam.

The procedure can be used successfully also for textured thin films provideda sufficient number of pattern/data is collected to characterize quantitatively thetexture [3–5]. This is necessary when the texture is strong as it happens oftenin the thin film production. Otherwise, for the errors connected to the adoptionof a simplified texture models and only one pattern, we remind at a subsequentparagraph.

21.3 Clay Materials and Turbostratic Disorder

The quantification of clay materials or in general of phases having stacking faultslike turbostratic disorder (e.g. graphitic materials) has been always a big problem asthe traditional Rietveld method is not able to fit the pattern with the same accuracyas for other phases. Indeed, also other methods (internal standard, RIR or PONKCS;see the work of Madsen in this book), if they can ameliorate the analysis, they donot provide a satisfactory solution, most of the time. In 2004 Ufer et al. [6, 7] haveintroduced an elegant way to treat the turbostratic disorder as encountered in manyclay materials. This permits a good fitting of the diffraction pattern using a simpleto setup model where it is only necessary to identify the faulting axis and fit theanisotropic crystallite size broadening. This has been used to successfully quantifyclay materials [8] as you can see in Fig. 21.1 and carbon-carbon composites. Thiskind of modeling is present in both BGMN and Maud Rietveld software.

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Fig. 21.1 Rietveld phases quantification of clay minerals using the model of Ufer et al. [6] forboth the Ca-montmorillonite and the Opal-CT

21.4 Texture and Preferred Orientation

Preferred orientations are one of the main causes of errors in phase quantificationby diffraction as enlightened in another part of this book. The best method in suchcase is to try to avoid them by changing the experiment or re-preparing the sample.Not always this is successful or can be done, as for example in the case of a bulkor a film. When the texture of the sample cannot be eliminated completely by theexperimental procedure, we need to choose an analysis strategy to get the phasequantities as much accurate as possible. We have ideally three choices:

• Not apply any correction for preferred orientations• Apply the texture factors computed from the Orientation Distribution Function

(ODF) for this sample• Fit and apply some corrections based on a simplified texture model.

To examine the effect of each method we have constructed a simple case with aknown amount of phases and ODF. We can test the different corrections and theireffect on the quantitative phase results. The test is based on a Cu-Fe bulk samplewhere the two phases were co-laminated for a certain extent. The original contentis 1/3 in volume of Fe and 2/3 of Cu. The sample in cube form has been analyzedat the IPNS neutron TOF source in Argonne. A sufficient number of TOF patterncovering different orientation of the sample has been collected in order to obtainthe true ODF for both phases. The complete analysis has been done on all patternsusing the Rietveld method with the EWIMV texture model [5]. This full analysis

21 Quantitative Phase Analysis: Method Developments 237

Fig. 21.2 Rietveld fit of all patterns of bank 2. Rwp D 18%

Fig. 21.3 Pole figures (recalculated from ODF, on the right) for the Fe-Cu sample as obtainedfrom the full refinement of all patterns by Rietveld C EWIMV; on the left the measurement pointof each pattern is reported as texture measurement angles (so-called pole figure coverage). Differentsymbols indicate patterns coming from different banks (detectors)

confirmed the 1/3:2/3 volume fraction for the Fe and Cu phases (0.667(2) % vol. Cuand Rwp D 18%) and the overall fitting for only one bank is reported in Fig. 21.2.The EWIMV method provides us the ODF from which some pole figures have beencalculated and are shown in Fig. 21.3 along with the measurement points in theorientation space. As we can see the texture is the typical one for such laminatedphases. It is not a strong texture and nearly orthorhombic.

For our test we will use the Rietveld method for the phase analysis but withonly one pattern in the center of the pole figure to be in the same usual conditionsof a Bragg-Brentano experiment. In Bragg-Brentano the usual measurement only

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Table 21.2 Result of quantitative analysis on Fe-Cu sample by refinementof a single pattern using different preferred orientation correction models

0.3–2.2 A range 0.79–2.2 A range

Type of P.O. correction % vol. Cu Rwp (%) % vol. Cu Rwp (%)

ODF corrected 67.8(14) 14.5 66.5(19) 15.6No correction 65.5(25) 31.5 63.3(37) 35March-Dollase 58.3(20) 25 57.3(38) 29Fiber harmonic, Lmax D 4 67.4(23) 27.5 62.7(41) 30.5Fiber harmonic, Lmax D 6 70.0(13) 13 66.0(25) 14.5Fiber harmonic, Lmax D 8 71.4(14) 12.7 70.2(40) 14.4

covers the center of the pole figure and consequently even if we spin the sample thisdoes not decrease the texture and only improves the statistic. Moreover measuringonly one point in the pole figures, even if for more reflections, it does not permitto discriminate the kind of texture or measure the complete ODF and in general thenormalization cannot be done. All the texture models we can apply in this case arebased on a fiber texture with the axe in the center of the pole figure. In Table 21.2the results for different models are summarized. Also two different ranges in d-spacewere used to check the effect of the number of reflections. Not always the best fitcorresponds to the best result in the quantitative analysis. Obviously correcting forthe true ODF should give a good result even if only one pattern is used. But whenthis is not possible the test shows that using no correction at all gives the worse fit butthe results are acceptable. Not always this is true for the other models. The March-Dollase produces a better fit but a wrong quantitative analysis. For the harmonicmethod this depends on the range and expansion used even if the texture is notstrong. This is due to the fact that we suppose a certain fiber texture when it is not.The advice is that when we do not know the texture type, applying no correctionsfor the texture has more probability of giving a good result even if the fit is worse(Table 21.2).

If the texture is stronger the use of the wrong model (and no simplified models,harmonic included, are correct in this case) will cause a worse result.

21.5 Microabsorption

Microabsorption or absorption contrast affects the quantitative analysis when wehave large grains of a highly absorbing phase with other low absorbing phases. Insuch cases we have few alternatives to avoid or reduce absorption contrast effects:

• Grind the powder to reduce the grain size below a certain level (usually 1 �m).• Change the wavelength or radiation to a more penetrating one.• Apply a model to correct the absorption contrast in the analysis (this requires the

knowledge of the grain sizes of the phases).

21 Quantitative Phase Analysis: Method Developments 239

Most of the time only the third option is feasible and the most used correctionis the so-called Brindley microabsorption model [9, 10]. Applying it to the formulafor quantitative analysis by the Rietveld method [10–12] we get for a mixture of Nphases:

wi D Ii �i

�i

PNj D1

Ii �j

�j

where I is the scale factor from the Rietveld fitting, wi is the weight fraction forphase i, �i the density and � i the absorption contrast effect. Brindley has reportedsome tables for the absorption contrast effect from which we can extrapolatea function of a form similar to the absorption characteristic [13]. For sphericalparticles the Brindley table can be fitted well by [13]:

�i D �0:00229 C 2:054e � .�i � N�/ Ri C 0:50356

0:69525

where �i is the linear absorption coefficient for phase i, � the mean linear absorptioncoefficient of the mixture and Ri the mean particles radius. Similar formulas can beobtained also for other shapes correction.

21.6 Amorphous and Polymer Crystallinity Determination

Le Bail in 1995 [14] published how by diffraction it is impossible to distinguishbetween a real amorphous and a nanocrystalline structure without any long rangeorder. So there is always the possibility to reproduce an amorphous diffractionpattern using a so-called pseudo-amorphous model in which the short range orderis simulated by a particular crystal structure and the long range order is loosedby small crystallite sizes and/or high value of r.m.s. microstrain. The advantageof the description is that the short range order can be refined much better in theRietveld framework instead of relying uniquely on Monte Carlo techniques. In 1998Lutterotti et al. [15] demonstrated how this pseudo-amorphous model can be usedin the Rietveld to accurately measure the amorphous content without any internalor external standard. The classical formula developed for Rietveld phase analysis[11, 15] works perfectly to give the amorphous content as a normal phase if themodel is sufficiently accurate in describing the short range order. The benefits ofthis method are that no calibration, internal or external standard are needed and itcan be applied also to bulk samples. Limitations are that it requires a model for thepseudo-amorphous and the background level should be estimated correctly. For thelast problem it is advisable to use a monochromator in the diffracted beam to lowerthe background as much as possible.

The methodology can be used for crystallinity determination in polymers. Anexample is shown in Fig. 21.4 for polypropylene where the same crystal structure

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Fig. 21.4 Crystallinity determination on polypropylene by Rietveld fitting

Fig. 21.5 Quantification of the nifedipine amorphization inside the PVP matrix by ball milling.On the left the Rietveld fit using the PVP modeled using the cell of the crystalline PVP with thePONCKS method and the pseudo-structure of the nifedipine for the amorphous part; on the rightthe nifedipine amorphous content as determine by the Rietveld fit

is assumed for both the amorphous part and the crystalline part. This assumptionis quite reasonable in the polymer case and it gives results in agreement with DSCmeasurements. The same hypothesis do not work always in the case of ceramicsor alloys, the silica glass structure by Le Bail is an example. The method requiresthat the pseudo-crystal structure is known. Up to now very few of them have been

21 Quantitative Phase Analysis: Method Developments 241

Fig. 21.6 Amorphization of the griseofulvine in the PVP matrix for different milling times. Thevalues of griseofulvine crystallinity are reported in Table 21.3

Table 21.3 Griseofulvine crystallinity after milling at different times

Crystallinity (%) 100 70 60 51 50 40 46 50 35 32 17Milling time (min) 0 5 10 15 20 25 35 45 60 90 120

found or tested [14, 16, 17] but in the case the pseudo-structure is unknown anda pure sample is available the same procedure as in the PONCKS can be usedto calibrate the intensities. This last methodology has been used to determine thephase quantities with the presence of two different amorphous structures [18]. Twoexamples are reported in Figs. 21.5 and 21.6 and Table 21.3. In both examplesan incipient (griseofulvine and nifedipine) in crystalline form is amorphized byball milling with the help of a polymer (PVP) to stabilize the amorphous form.In such case the procedure has been able to quantify the incipient crystallinity withthe presence of a second amorphous, the PVP polymer by the fact that the twoamorphous phases have a different pattern.

Those last examples are a further proof of how flexible is the Rietveld methodand how a correct quantification can be reached using the appropriate model.

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References

1. Lutterotti L, Scardi P, Tomasi A (1993) Application of the Rietveld method to phase analysisof multilayered systems. Mater Sci Forum 133–136:57–62

2. Simek D, Kuzel R, Rafaja D (2006) Reciprocal-space mapping for simultaneous determinationof texture and stress in thin films. J Appl Crystallogr 39:487–501

3. Ferrari M, Lutterotti L (1994) Method for the simultaneous determination of anisotropicresidual stresses and texture by X-ray diffraction. J Appl Phys 76(11):7246–7255

4. Cont L, Chateigner D, Lutterotti L, Ricote J, Calzada ML, Mendiola J (2002) Combined X-raytexture-structure-microstructure analysis applied to ferroelectric ultrastructures: a case studyon Pb0. 76Ca0. 24TiO3. Ferroelectrics 267:323–328

5. Lutterotti L (2010) Total pattern fitting for the combined size-strain-stress-texture determina-tion in thin film diffraction. Nuclear Inst Methods Phys Res B 268:334–340

6. Ufer K, Roth G, Kleeberg R, Stanjek H, Dohrmann R, Bergmann J (2004) Description of X-ray powder pattern of turbostratically disordered layer structures with a Rietveld compatibleapproach. Zeitschrift fur Kristallographie 219:519–527

7. Ufer K, Kleeberg R, Bergmann J, Curtius H, Dohrmann R (2008) Refining real structure param-eters of disordered layer structures within the Rietveld method. Zeitschrift fur Kristallographie27(Suppl):151–158

8. Lutterotti L, Voltolini M, Wenk HR, Bandyopadhyay K, Vanorio T (2010) Texture analysis ofa turbostratically disordered Ca-montmorillonite. Am Miner 95:98–103

9. Brindley GW (1945) A theory of X-ray absorption in mixed powders. Philos Mag 36:347–36910. Taylor JC, Matulis CE (1991) Absorption contrast effects in the quantitative XRD analysis of

powders by full multiphase profile refinement. J Appl Crystallogr 24:14–1711. Hill RJ, Howard CJ (1987) Quantitative phase analysis from neutron powder diffraction data

using the Rietveld method. J Appl Crystallogr 20:467–47412. Bish DL, Howard SA (1988) Quantitative phase analysis using the Rietveld method. J Appl

Crystallogr 21:86–9113. Elvati G, Lutterotti L (1998) Avoiding surface and absorption effects in XRD quantitative phase

analysis. Mater Sci Forum 278–281:69–7414. Le Bail A (1995) Modelling the silica glass structure by the Rietveld method. J Non-Cryst

Solids 183(1–2):39–4215. Lutterotti L, Ceccato R, Dal Maschio R, Pagani E (1998) Quantitative analysis of silicate glass

in ceramic materials by the Rietveld method. Mater Sci Forum 278–281:87–9216. Le Bail A (2000) Reverse Monte Carlo and Rietveld modeling of the NaPbM2F9 (M D Fe, V)

fluoride glass structures. J Non-Cryst Solids 271:249–25917. Lutterotti L, Campostrini R, Di Maggio R, Gialanella S (2000) Microstructural characterization

of amorphous and nanocrystalline structures through diffraction methods. Mater Sci Forum343–346:657–664

18. Lutterotti L, Bortolotti M, Magarotto L, Deflorian C, Petricci E (2005) Rietveld structuraland microstructural characterization of pharmaceutical products using the program Maud.Presented at PPXRD-4, Barcelona, Spain