In situ diffraction analysis of single grain behaviour during tensile straining of polycrystals

Transcription

1 In situ diffraction analysis of single grain behaviour during tensile straining of polycrystals Dissertation zur Erlangung des Grades Doktor-Ingenieur der Fakultät für Maschinenbau der Ruhr-Universität Bochum von Marcin Mościcki aus Kędzierzyn-Koźle, Polen Bochum 00

2 Dissertation eingereicht am: 3. März 00 Tag der mündlichen Prüfung:. Mai 00 Erster Referent: Prof. Dr. -Ing. Anke Rita Kaysser-Pyzalla Zweiter Referent: Prof. Dr. András Borbély

3 ABSTRACT Work over the last decade at third generation synchrotron sources led to the development of advanced experimental techniques making possible nondestructive investigation of the internal structure of materials at different length scales. From the point of view of structural materials, consisting mainly of polycrystals, the micro and meso-scales are of significant importance, since only investigations at this level give the unbiased information necessary to understand grain interactions or texture evolution during deformation. The three dimensional X-ray diffraction (3DXRD) method developed at the European Synchrotron Radiation Facility enables for example to study the response of single grains to external stresses, 3D grain growth during recrystallization or phase transformations. In the present work an original framework dedicated to 3DXRD analysis is developed and presented. The software operates on a new evaluation procedure enabling characterization of orientation, position and strain tensor of single grains within the bulk of a polycrystalline sample. Presented methodology includes improved calibration technique using reference powder and a single crystal samples. A new Friedel pair based indexing method is introduced. Considering the symmetry properties of Friedel-pairs the contributions to reflection spot positions arising from grain orientation and position could be clearly separated. The developed GRIP software allows a good evaluation of experimental patterns enabling automatic characterization of several dozens grains embedded within polycrystalline sample. Depending on the number of reflections considered the accuracy of grain orientation can be less than 0., and the position of the center of mass of the grains can be accurate within one-third of the pixel size of utilized detector. The developed method allows obtaining the grain resolved strain tensor, too. However, due to low spatial resolution of present detectors the single grain strain tensors have relatively large errors. The average tensor components in contrast are in good agreement with the applied macroscopic strain field.

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5 ACKNOWLEDGEMENTS Several people played a decisive role supporting me on the way of my dissertation. First of all I would like to express my gratitude to Prof. Anke Kaysser-Pyzalla for her extensive professional support, encouragement and assistance from the beginning to the end in the course of pursuing these doctoral studies. My deepest gratitude goes also to Prof. András Borbély, whose professional advice, precious suggestions and knowledge accompanied me constantly while working on this project. I would also like to express my deep appreciation to Prof. Romuald Będziński, to whom I am indebted for the opportunity given to me to start this Doctorate. Furthermore, I would like to acknowledge Dr. Haroldo Pinto, who introduced me into the topic of three-dimensional X-ray diffraction and offered valuable support at experiments. I must also thank Mr. Benjamin Breitbach for his assistance and time spent at concerned beamlines. Moreover, my gratitude goes to my long-standing friend and officemate Krzysztof Dzięcioł who constantly supported me in programming matters. My sincere thanks go to many friends and colleagues at Vienna University of Technology and MPIE for scientific discussion, advice and continuous support always so greatly appreciated. Finally I would like to thank the staff of the beamlines ID, HARWI and F, particularly Dr. Jonathan Wright, Dr. Thomas Lippmann and Dr. Carsten Paulmann, the staff of Risoe National Laboratory, in particular Dr. Jette Oddershede, Dr. Soeren Schmidt and Dr. Lawrence Margulies, for their valuable assistance during experiments. Last but not least my deepest gratitude goes to Rosario Maccio who supported and encouraged me during the last three years of my dissertation. Without her encouragement I would not be able to successfully accomplish my thesis.

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7 To my parents

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9 TABLE OF CONTENT INTRODUCTION... 3DXRD METHOD DESCRIPTION OF DIFFRACTION GEOMETRY CRYSTALLOGRAPHIC ORIENTATION EQUILIBRIUM OF MACROSTRESSES ELASTIC STRAIN TENSOR GRAIN INTERACTIONS... 3 INDEXING METHODS GRAIN REFLECTIONS INDEXING PROGRAM (GRIP) IMAGE ANALYSIS SPATIAL DISTORTION BACKGROUND AND DARK FIELD SUBTRACTION CONVERSION TO POLAR COORDINATE SYSTEM ω -η -θ - SPACE REFLECTION RECOVERY FRIEDEL PAIR BASED INDEXING METHOD EVALUATION OF GRAIN ORIENTATION REFINEMENT OF GRAIN ORIENTATION BASED FRIEDEL PAIRS RESULTS OF MODEL STRUCTURES SIMULATION OF BRAGG REFLECTION SPOTS TOLERANCES FOR INDEXING INDEXING ACCURACY CALIBRATION OF THE EXPERIMENTAL SETUP INTRODUCTION CALIBRATION USING A REFERENCE POWDER SAMPLE ELLIPSE FITTING OF DEBYE-SCHERRER RINGS APPLICATION OF THE ELLIPSE FIT TO EXPERIMENTAL DATA CALIBRATION USING COORDINATE SYSTEM TRANSFORMATION CALIBRATION WITH SINGLE CRYSTAL EXPERIMENTAL RESULTS EXPERIMENTAL SETUP MATERIALS SAMPLE GEOMETRY HEAT TREATMENT DEFORMATION DEVICE ANGLE DISPERSIVE SYNCHROTRON SETUPS EXPERIMENTAL PROCEDURES RESULTS ON WIRE SPECIMENS STAINLESS STEEL GRADE STAINLESS STEEL GRADE RESULTS ON CYLINDRICAL SPECIMENS COPPER (HARWI ) COPPER (ID)... 05

10 7.3.3 STAINLESS STEEL GRADE IRON ALUMINIUM (FE AL) (ID) DISCUSSION CONCLUSIONS REFERENCES... 37

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13 Introduction Introduction A substantial fraction of industrial components consists of polycrystalline materials. The physical properties of constitutive crystals are related to their crystallographic nature and usually determine the macroscopic behavior of the component. Therefore, the knowledge of grain orientation distribution (texture) may allow the selection of adequate technologies, reducing production costs and to increase durability of engineering products. The deformation of polycrystalline structures and particularly the prospect of modelling deformation phenomena is an ongoing subject of technological research. The deformation of a single grain and subsequently the polycrystalline matter is actuated by slip and twinning. Both phenomena take place at the atomic level and are driven by dislocation movement, which occurs on particular planes leading to the glide of neighbouring crystal parts along each other. The predisposition of a material to glide and the directions of its movements are strictly dependent on the crystallography of the host grain, but also on the interaction with the neighbouring matter. Thus for instance in the case that a dislocation finds its way out of a grain at its surface a slip step occurs entailing a shape change of the sample. Such phenomenon accounts for a chain of reactions acting locally in the sample. Due to the fact that two neighbouring grains ought to be adjacent to each other and contribute to the local stress equilibrium in the material, they will mutually affect its behaviour. This mutual influence happens to be termed as grain-to-grain interaction. The orientation difference between neighbouring grains and its influence on grain dynamics during deformation is here of particular interest. In the last decade the study of grain-to-grain interactions during static or dynamic loading is particularly increasing due to the frequent use of miniaturized components [Mur 03] and the advent of novel nanostructure materials. So far several models have been proposed to describe the elastic and plastic deformation behaviour of polycrystals, however, all these models rely on assumptions and simplifications regarding the grain to grain interactions. The upper-bound Taylor model [Tay 38] and lower-bound Sachs model [Sac 8] represent

14 Introduction the two major approaches to polycrystal plasticity. Mentioned bounds represent a way to achieve either the compatibility condition, that is the accommodation of the deformation between neighbour grains, or local equilibrium conditions, which means the stress being balanced in the grains and thus being homogenous throughout the body. Additional approaches based on self consistent one (SC) or n-site schemes have also been proposed in the last years e.g. [Din 00, Mer 09, Sol 0]. Those models giving a more factual image of the processes governing polycrystal plasticity have shown to be effective in simulating the plastic deformation and texture in low symmetry aggregates [Koc 98]. The models assume interactions between grains (one site modelling) or compact arrangements of grains (N-site models) to be similar as the interaction with a surrounding homogeneous equivalent medium, which represents the average properties of the bulk. All mentioned models predict the activated slip systems, the texture evolution and various mechanical properties. Results obtained by use of various schemes differ due to diverse assumptions determining the models. For instance the fully constrained (FC) Taylor model where all strain rates are assumed equal is meant to fail when strain heterogeneity is involved. In those cases the selfconsistent models prove to be more successful. They explicitly account for the specific neighbourhood of the grain. Nevertheless results obtained with various approaches differ from experimental findings. All models, however, are marked by various discrepancies with experiment. These discrepancies are mainly related to texture sharpness, its rate compared to the experimental structures and system activity overestimation [Koc 98]. Thus the modelling of grain dynamics lacks in local scale dynamic data. It appears that for quantitative comparisons between models and experiments it is mandatory to be able to measure the initial texture and textures at distinct strains. Instead of such a statistical comparison, however, it would be more helpful if the plasticity orientation change could be performed at the single grain level. The role of I st kind stresses, extending over many grains, on the mechanical material performance has been already extensively studied e.g. [Scho 97]. On the other hand, much less is known about the effect of micro stresses arising from grain interactions. Micro stresses act within small sample volumes ranging to couple of grains, however, within these small zones they might reach significantly high values [Beh 00, Beh 94]. Since the micro-stress state within a single grain of a polycrystal appears to be an important factor with respect to grain dynamics or material fatigue

15 Introduction 3 (e.g. crack initiation), special attention should be given to the effect of grain boundaries, which directly influence the internal (grain) deformation mechanisms [Rey 8, Chen 98]. In order to assess and verify the impact of grain-to-grain interaction, the transition from elastic to plastic deformation has to be first understood. For that purpose experimental high-resolution single crystal diffraction techniques [Rei 89, Rey 8, Chen 98, Lu 03] need to be worked out. The classical X-ray and neutron scattering, due to employment of modern beam optics, various diffractometer geometries and improved detectors, enable to measure the lattice cell parameters with a high precision. Various techniques have been developed over the past years to perform stress measurements e.g. energy dispersive analysis taking advantage of white synchrotron radiation eg. [Ste 05, Gen 07] or single exposure method for biaxial stress analyses eg. [He 97]. Additionally methods using neutrons for scanning through the sample bulk have been derived. Nevertheless all of them represent indirect approaches while averaging the result over a large number of crystallites, on one site, and allowing just conditional measurement of micro components, on the other. It is mainly due to the imposed restrictions which reside either in low penetration depth of the beam or in limited collimation capacity. With increasing power of diffraction experiments dedicated instrumentation and ongoing improvement of X-ray optics, new measurement techniques were developed in order to characterize the local crystalline structure. Concurrently the emergence of D detectors and their successful application in the field of materials characterisation contributed to their growing popularity and importance. The biggest testimony of that is the fact of awarding in 009 the invention of charge-coupled devices (abbreviated as CCD s and generally used nowadays in D detectors) the Nobel Prize for Physics. Being a substitution to the so far widely used scanning point detectors or one dimensional position sensitive detectors [Pre 98, He 97] the D offers new possibilities for local strain studies. Both, the growing efficiency of X-ray detectors and the developments at synchrotron facilities allow today experiments at high photon energies (50-80 kev) eg. [Sha 98, Schn 89, Fre 95, Lor 97, Pou 03-, Mar 04] simultaneously providing good spatial resolution of a few micrometers in samples with millimetre dimensions. The first feasibility studies on local residual strains in polycrystalline materials and metal-matrix composites were performed by [Pou 97-, Pou 97-, Lor 97, Mar 0]. Their local character was achieved by use of slits defining

16 Introduction 4 the incoming and outgoing beam and thus determining the investigated gauge volume. In those studies the magnitudes of various strain components, which were averaged over groups of several grains and evaluated in the manner of the powder method, could be successfully measured with an accuracy of ε = x0-4. Therefore the deep embedded grains in the sample could be mapped for the first time and the residual strains connected to the grains quantified. A new three dimensional X-ray diffraction (3DXRD) method was developed by researchers from the Risø National Laboratory (Denmark) at the European Synchrotron Radiation Facility (ESRF) [Pou 04]. 3DXRD is a method developed for structural characterisation and facilitates in situ studies. It was successfully applied in various fields of material science: Marguiles [Marg 0, Marg 0] publishes the first feasibility studies with regard to the lattice strains of a grain embedded in a polycrystalline sample and successfully applies the technique to in situ studying the texture evolution during tensile deformation [Marg 0]. At the same time a method for measuring local strains and crystallographic orientation in polycrystalline samples has been also developed based on X-ray microbeams with broad-bandpass energy [Chu 99]. The so far existing methods require a beam centre precisely adjusted with respect to the centre of subsequent grains. The so called tracking is therefore done by sorting out reflections raised by corresponding grains (by a beam having approximately the size of the grain) and adjusting the position based on a maximum intensity criterion. Using the tracking technique, grain orientation change and elastic strain variation during tensile straining could be observed in several grains at various deformation states e.g. [Mar 04]. The applied alignment determined the accuracy of strain estimation, but the procedure turned out to be time consuming and impeded studies on large number of embedded crystallites. At the same time methods branched off providing increased reciprocal space resolution and allowing submicron level studies of dislocation structures [Jak 06]. So, the response of single grains to external stresses in terms of grain interactions could be analysed and verified by finite element modelling [Lie 04] and a crucial step was set in observing clearly identifiable sub-grains and their formation during plastic deformation [Jak 06]. Introducing novel algorithms, which simplified the acquisition technique, blazed the trial to simultaneous observations of appreciable numbers of

17 Introduction 5 neighbouring crystallites. New indexing algorithms allowed a simultaneous indexing of a couple of hundred grains in the illuminated cross section while skipping the unwanted adjustment of grains of interest in the rotation centre. In this way for the first time it became possible to verify various plasticity models at the single grain level, observing orientation changes in several dozens of grains embedded in polycrystalline aluminium [Win 08]. In most recent approaches in-situ measurements of grain specific strain tensors in titanium alloys were successfully performed [Lie 09] with a strain accuracy of x0-4 with a 00µm pixel size far field detector. The same precision was achieved while investigating the evolution of the axial strain component in a Cu tensile sample [Odd 09]. The simultaneously followed grains reached an approximate number of 800 whereas the estimated errors related to the grain positions and orientations amounted to 0.05 and 0µm respectively. The 3DXRD method was also successfully applied to investigate different phenomena such as 3D grain growth during recrystallization [Schm 04] or phase transformations [Offe 0]. In-situ investigation of materials under loading enables new insights into the kinetics of governing physical processes and the determination of local strains and stresses in single grains during loading represents one of the major challenges that are expected to be solved by the 3DXRD technique [Marg 0]. From the point of view of structural materials the micro and meso-scales are of significant importance, since only investigations at this level give the unbiased information necessary to understand grain interactions. This clear message entails the efforts determining the aim of this work. The main effort in the present work is placed on development of a novel 3DXRD method associated with an improved indexing technique dedicated to determination of deformation characteristics within individual crystallites of polycrystalline materials. The emphasis was placed on in-situ type of 3DXRD experiments which take advantage of monochromatic focussed high energy synchrotron radiation combined with large area X-ray detectors. Until 006, when the project started, various data on the topic of orientation tracking and axial strain tensor component recovery in single grains of polycrystalline sample has been published, eg. [Pou 97-, Pou 97-, Lor 97, Marg 0, Marg 0, Pou 04] nevertheless no free accessible data analysis software was available. From these facts the objectives of the present work are emerging: The necessity for experimental data for optimization of the 3DXRD method gives rise to the first objective. The focus of the studies needs to be placed on the accuracy of

18 Introduction 6 recovered parameters, particularly the components of grain resolved strain tensor. Of interest is a routine improving efficiency of indexing and developed considering high accuracy of estimated parameters. Literature on experiments performed with the 3DXRD method usually reports the assumption of a perfect detector plane alignment i.e. to be perpendicular to the incoming X-ray beam, and passing over in silence details on eventual deviations. However, knowledge of systematic errors related to experimental setup is necessary for accurate evaluation of grain related crystallographic parameters since a perfect alignment can never bee reached. This gives rise to the second objective of the following study entailing the development of accurate calibration methodology dedicated to 3DXRD experimental setups. The final aim is directly linked with the previous two and entails obtaining of relevant information about the orientation change and grain resolved strain in single grains of various specimens subjected to uniaxial plastic deformation.

19 3DXRD method 7 3DXRD method Three-dimensional X-ray diffraction, abbreviated as 3DXRD, is a method created with the aim of structural characterisation of crystalline matter. It accounts for a novel tool non-destructive characterisation of grains embedded in polycrystalline mater. The lattice of each grain is locally described by six parameters of its crystallographic unit cell (three base vectors a, b, c and three angles α, β, γ between the base vectors) and its orientation in sample frame. Orientation of the grain is typically expressed by three angles ϕ, φ, ϕ termed as Euler angles. Furthermore the center of mass of each grain is translated with respect to sample frame by a vector r = (x, y,z ). The crystal lattice of investigated grain is related to the laboratory s s s frame by a Bragg reflection acquired with a D-detector. The diffraction phenomena attributed by this Bragg reflection is in the reciprocal frame of the crystal described by a vector q defined by a set of indices (h, k, l). That reflection is related, assuming a perfectly aligned detector, through twelve specified above parameters. The task set to 3DXRD is thus not trivial and consist in determining the spatial variation of orientation, positions and strains operating in this dimensional space. The determination is although impeded by instrumental limitations and local defects, associating a distribution function to each space point. Impurities, lattice defects, possible type III residual strains, small grain size and shape imposed absorption characteristics, cause that a single grain is typically composed of small mosaic blocks (mosaicity of crystal), revealing a certain orientation distribution. It entails that at a beam diffracted by the lattice of a grain is characterised by an intensity distribution rather than a Dirac impulse. In addition the measured Bragg reflection intensity represents a spectrum described by the convolution of various instrumental and physical factors describing the utilized diffractometer and the sample. Any change denoting the impact of external conditions (local or global), can be measured via diffraction methods. Diffraction measurements on single crystals are

20 3DXRD method 8 generally described by concepts of scattering vectors related to the reciprocal space lattice. The main challenge of the 3DXRD method is the determination of the orientation and elastic strain [Pou 04] locally in single grains embedded in the crystalline specimen. It can be accomplished by linking the indexed diffractions with the reciprocal space of each grain. The 3DXRD is based on the principles and geometry of the rotation method [Gia 0, Bun 44, War 90] being its further development. In case of the 3DXRD method the single crystal diffraction principles are shifted into the polycrystalline regime. The typical geometry of a 3DXRD method is sketched in the Fig... A polycrystalline sample is here mounted on the stage which is rotated about the vertical axis Oz L, perpendicular to the incoming monochromatic beam. Grains of the sample which meet the Bragg condition produce diffracted intensities, which are recorded with an area detector. The detector screen is perpendicular (in ideal case) to the incoming beam described by the wave vector K. The scattered beam with wave vector K excites the photon-sensitive detector screen at the spot of intersection. Using the Ewald sphere construction it can be determined whether a certain reciprocal space node is recorded on the detector within a certain ω interval. Fig..a displays the Ewald construction (continuous circle with center A) rotated by a small angle ω around point O fixed in the reciprocal space to position A (dashed circle with center A ). Cross-section of those circles (A and A ) defines two crescent shapes (marked red in the Fig..a) that contain all the reciprocal lattice regions contributing to the diffracted intensity recorded on the detector. Since, as already mentioned, real crystals usually are not perfect, the diffraction phenomenon isn t represented in the reciprocal space by a point but rather a volume. It implies that a reflection can not be completely recorded until its reciprocal volume doesn t passes completely through the Ewald sphere. The radial and azimuthal intensity distribution of Bragg reflections captured on the detector can be quite accurately described by a convolution of the Gaussian and Lorentzian distribution e.g. in [Sán 97, Ida 00, Végh 05]. For small rotation angles ω a substantial number of reflections recorded on the subsequent frames are partially recorded reflections. Therefore to assure that the total intensity of the reciprocal nodal volume is recorded, the crystal is constantly rotated or oscillated in steps of ω. This allows mapping of continuous fractions of the reciprocal space on discrete frames Fig..b. The partially recorded reflections, which appear on successive diffractograms should then be analysed with respect to their intensity

21 3DXRD method 9 distribution and integrated to obtain the total diffracted intensity of the reflection [Arn 77]. Fig.. Mapping the reciprocal space: a) schematic concept of the rotation method - intersection of crescent red areas and reciprocal points display the acquired piece of reciprocal space when rotating the sample by ω; b) fractions of reciprocal space acquired on subsequent discrete frames Thus the measurement made according to the rotation method, maps sections of reciprocal space in a discrete three-dimensional space in which one dimension is described by the discrete sample rotation interval ω. Extending the above idea to aggregates of crystals embedded in the bulk of a sample an identification of single grains can be achieved. Experiments on polycrystalline matter can be therefore performed under the conditions of appropriate penetrating ability depending on photon energy of the beam and diverse material parameters (atomic number, density, thickness etc.). Various set-ups can be used for the acquisition purpose. With respect to the beam size, either localized investigations can be carried out, or the whole cross-section of the sample can be probed at once. Localised studies can be performed using a pencil beam, where the illuminated area is smaller that the structural element of the sample. In case where the whole cross-section is probed either a box beam or a line beam can be applied [Pou 04]. While the box beam allows investigating the desired threedimensional fragment of the sample in a single ω-scan, the line beam requires a series of ω-scans in which the same fragment has to be sliced along the ω axis by the beam strongly confined in the ω-axis direction. 3DXRD experiments can be performed in

22 3DXRD method 0 three different configurations [Pou 04] with respect to both, experimental set-up and operational parameters of available detectors. Firstly experimental configuration was achieved with a high spatial resolution near field detector, e.g. in [Lud 09, Pou 03-]. Typical pixel size of such detector amounts to 6µm. Sample is measured according to the method described above at several sample-detector distances varying in-between 3-8 mm. The method delivers accurate grain position data adequate for 3D grain-structure reconstructions. Low to medium spatial resolution detector (pixel size of µm) is set at a distance of mm in case of the second configuration, e.g. in [Lau 00, Rod 07, Nie 0]. The sample detector distance depends in this case on the required number of reflection orders (Debye-Scherrer rings) and used photon energy. Both are optimized in terms of getting the required angular resolution. The radial resolution resulting from the detector resolution is of medium value and amounts to x0-4 rad. The third configuration is achieved similarly as in the second case with detectors of medium spatial resolution (channel size of 50 µm), but placed in the far field at the distance of about 500 mm, e.g. in [Jak 06, Jak 08, Pan 09, Pan 04]. At that distance just one peak of the diffraction pattern is measured. The setting assures a high angular resolution restricted in practice only by the quality of the monochromator. The choice of the appropriate experimental set-up depends on the aim of the experiment. While the first configuration is well suited for reconstruction of grain shapes, including their precise position in the sample, the second and third configuration are rather dedicated to estimate grain orientations and strains, respectively. Moreover the third configuration benefits of high resolution peak profiles based upon single grains reflections. Thus this setup is well applicable for observations of sub-grain level strains and various crystallographic defects e.g. dislocation walls [Jak 08].. Description of diffraction geometry Three-dimensional X-ray diffraction is based upon single crystal methods. The methods already derived [Bus 67] are well applicable in this case. Fig.. shows a schematic drawing of the typical 3DXRD experimental set-up. Four Cartesian coordinates systems are defined, which are bound subsequently to the laboratory

23 3DXRD method Fig.. Schematic drawing of the experimental setup system, rotation table, sample and single grain (crystal). Respectively the subscripts L, ω, S and C will be assigned to these systems. During a measurement the sample, associated with sample coordinate system, is rotated in steps of ω around the vertical axis Oz of the laboratory frame. The rotated ω system is rigidly attached to the ω turntable. At given ω positions (or small intervals in case when orientation gradients are present) the (h, k, l) lattice planes of given grains fulfil the Bragg condition and give rise to diffraction spots recorded by a D detector placed at distance D behind the sample. To observe a reflection on the detector plane it is necessary that the angle θ satisfies the Bragg equation (Fig..3): π K K = = λ ( e e ) g hkl (.) where e and e denotes the unit vectors in the direction of the primary and diffracted beam. G hkl denotes the diffraction vector normal to the planes hkl and of a length equal to the reciprocal of the hkl lattice planes spacing. The relations are represented graphically in Fig..3a. The scattered intensity is integrated during each ω step (for the reasons described in chapter ) after which a detector image is saved. The vector g hkl is connected with the scattering event in the reciprocal space will be denoted in

24 3DXRD method Fig..3 Relations describing a) the vector representation of Bragg law b) association of the g C with laboratory system crystal Cartesian system by discussed in [Bus 67]: g C. Those two systems are related by a transformation g = B (.) C g hkl where: b b cos β 3 b3 cos β B = 0 b sin β 3 b3 sin β cosα (.3) 0 0 b 3 sin β sinα In the eq. (.3) the a i s, α i s, b i s and β i s represent respectively the direct and reciprocal lattice parameters and besides: cosα = cos β cos β cos β 3 sin β sin β (.4) The vector g C is linked to the laboratory system by set of relations expressed by rotation matrixes. The crystallographic orientation of a grain with respect to the sample system is described by the orientation matrix U, which allows transforming the diffraction vector g C from the crystal system into the sample coordinate system g S :

25 3DXRD method 3 g S gc = U (.5) The sample system is rigidly fixed to the sample and in the case of investigations described in this work it coincides with the laboratory coordinate system (at ω = 0): g L = Sg S (.6) Matrix S in eq. (.6) represents therefore the identity matrix and g = g will be L S considered in the following. As already described the ω system rotates around the vertical axis Oz of the laboratory frame. The position of any scattering vector g ω in the Laboratory system when the Bragg position is fulfilled (at specific ω) can be obtained based on a rotation of g S by an angle ω. This rotation is described by the usual rotation matrix Ω z having its axis identical with Oz: g ω =Ω z g S (.7) The diffraction vector in the laboratory system can be simply given in terms of the Bragg angle θ and the azimuthal angle η of the diffraction spot Fig..3b (η is measured clockwise from the projection of the rotation axis on the detector, when looking downstream the beam from the sample towards the detector [Pou 04] Fig..): where: Based on simple geometric considerations sketched in (Fig..) the vector describing the position of a diffraction spot with regard to the beam center on the detector is given by: g sinθ = cosθ sinη cosθ cosη g L (.8) 4π sinθ g = (.9) λ r d

26 3DXRD method 4 r d = rω + e Kt De (.0) Parameter T r ω = ( x ω, yω, z) used in eq. (.0) points to the center of mass (CM) position of the grain in the laboratory frame T r = ( x, y, z) when the Bragg condition is fulfilled. T e = (,0,0) and e K are unit vectors in the direction of the incoming and diffracted beams, respectively. The parameter t is determined from the intersection condition between the scattered wave vector K and the detector plane. Usually the grain position T r = ( x, y, z) in the sample coordinate system (at ω = 0) is of practical interest, which means that r ω should be related to r through the matrix describing the rotation of the grain around the vertical axis Oz: x y z ω ω x cosω sin ω 0 x = Ω z ( ω) y = sinω cosω 0 y (.) z 0 0 z The direction of the scattered intensity K = K +, which can be written in terms of the unit vectors: g ω e K is calculated from the Bragg s law e K = e + sinθ e ω (.) where e ω is the unit vector along the diffraction vector in the laboratory frame gω and e represents the unit vector along the K which again denotes the incoming wave vector, θ represents the Bragg angle. G ω is linked to the reciprocal lattice vector 04, Win 04]: T g hkl = ( h, k, l) through the formula [Pou ( Bg hkl ) Rg C g = Ω SU Bg = R = ω z hkl (.3) where g C is the diffraction vector in the Cartesian grain system and is obtained from g hkl through the transformation matrix B [Bus 67], eqs.(.)-(.3). U is the orientation

27 3DXRD method 5 matrix of the grain (defined according to the passive convention) and S relates the sample coordinate system to the laboratory system. For simplicity we introduce the matrix R denoting the product of three orthogonal matrices Ω, S and U : z R = Ω z SU (.4) Introducing eqs. (.), (.) and the normalized eq. (.3) into eq. (.0) a system of three equations is obtained, which represents the basis of so called forward simulation. Based on it the spot locations (u, v) with regard to the beam-centre BC can be predicted as a function of position and orientation of the grains. In case of a perfectly aligned detector, having its plane perpendicular to the direct beam and pixel columns lying parallel to the rotation axis of the sample, eq. (.0) becomes: D x cosω y sinω γ h u = xsinω + y cosω sinθ R γ k t v z 0 γ l (.5) where ( C T γ h, γ k, γ l ) = g / g is the unit vector along g C. The detector spot coordinates (u, v) are given by: ( R γ + R γ R γ ) t v = z + sin + (.6) θ 3 h 3 k 33 ( R γ + R γ R γ ) t u = xsin + y cosω + sinθ + (.7) ω h k 3 l l where D x cosω + y sin ω t = (.8) + sinθ ( R γ + R γ + R γ ) h k 3 l The rotation angle ω in eqs. (.7)-(.8) is determined by the Bragg s law, which imposes the geometrical relationship between e ω and e : e eω = sinθ (.9)

28 3DXRD method 6 Independent if the rotation axis of the sample is aligned along Oz or not, eq. (3..0) can be reduced to the following form: where a and b are constants depending on sample rotation axis and γ h, γ k, γ l and tilt components of the c = sinθ. If the angle between a diffraction vector and sample rotation axis is larger than the Bragg angle θ, eq. (.0) has two solutions in the [0, π] interval which is given by: a sin ω + b cosω + c = 0 (.0) ca ± a b a + b c cb ± a a + b c ω, = arctan, (.) a( a + b ) a + b In fact the rotation axis of the sample in most of the cases can not be assumed to be aligned along Oz axis of the laboratory coordinate system. Mostly an inclination is incorporated which has to be considered in more general equations to account for precise scattering vector positions. The problem will be in detail described in chapter 6.4 dedicated to calibration methods, where a second form of the eq. (.) will be derived.. Crystallographic orientation Fig..4 The definition of the Euler angles ϕ, Φ, ϕ

29 3DXRD method 7 The orientation of crystallites in a polycrystalline material can be described in various ways by various representations. For the purpose of this work to identify orientation of a crystallite with respect to the sample the Eulerian angles have been chosen. Euler notation uses three angles (ϕ, Φ, ϕ ) representing three successive rotations about three axes of Cartesian coordinates system. Sequence of those rotations can be chosen freely. In this work, as proposed in [Bun 83], the variant ZXZ of Euler representation was chosen. Following this rotation pattern a freely oriented crystal coordinate system can be brought to a position in which its axes are parallel to those of a sample coordinate system executing the following (as shown in Fig..4): - first rotate the crystal system around its axis Z through the angle ϕ, - then about the X axis of its new orientation through Φ, - finally around the newly oriented Z through the angle ϕ. The matrix representation of the orientation can be thus achieved by multiplication of three orthogonal matrices Z g ϕ, X g Φ and Z g ϕ where: Multiplication of matrices of eqs. (.)-(.4) yields: where abbreviations cϕ and sϕ stand respectively for cosϕ, sinϕ. The way how equation (.5) was derived assures that the rows of U represent the Φ Φ Φ Φ Φ + Φ Φ Φ + Φ = = Φ Φ c s c s s s c c c c s s c c s s c s s c s c c s c s s c c U Z X Z ' ),, ( ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ g g g (.5) Φ Φ Φ Φ = Φ cos sin 0 sin cos X ' g (.3) = cos sin 0 sin cos ϕ ϕ ϕ ϕ ϕ Z g (.4) = cos sin 0 sin cos ϕ ϕ ϕ ϕ ϕ Z g (.)

30 3DXRD method 8 direction cosines of crystallographic unit cell axes in the sample frame. It is common to represent the three angles (ϕ, Φ, ϕ ) contained in the orientation matrix (eq. (.5)) as Cartesian coordinates in Euler space. That implies that each point in the Euler space represents a particular orientation, which is convenient for graphical representation of crystal orientations. For Φ = 0 and Φ = 80 the orientation is represented exclusively by the sum ϕ +ϕ. Therefore all the points in the space for which ϕ +ϕ = constant describe the same orientation and reveal the singularity of Euler space. Additionally the space is characterised by additional feature it is periodic in all three dimensions and satisfies the following identity [Bun 83]: { ϕ + π, π Φ, ϕ + π} = g{ ϕ, Φ ϕ } g (.6),.3 Equilibrium of macrostresses In crystal structures any atom within its unit cell occupies a position at which it achieves minimum potential energy. According to [Nye 85] if no external force is acting, the divergence of the stress tensor within the crystal should equal zero: 3 j= σ x ij j = 0 (.7) An external load imposed to the ordered arrangement of atoms will cause their relative displacements and effect the deformation of the structure. The relative movement of the atoms will follow a scheme, which leads to a static equilibrium between the applied load and the matter of the crystal. If the element is restricted by any surface boundaries, any forces acting on that surface will also contribute to the mentioned force balance. If the surfaces of the crystal are free of any force that the boundary conditions can be written as[noy 87]: σ 0 (.8) ij n j = where n j is the unit vector in the x j direction on the boundary (surface normal). Relying upon eqs. (.7)-(.8) a conclusion can be drawn which, if both of the equations are satisfied, states that the average of all the stresses existing within the material amounts

31 3DXRD method 9 to 0 [Noy 87]: V σ dv ij = 0 (.9) where V represents the total volume of the considered body. Mechanical parts are mostly composed of polycrystalline materials with randomly oriented grains and usually of various phases. Thus any interaction, for instance an external load applied to a part of the volume causing a non-elastic strain occurring in some of the grains, will give rise to residual stresses. The term residual stress represents those remaining stresses in the engineering component after the removal of the external load. Due to the so called elastic incompatibility or differential plastic deformation [Noy 87] the residual stresses will vary across the component and will reveal micro-stress fields of different magnitudes within and around subsequent grains [Wit 0-, Wit 0-]. Fig..5 Schematic representation of residual stresses of different type in a multiphase (α, β) polycrystalline sample Due to the subdivided structure of plastically deformed crystals it is appropriate to classify residual stresses in different types as: residual stresses of I, II and III type as shown in the Fig..5 ( [Mac 73]). Residual stresses of I st kind (σ I ) act over a large area of the component, which should consist of a statistically representative amount of different oriented crystallites and of different phases. The average is then homogeneous and constant in different phases of

32 3DXRD method 0 the material. Type II residual stresses (σ II ) are constant over single grains of the material. They exist due to deviations of the local crystallography and physical parameters of the single crystallites. The deviations are subjected to crystallites in sense of mean values over the whole crystallite volume. Type III residual stresses (σ III ) occur at the subgrain level and exist due to inhomogeneity of the composition, dislocation arrangement within single crystallites. The term micro residual strain refers to the sum of the stresses of the I st and the II nd type..4 Elastic strain tensor Deformation of a crystal can be described in terms of change in unit cell vectors. The three-dimensional reciprocal lattice can be mapped onto the three-dimensional Cartesian coordinate system and expressed by a symmetrical second-rank tensor [Sch 78]. That representation applies to both, strains and stresses. The conventional orientation of the crystallographic coordinate system relative to a Cartesian system can be obtained by a transformation matrix described in analogy to eq.(.3) as [Pou 04]: a b cosγ c cos β A = 0 b sin β csin β cosα (.30) 0 0 csin β sinα Parameters a, b, c denote three lattice constants of a crystal unit cell, α * denotes the lattice angle in reciprocal space whereas β and γ stand for lattice angles in direct space. Referring to eqs.(.4)-(.5), the strain state of a single crystallite can be obtained on the way of optimization twelve parameters (x, y, z, ϕ, Φ, ϕ, a, b, c, α, β, γ -assuming a perfectly calibrated setup and detector) through the consideration comprised in [Sch 78]. Utilizing eq.(.30) strain can be expressed by the relation [Pou 04]: ε ij = ( Tij + T ji ) I ij (.3)

33 3DXRD method where I describes an identity matrix and T is given by: T = (.3) AA 0 where again A 0 denominates an reference lattice. However the method described in the following work is relayed upon a simplified assumption. In case of hard X-rays, when the measurement occurs under small Brag angles, the strains are linearly related to the relative shift in θ between the reflections of the unstrained and strained lattice [Pou 04]. The strain is thus determined directly from the measured theta shifts and can be expressed by following formulae: ε hkl hkl hkl hkl hkl d d 0 sinθ 0 sinθ = = hkl (.33) d sinθ hkl 0 ε hkl denotes the strain measured in crystal system perpendicular to the lattice plain hkl, d hkl and d 0 hkl stand for lattice spacing in particular hkl in strained and reference sample respectively. The lattice spacing is evaluated from indexed diffraction vectors of subsequent orders measured at sample. θ hkl and θ 0 hkl denote measured Bragg angle for diffraction vectors corresponding to the lattice plane hkl in strained and unstrained sample, respectively. Since the lattice of a crystal changes with direction the elastic modulus also varies with the direction. Thus the Hooke s law for anisotropic materials can be expressed in terms of tensors and has the form [Noy 87]: ε = (.34) ij Sijklσ kl or expressing the stress in terms of strain: σ = (.35) ij Cijklε kl Parameters S ijkl and C ijkl represent respectively the compliance and stiffness moduli of the crystal and denote tensors of the fourth rank. Additionally the transformation of strain and stresses between mutually disoriented coordinate systems takes advantage of the transformation rule for the second rank tensors:

34 3DXRD method σ ' = a a σ (.36) ij ik jl kl Since the unit cell parameters and the Hooke s law are defined in the crystal system of the grain the strain and stress will be obtained primordially in this system. Eq. (.36) allows their transformation to any desired frame, such as laboratory or sample coordinate system..5 Grain interactions Applying an external force to a polycrystalline material will cause deformation of its constituent grains. While the stress reaches the yield stress plastic slip occurs on specific slip planes of the crystal. Yielding takes place through slip of dislocations, which move in the direction of their Burgers vectors. To characterize plastic slip, the resolved shear stress (the projection of the applied stress onto the slip plane in the direction of the Burgers vector) should be known. Plastic yielding is generally associated to a critical shear stress value (the critical resolved shear stress) [Koc 98] as emphasized by the generalized Schmid Law: m S C S : σ = τ (.37) where the inner product m s :σ c represents the projection of the applied stress onto the slip direction. The superscript s implies a particular slip system in a particular crystal and the superscript c denotes quantities describing single crystal behaviour. In its essence the Schmid Law says that the component of the applied stress which drives the plastic work determines the kinematic behaviour [Koc 98]. The resolved shear stress on the slip planes of a particular crystal system under uniaxial compression or tension can be related by a geometric function referred to as Schmid factor m s, derived for particular slip system from the following relationship: S i = sinφ cosλ (.38) The geometrical relations are displayed in the Fig..6. Strain ε appearing in the direction of acting force F can be resolved on particular slip plane as ε n (F n ) and

35 3DXRD method 3 subsequently in a given slip direction (e.g. most favourably oriented system) as ε nn (F nn ). Fig..6 Description of the grain an one of its slip planes by a sectioned cylinder shape. Schematic representation of strain ε resolved on the slip direction of the slip plane described by its normal vector n. Constantly growing strain, within the grain, taking place on insufficient number of slip planes leads to interactions with surrounding neighbours via back-stress formation. That back-stress leads to activation of new slip systems driving the deformation of the grain [Koc 98] and its crystallographic rotation at the same time.

36 Indexing methods 4 3 Indexing methods Indexing accounts for identifying crystallographic orientation of single grains when measurements with X-rays are performed. The term indexing expresses the identification of the reciprocal lattice from a cloud of reciprocal lattice points recorded during the measurement. Majority of the first three-dimensional indexing methods, for instance those used in INDEX [Enr 00], DIRAX [Dui 9] or RAMCEL [Enr 00], are based on the principle of maximizing the number of observed reciprocal space vector matching the initial reciprocal cell vectors. This can be realised in various ways, operating either in real or reciprocal space. Indexing in structural characterisation of materials is always connected with preliminary analyses. The subsequent steps of the preliminary analysis are shown in Fig. 3.. Fig. 3. Flow chart of subsequent procedures proclaiming the tracing and indexing algorithm Image analysis constitutes the first step of every indexing process. At this point the Bragg spot information is extracted from acquired diffractograms and the spot parameters are transferred to subsequent analysis step. The so called ray tracing identifies the centre of mass positions (CM) of observed Bragg spots and estimates parameters necessary to calculate normalised scattering vectors, which establish the indexing input. Ray tracing module carries out the assignment of scattering vectors into the laboratory system. The arrangements are performed in a way assuring that the reflections data meet the input requirements of the particular indexing routine. The indexing itself can be performed utilizing various approaches. The direct space method

37 Indexing methods 5 [Cle 84, Jac 76] for instance operates in the direct lattice space and relays on the assumption that a scalar multiplication of a true direct vector and any true reciprocal vector results in an integer. Thus the tentative direct space vectors of an initial (guess) unit cell are compared to reciprocal vectors observed during experiment. If the great majority of the reciprocal vectors meet the assumption with respect to the inspected direct vector, the inspected direct vector is assumed to be a true one. The set of obtained direct vectors gives then the basis to build up a final cell. The direct space method is easy and quite neat in realisation, although, like all of the first indexing methods, it is dedicated to the single crystal diffractometry. It means that it is able to handle rather low number of reflections, where all or majority of them belongs to one single grain. The development of high brilliance third generation synchrotron radiation sources together with progress made in X-ray optics started a new trend in the field of material studies. Available spatial resolution routed the grain investigations on a sub-micron level, making of X-ray diffraction a promising tool for strain and texture oriented analyses. Although measurements of polycrystalline samples bring along additional problems linked to the complexity of the new measurement techniques. The indexing method applied in the program ORDEX [Chu 99] is one of the first developed for purposes of polycrystalline sample characterisation. The algorithm has been dedicated to white-beam Laue microdiffraction method. When caring out measurements with microbeams, sample rotation may cause changes in illuminated volume of single grain. Thus Laue diffraction is usually applied when rotation of the sample has to be avoided. Method dedicated to that kind of experiment is comprised in ORDEX. It operates on Bragg plane normals q ) which are determined based on the incident and diffracted wave vectors (in respectively K r and K r Fig..). In order to identify a grain all observed reflections are compared with respect to their mutual angular relationships. Reflections are told to be valid if the mutual angular relationship meets the estimated values within tolerances given by limit distortion of the cell. In this way all reflections from a single grain can be identified and indexed. ORDEX is able to process overlapping Laue patterns, however, it deals with rather small amount of data (~0 reflections) [Chu 99] without engaging in splitting reflections data into subgroups associated with individual grains. This feature displays one of the strongest issues of that method. It is recently required that investigation techniques allow either, the