X-Ray Tomography in Material Science

X-Ray Tomography in Material Science

0 HERMES Science Publications, Paris, 2000 HERMES Science Publications

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ISBN 2-7462-0115-1 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission, in writing, from the publisher.

Disclaimer While every effort has been made to check the accuracy of the information in this book, no responsability is assumed by Author or Publisher for any damage or injury to or loss of property or persons as a matter of product liability, negligence or otherwise, or from any use of materials, techniques, methods, instructions, or ideas contained herein.

X-Ray Tomography in Material Science

Jose Baruchel Jean-Yves Buffiere Eric Maire Paul Merle Gilles Peix

•cience

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Authors

ANDERSON P., Department of Biophysics in Relation to Dentistry, St Bartholomew's and The Royal London Scool of Medecine and Dentistry, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK BABOT D., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France BARUCHEL J., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France BELLET D., Laboratoire GPM2, INPG, BP 46, 38402 Saint-Martin-d'Heres BENOUALI A.-H., Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium BERNARD D., ICMCB, CNRS, 87 avenue du docteur Albert Schweitzer, 33608 Pessac, France BLANDIN J.-J., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France BOLLER E., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France BOUCHET S., Ecole des mines , ENSMP, 35 rue St Honore, 77300 Fontainebleau, France BRACCINI M., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France BUFFIERE J.-Y., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621 Villeurbane Cedex, France CLOETENS P., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France DAVIS G., Department of Biophysics in Relation to Dentistry, St Bartholomew's and The Royal London Scool of Medecine and Dentistry, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK DEGISCHER H.P., Institute of Materials Science and Testing, Vienna University of Technology, Karlsplatz 13, A-1040 Wien

6

X-ray tomography in material science

DERBY B., Manchester Materials Science Centre, UMIST and the University of Manchester, Grosvenor Street, Manchester, Ml 7HS,UK DUVAUCHELLE P., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France ELLIOTT J., Department of Biophysics in Relation to Dentistry, St Bartholomew's and The Royal London Scool of Medecine and Dentistry, Queen Mary and Westfield College, Mile End Road, London, El 4NS, UK FOROUGHI B., Institute of Materials Science and Testing, Vienna University of Technology, Karlsplatz 13, A-1040 Wien FREUD N., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France FROYEN L., Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium GuiGAY J.-P., University of Antwerp, RUCA Groenenborgerlaan 171, B-2020 Antwerp, Belgium HEINTZ J.-M., ICMCB, CNRS, 87 avenue du docteur Albert Schweitzer, 33608 Pessac, France JOSSEROND C., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France JUSTICE I., Department of Materials, University of Oxford, Parks Rd, Oxford, OX1 3PH, UK KAFTANDJIAN V., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France KOTTAR A., Institute of Materials Science and Testing, Vienna University of Technology, Karlsplatz 13, A-1040 Wien LUDWIG W., European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble, France MAIRE E., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621 Villeurbane Cedex, France MARC A., LETI-CEA/Grenoble, 17 rue des martyrs, 38054 Grenoble Cedex 9, France MARTIN C.F., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France PEK G., Laboratoire CNDI, INSA, Batiment 303, 69621 Villeurbane Cedex, France PEYRIN F., CREATIS, INSA-Lyon, 69621 Villeurbane, France ROBERT-COUTANT C., LETI-CEA/Grenoble, 17 rue des martyrs, 38054 Grenoble Cedex 9, France SALVO L., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France SAVELLI S., GEMPPM INSA Lyon, 20 avenue Albert Einstein, 69621 Villeurbane Cedex, France SCHLENKER M., CNRS, Laboratoire Louis Neel, BP 166, F-38042 Grenoble, France SUERY M., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France

Authors

7

VAN DYCK D., University of Antwerp, RUCA Groenenborgerlaan 171, B-2020 Antwerp, Belgium VERRIER S., Genie physique et mecanique des materiaux, ENSPG-UJF, BP 46, F-38402 Saint-Martin-d'Here Cedex, France VIGNOLES G.-L., LCTS, CNRS-SNECMA-CEA, Universite Bordeaux 1, 3 allee La Boetie, F-33600 Pessac, France WEVERS M., Department of Metallurgy and Materials Engineering, Katholieke Universiteit Leuven, De Croylaan 2, B-3001 Heverlee, Belgium

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Table of contents

Foreword

13

Chapitre 1. General principles

G. PEIX, P. DUVAUCHELLE, N. FREUD 1.1. Introduction 1.2. X and gamma-ray tomography: physical basis 1.3. Different scales, different applications 1.4. Quntitative tomography 1.5 Conclusion 1.6. References

15 15 16 20 23 26 26

Chapitre 2. Phase contrast tomography

P. CloETENS, W. LUDWIG, J.-P. GUIGAY, J. BARUCHEL, M. SCHLENKER, D. VANDYCK 2.1. Introduction 2.2. X-ray phase modulation 2.3. Phase sensitive imaging methods 2.4. Direct imaging 2.5. Quantitative imaging 2.6. Conclusion 2.7. References

29 29 30 32 38 38 42 43

Chapitre 3. Microtomography at a third generation syncrotron radiation facility

J. BARUCHEL, E. BOLLER, P. CLOETENS, W. LUDWIG, F. PEYRIN 3.1. Introduction 3.2. Syncrotron radiation and microtomography

45 45 46

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X-ray tomography in material science

3.3. Improvement in the signal to noise ratio in the 3D images 3.4. Improvement in the spatial resolution 3.5. Quantitative measurement (absorption case) 3.6. Present state of "local" tomography 3.7. Sample environment in microtomography 3.8. Phase Imaging 3.9. Other new approaches in microtomography 3.10. Conclusion 3.11. References

49 50 51 53 54 55 56 57 57

Chapitre 4. Introduction to reconstruction methods C. ROBERT-COUTANT, A. MARC

61

4.1. Introduction 4.2. Description of projection measurements 4.3. Backprojection 4.4. Projection-slice theorem 4.5. Fourier reconstruction methods 4.6. Filtering in Fourier methods 4.7. ART-type methods 4.8. Conclusion 4.9. References

61 62 65 66 67 69 70 74 74

Chapitre 5. Study of materials in the semi-solid state

S. VERREER, M. BRACCINI, C. JOSSEROND, L. SALVO, M. SUERY, W. LUDWIG, P. CLOETENS, J. BARUCHEL 5.1. 5.2. 5.3. 5.4. 5.5. 5.6.

Introduction Experimental device and procedure Results on Al-Si alloys Results on Al-Cu alloys Conclusion and perspectives References

,...

77 77 79 80 85 86 87

Chapitre 6. Characterisation of void and reinforcement distributions by edge contrast

I. JUSTICE, B. DERBY, G. DAVIS, P. ANDERSON, J. ELLIOTT 6.1. Introduction 6.2. Dual energy X-ray microtomography 6.3. Experimental materials 6.4. Results and discussion 6.5. Conclusions 6.6. References

89 89 90 92 94 100 101

Table of contents

11

Chapitre 7. Characterisation of MMCp and cast Aluminium alloys

J.-Y. BUFFIERE, S. SAVELLI, E. MAIRE 7.1. 7.2. 7.3. 7.4. 7.5.

103

Introduction Experimental methods Results and discussion Conclusion References

103 104 107 112 113

Chapitre 8. X-ray tomography of Aluminium foams and Ti/SiC composites

E. MAIRE, J.-Y. BUFFIERE

115

8.1. General introduction 8.2. Aluminium foams 8.3. Titanium composites 8.4. General conclusion 8.5. References

115 116 121 124 125

Chapitre 9. Simulation tool for X-ray imaging techniques

P. DUVAUCHELLE, N. FREUD, V. KAFTANDJIAN, G. PEIX, D. BABOT 9.1. 9.2. 9.3. 9.4. 9.5. 9.6.

127

Introduction Background Simulation possibilities Simulation examples in tomography Conclusions and future directions References

127 128 129 132 135 136

Chapitre 10. Micro focus computed tomogrgraphy of Aluminium foams A.-H. BENAOULI, L. FROYEN, M. WEVERS 10.1. 10.2. 10.3. 10.4. 10.5. 10.6.

Introduction Production process of Aluminium foams Mechanics of foams Non-destructive investigation of Aluminium foams Conclusion References

139 139 140 142 144 151 152

Chapitre 11. 3D observation of grain boundary penetration in Al alloys W. LUDWIG, S. BOUCHET, D. BELLET, J.-Y. BUFFIERE

11.1. Introduction 11.2. Experimental set-up 11.3. Result 11.4. Conclusions 11.5. References

.*.

155

155 157 158 160 163

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X-ray tomography in material science

Chapitre 12. Determination of local mass density distribution

H.P. DESISCHER, A. KOTTAR, B. FOROUGHI 12.1. Introduction 12.2. Material 12.3. X-ray radiography 12.4. Result 12.5. Application of the mean local density distribution 12.6. References

165 165 166 166 168 172 175

Chapitre 13. Modelling porous materials evolution

D. BERNARD, G.-L. VIGNOLES, J.-M. HEINTZ 13.1. Introduction 13.2. Evolution of sandstone reservoir rocks by pressure solution 13.3. C-C 13.4. Ceramics sintering 13.5. Conclusions and forthcoming works 13.6. References

177 177 179 185 187 190 191

Chapitre 14. Study of damage during superplastic deformation

C.-F. MARTIN, J.-J. BLANDIN, L. SALVO, C. JOSSEROND, P. CLOETENS, E. BOLLER 14.1. Introduction to damage in superplasticity 14.2. Usual techniques of characterisation 14.3. Experimental procedure 14.4. X-ray microtomography results 14.5. Quantification of the coalescence process 14.6. Conclusions 14.7. References

193 193 197 198 199 200 203 204

Foreword

This book collects the texts of the lectures given during the Workshop on the application ofX-Ray tomography in material science which was organised by the Groupe d'Etudes de Metallurgie Physique et de Physique des Materiaux (GEMPPM) in Villeurbanne on October 28-29 1999. Researchers from several European universities, research centres and companies attended the lectures which were given by experts in both materials science and X-ray tomography. The workshop was subsidised by the INSA Lyon, the MMC Assess european network and the Region Rhone Alpes and we would like to acknowledge their support. The scope of this European workshop was to provide material scientists with a detailed presentation of X-Ray tomography techniques, including the latest developments, and to present recent applications of these techniques in the field of structural materials. The interest of material scientists in X ray tomography arises from two facts: 1) most structural materials are opaque, and 2) it is of very crucial importance to observe what occurs in the bulk of materials when they are subjected to a mechanical loading. The apparent contradiction between these two facts has been overcome by recent progress in X Ray tomography which has allowed 3D non destructive images of structural materials, with a resolution around 1 micron, to be achieved. Synchrotron radiation sources are necessary to record these very high resolution images. Moreover, the phase contrast images, easily obtainable with X ray sources emitting photons with a high spatial coherence, even permits the visualisation of features with weak attenuation differences. This technique is especially well adapted for studying metal matrix composites which are among the most promising structural materials and for which damage development under stress is of crucial importance. Within this framework, the workshop was divided into two parts. The first one included a global description of the technique itself, an introduction to the

14

X-ray tomography in material science

reconstruction algorithms, and an overview of the new possibilities offered by synchrotron X ray sources with an emphasis on the phase contrast images. The second part was devoted to the presentation of some examples of the application of X-Ray tomography to investigating micro-heterogeneous structural materials. The use of synchrotron and laboratory X-Ray sources was illustrated. The workshop was a stimulating event which has given scientists with various backgrounds the opportunity to discuss and exchange ideas and experiences. We do hope that this book will bring useful information to material scientists looking for new characterisation methods in their research fields. The organisers, Jose Baruchel Staff Scientist, Group Leader ESRF Jean-Yves Buffiere Maitre de conferences INSA Lyon Eric Maire Charge de recherches INSA Lyon Paul Merle Professeur INSA Lyon Gilles Peix Maitre de conferences INSA Lyon

Chapitre 1

General principles

Among the different methods allowing to obtain, in a non-invasive way, the image of a slice of matter within a bulky object, X-ray transmission tomography is widely used in both the medical and the industrial fields. In the latter case, defect detection, dimensional inspection as well as local characterization are possible. Non destructive testing, process tomography and reverse engineering are thus feasible. A wide range of sizes can be 1 mm small inspected, starting from a sample, up to a whole rocket motor (several meters in diameter). The present paper describes the physical basis and give examples of some industrial applications. The main reconstruction artifacts are described.

1.1. Introduction Tomography is referred to as the quantitative description of a slice of matter within a bulky object. Several methods are available, delivering specific images, depending on the selected physical excitation: - ultrasonics, - magnetic field (in the case of nuclear magnetic resonance imaging), - X and gamma-rays (y rays), - electric field (in the case of electrical impedance or capacitance tomography). In the field of industrial non-destructive testing (NOT), as well as in the field of materials characterization, X-ray or y-ray tomography is mostly used today. Tomography is a relatively "new" technique. The very first images were obtained in 1957 by Bartolomew and Casagrande [BAR 57]: they characterized the density of

16 X-ray tomography in material science particles of a fluidized bed, inside a steel-walled riser. The first medical images were performed by Hounsfield in 1972, and most industrial applications were developed much later, in the 1980's. This slow development can be explained by the huge amount of data to handle, and thus by the need for high speed and high memory computers. Industrial benefits of what is called computed tomography (CT) today are numerous. This is due to the wide range of potential applications, starting from the small sample, 1mm in size, dedicated to the characterization of advanced composite materials, and displayed in three dimensions with a one micrometer (urn) voxel size, up to the single slice image, across a 1 meter diameter riser, with a five centimeter pixel size. 1.2. X and gamma-ray tomography: physical basis 1.2.1. Different acquisition set-ups The simplest set-up consists in detecting the photons which are transmitted through the investigated object (Fig. 1.1): transmission tomography delivers a map of u, the linear attenuation coefficient, quantity which is in turn a function of p (the density) and Z (the atomic number).

Figure 1.1. X-ray transmission tomography

The clear separation between p and Z implies to perform either bi-energy tomography or scattered photons tomography [ZHU 95, DUV 98] (Fig. 1.2). This last technique is based on the clear differentiation between Compton and Rayleigh scattered photons. The ratio between those two measured quantities is purely proportional to Z and is not affected by the density. The third possibility is to detect photons emitted by the investigated object itself. Such is the case when gamma-ray sources are distributed inside a nuclear waste container, for instance. Emission tomography is thus performed [THI 99] (Fig. 1.3). An alternative is encountered when the distributed source is a positon emitter: the

General principles

17

local positon annihilation delivers pairs of 0.51 MeV annihilation photons which are detected outside. This is the PET technique, used in the medical field.

Figure 1.2. Scattered photons tomography

Figure 1.3. Emission tomography

1.2.2. X-ray transmission tomography The present paper will be focused on transmission tomography, which is widely used in both industrial and medical fields. It is based on the application of equation [1], known as the Beer-Lambert law, or attenuation law. Figure 1.4 describes the basic experimental set-up for transmission tomography inside a single slice.

N l=

Ar0exp[-

v(x,yi)dx]

[1]

path

Measuring the number N0 of photons emitted by the source and the number N, of photons transmitted throughout a single line across the sample allows to calculate the integral of ja along the considered path: [2] N

l

path

The term ((x,y) represents the value of the linear attenuation coefficient at the point (x,y). Repeating such a measurement along a sufficient number of straight lines within the same slice delivers the Radon transform of the object. Radon demonstrated in 1917 the possibility to find an inverse to that transform and thus to reconstruct the n(x,y) map of the slice [KAK 87].

18 X-ray tomography in material science As industrial tomography makes frequently use of an X-ray generator, we will focus our discussion on that kind of experimental set-up. Nevertheless, some comments will be made on gamma-ray tomography.

Figure 1.4. Physical basis of transmission tomography inside a slicef

1.2.3. The linear attenuation coefficient Transmission tomography delivers a map of (x,y), the linear attenuation coefficient, which is correlated to i) the photon energy E, ii) the density p and iii) the atomic number Z of the investigated material. Figure 1.5 displays the dependance between those quantities for carbon (Z=6) and iron (Z=26). It must be noticed that the quantity displayed on Fig. 1.5 is in fact the ratio /p, the mass attenuation coefficient.

Figure 1.5. Value of the mass attenuation coefficient for carbon and iron

Two main domains appear in Fig. 1.5. Below 200 keV, the photoelectric effect dominates and jj/p is sharply dependant on E and Z. Equation [3] is often used to describe this behaviour [ATT 68]:

General principles

19

[3]

where K is a constant. Such an equation implies that, for any given photon energy, is proportional to p and to Z4. Performing images in the photoelectric domain implies two main characteristics: - a comparison of p between two areas of the object (or between two objects) can be achieved only in the case when Z is constant (same atomic element or same composition), - a change in p between two areas can be cancelled by a change in Z in the opposite direction. It thus appears that a clear separation between Z and p can not be obtained, in the photoelectric domain, unless two tomographic images are performed, using two different energies. Within the Compton domain, above 200 keV, u can be considered as weakly dependant on Z and on photon energy. Tomography thus delivers an information which is nearly proportional to p. However, due to the higher photons energy, and hence to the lower value of u, the contrast within the object image is lower, as can be derived from Beer-Lambert law.

1.2.4. Different experimental set-ups In the field of industrial tomography, three different configurations are mainly encountered. They are displayed on figure 1.6.

Figure 1.6. Different experimental set-ups in the field of industrial tomography: a) first generation scanner, b fan-beam scanner, c) cone-beam scanner

20 X-ray tomography in material science Figure 1.6.a corresponds to the simplest experimental set-up. A single sensitive element is used and a rather long scanning time is needed, as the acquisition of a single linear "projection" needs a set of elementary translations. Successive projections are then acquired, corresponding to different value of the angle of rotation. A half turn is sufficient to reconstruct the image of a slice. Figure 1.6.b implies the use of a linear array. Acquisition is shorter, as a whole linear projection is acquired at a time. A complete turn is needed since the beam diverges. Figure 1.6.c makes the best use of the X-ray cone-beam; one turn of the object is needed. The Feldkamp algorithm [PEL 84] allows the direct 3D reconstruction of the whole object.

1.3. Different scales, different applications 1.3.1. Industrial tomography The main application in the field of X-ray tomography is Non-Destructive Testing (NOT) of manufactured components, i.e. detection of internal defects. Among other issues there are i) "reverse engineering", whose purpose is the geometrical inspection of a component, in such a way to assist the design, ii) local characterization of materials (density measurements, for instance) and Hi) process tomography, able to deliver some kind of control on a continuous manufacturing process. As industrial applications involve a broad range of sizes and a great variety of materials to be inspected, the corresponding devices may be very different. 1.3.1.1. Different photon sources Inspection of small components can be performed using a standard industrial Xray tube (160 kV for instance). Much attention must be paid to the stability of both the high-voltage and the anode current, because the consecutive projections must be acquired within constant conditions. A focus size within the range 1 to 3 mm is acceptable. Inspecting heavier components may require a 450kV tube, or even a linear accelerator. Two different high capacity scanners were constructed by the french Atomic Energy Commission (CEA-LETI, Grenoble). A 420 kV X-ray generator in the first case and an 8 MeV linear accelerator in the other case allow the complete inspection of a whole (empty) rocket motor, up to 2.3 meters in diametre, of a nuclear waste container or of a whole car engine. Gamma-ray sources can be used, in spite of the very low emitted photon flux. The Elf Research Centre (Solaize-France) uses a cesium 137 source with an activity up to 18 GBq (gigabecquerels). The high monochromatic energy (662 keV) delivered by the source allows to map the density of solid particles inside a fluidized

General principles

21

bed, through the steel wall of the riser (0.85 meter in diametre). A single source and a single detector (Nal) are used, thus constituting a first generation tomograph, as shown in Fig. 1.6a. The scan lasts 3 hours [BER 95]. The University of Bergen and the Norsk-Hydro Company built a static device using a set of five americium 241 sources (energy: 60 keV) distributed around a pipe [JOH 96]. A linear array comprising 17 semiconductor detectors is set opposite to each source, allowing a near real-time (0.1 second) imaging of the slice. The purpose is to visualize the liquid components (oil, water) apart from gas within a pipe. This application is an example of process tomography, i.e. fast imaging dedicated to the control of a manufacturing process. 1.3.1.2. Different families of detectors Four main families of detectors can be found: 1. gas ionisation detectors were used in the early medical scanners. They are still in use today in some industrial applications. Their main characteristic is their high dynamic range. Filled with gas having a high atomic number, they can be used even with high energies. Linear arrays are available. 2. image intensifiers (I.I.) are used in "desktop" scanners for industrial NDT of small components. Their low dynamic range and the inherent distortion of the image need some care. Significative 3D images can nevertheless be obtained. 3. scintillation detectors, composed of a fluorescent material (e.g. gadolinium oxysulphide Gd2O2S, or caesium iodide Csl) are nowadays widely used. Those detectors are of two kinds: i) the fluorescent material is directly coupled to an array of photodiodes [KAF 96] or of photomultipliers (in some cases the coupling is realized using tapered optic fibers), ii) the fluorescent material is spread on a screen, which is optically coupled to a CCD camera via a lens [CEN 99]. 4. arrays of semiconductors (e.g. CdTe or ZnCdTe), which allow a direct photon detection are promising. High energy applications are possible.

1.3.2. Microtomography Considering advanced materials characterization, the need of 3D images with a very high resolution (a few um) obtained through a non invasive method is growing. Figures 1.7 and 1.8 show two specific examples of 3D tomographic images performed with two different scanners, conceived and built in our laboratory [KAF 96] [CEN 99]. Such 3D images are then used by the researchers for the modelisation of the mechanical properties of materials, within finite elements models computations. For such applications micro-focus X-ray tubes, with a focus size in the range 5 - 1 0 micrometers, are used. A very low focus size allows to set the investigated object directly at the window of the tube. A geometrical magnification

22 X-ray tomography in material science can thus be obtained. Figure 1.9 shows that the magnification can be easily modified. A limit exists to the magnification: the geometrical unsharpness [HAL 92] must be kept lower than p, the size of the sensitive element of the detector (sampling step). In practice, this upper boundary to the magnification Gg can be computed according to equation [4], where represents the size of the focus: [4]

Figure 1.7. 3D rendered view of a tomographic image of a composite material with 400 yon glass balls inside an organic matrix. (Herve Lebail; Laboratory GEMPPM). The voxel size is set to 42 jjm

Figure 1.8. 3D rendered view of a tomographic image of an aluminium foam (density 0.06) (Eric Maire; Laboratory GEMPPM). The voxel size is 150 pm. The size of the sample is 3cm

Figure 1.9. According to the location of the investigated object between the focus and the screen, different geometrical magnifications are attained

General principles

23

Designing and building such a kind of scanner implies some care in at least three domains: - the low photon flux delivered by the micro-focus X-ray tube results in long exposure times; the camera must therefore deliver a very low noise, - the choice of the photon energy is important: low energy photons deliver images with an higher contrast, but also with an higher relative noise, - the accuracy of the mechanical setting must be better than the expected image resolution. Today, the most powerful tool involves the use of synchrotron radiation. The European Synchrotron Radiation Facility (ESRF-Grenoble) delivers a huge X-ray flux and thus allows very short exposure times. A complete scan can be acquired within a few minutes, with a spatial resolution down to 1 um. On beam-line ID 19, the source is located far from the working hutch (145 meters), thus delivering photons with a high spatial coherence. This property of the X-ray flux generates diffraction features which underline the edges within the sample, and thus highlighting sharp defects. Such a phenomenon, the so-called "phase contrast" [CLO 97], allows very small defects to be detected. As the beam is non-diverging, the resolution is set by the detector itself. Transparent luminescent screens are used, with a 5 jam sensitive layer of an yttrium-aluminium (YAG) or lutetium-aluminium (LUAG) garnet, epitaxially grown on a YAG monocrystal, 170 jim in thickness; they allows a high resolution (1 fim) and a 4% to 8% efficiency for 14 keV photons.

1.4. Quantitative tomography As mentionned earlier, tomography offers many possibilities. If the goal is just defect detection, the selected resolution must therefore be adjusted to the size of the details to be observed. Much attention must also be paid to the noise of the camera or, more precisely, to its dynamic range [CEN 99]. When the inspection's issue is the determination of the accurate size of some internal feature, or the local characterization of materials (density measurement for instance), then an increased attention must be paid to the reconstruction artifacts. They create artificial patterns inside the reconstructed slice (streak artifacts), or they locally modify the pixels values (cupping effect), and hence the quantitative result [ISO 99] [SCH 90]. In the following lines, we will describe the main physical mechanisms leading to erroneous reconstructions, as well as the shape of the corresponding artifact in the reconstructed image. - Beam hardening As an X-ray tube delivers a polychromatic spectrum, differential attenuation of photons within the investigated object leads to the rapid attenuation of the lowest

24 X-ray tomography in material science

energy photons, and hence to the gradual increase of the mean energy along the path. The reconstruction algorithm uses, for the reconstruction of any single point, experimental data corresponding to individual rays impinging the point of interest, but coming from different orientations. The corresponding information therefore corresponds to different attenuations, and hence different energies, and different values of ji. Two kinds of artifacts are generated by beam-hardening: i) cupping effect and ii) streaks. Cupping effect corresponds to measured values of \JL which are corrupted, thus preventing the measurement of the "true" density. As the measured values, inside an homogeneous sample, are lower at the center than at the edges, the name of cupping effect is generally used to describe this artifact. Projections can be corrected by acquiring an image of a step-wedge, made of the same material, in such a way to correlate the mesured attenuation to the true material thickness. Streaks artifact correspond to abnormal values along lines which correspond, inside the object, to high attenuation. Beam hardening artifacts can be avoided when using some filter, i.e. a metallic foil, directly set at the window of the X-ray tube and intended to pre-harden the spectrum [KAF 96]. Figure 1.10 displays an example of streaks inside the tomographic image of a set of six samples surrounded by air (Fig l.lO.a); the streaks are suppressed by the use of a copper filter, 0.1 mm in thickness (Fig. l.lO.b).

Figure 1.10. The reconstructed slice (l.lO.a) is corrupted by streaks due to beamhardening (l.lO.a). Filtration with a foil of copper, (0.1 mm) nearly suppresses the streaks (l.lO.b). The high voltage used for both images is 100 kV

Beam hardening is also avoided when using a monochromatic y-ray source. But it must be kept in mind that y-ray sources deliver a very low photon flux (typically one hundredth of the flux delivered by a tube). Tomography using synchrotron

General principles

25

radiation does not generates artifacts because a monochromator is always used, thanks to the huge X-ray flux. - Detector saturation To obtain a reconstruction which is free of defect, the signal delivered by every cell of the detector must be strictly proportional to the photon flux. Thus high values (approaching the upper limit of the digitization range) as well as low values (approaching the noise level) of the flux must be avoided. Streaks artifacts, similar to those obtained in the case of beam-hardening, are generated along lines which correspond to high attenuation. - Aliasing High (spatial) frequencies are encountered in the signal corresponding to every projection. They are due to the steep edges which are eventually present in the object. As the detector samples the signal (all along the projection) with a non-zero step, high frequencies corrupt the data, within the Fourier domain. Streaks are generated [KAK 87]. On figure 1.11, aliasing is visible at the corners of the objects. - Scattered photons Photons scattered by the sample or by its environment deliver a wrong information which leads to cupping effect. Collimation can improve the reconstructed image.

Figure 1.11. Aliasing at the corners

Figure 1.12. Ring artifacts

-111 corrected detector The signal delivered by every sensitive cell of the detector must be linearly spread between the offset level (corresponding to the absence of photons) and the gain level (corresponding to the non-attenuated flux). A bad correction of one cell will generate, in the reconstructed image a "ring artifact", i.e. the image of a ring,

26 X-ray tomography in material science

centered on the pixel corresponding to the location of the rotation axis. On figure 1.12 a great number of concentric rings are visible. - Spatial distortion of the detector Distortions of the projections, due for instance to the camera (e.g. distortions due to the lens) deliver artifacts which can be corrected by software. - Centering error The reconstruction requires the knowledge of the location of the projection of the center of rotation within the detector. Distortions are generated when the reference to the centre is erroneous.

1.5. Conclusions X and y-ray tomography allow a great number of potential applications. The measured quantity is in fact the linear attenuation coefficient \i, and not directly the density. A careful choice of the photons energy and the selection of a detector with a high dynamic range allows to lessen the noise to a reasonable level. Coefficient \JL can be estimated with an accuracy slightly better than 1%.

1.6. References [ATT 68] Anrx F.H.R., ROESCH W.C., Radiation Dosimetry, Academic Press, 1968. [BAR 57] BARTHOLOMEW R.N., CASAGRANDE, R.M., "Measuring solids concentration in fluidized systems by gamma-ray absorption", Industrial and Engineering Chemistry, vol. 49, n. 3, p. 428-431, 1957. [BER 95] BERNARD J.R., Frontiers in Industrial Process Tomography, Engineering Foundation, Ed. DM SCOTT& RA WILLIAMS, New-York, p. 197, 1995. [CEN 99] CENDRE, E. et al., "Conception of a high resolution X-ray computed tomography device; Application to damage initiation imaging inside materials", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 362-369, 1999. [CLO 97] CLOETENS P., PATEYRON-SALOME M., BUFFIERE J.-Y., PEK G., BARUCHEL J., PEYRIN F., SCHLENKER M., "Observation of microstructure and damage in materials by phase sensitive radiography and tomography", J. Appl. Phys., vol. 81, n. 9, p. 5878-5886, 1997. [DUV 98] DUVAUCHELLE P., Tomographie par diffusion Rayleigh et Compton avec un rayonnement synchrotron: Application a la pathologic cerebrale, these de doctoral, universite de Grenoble 1, 1998. [PEL 84] FELDKAMP L.A., DAVIS L.C., KRESS J.W., "Practical cone-beam algorithm", J. Opt. Soc., vol. 1, n. 6, p. 612-619, 1984.

General principles

27

[HAL 92] HALMSHAW R., "The effect of focal spot size in industrial radiography",flrif/s/i Journal of NOT, vol. 34, n. 8, p. 389-394, 1992. [HAR 99] HARTEVELD W.K. et al. "A fast active differencial capacitance transducer for electrical capacitance tomography", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 571-574, 1999. [ISO 99] iso/TC 135/SC 5 , ISO document "NDT Radiation methods- Computed tomography", Part I: Principles; Part II: Examination Practices, 1999. [JOH 96] JOHANSEN G.A, FR0YSTEIN T., HJERTAKER B.T., OLSEN O., "A dual

sensor flow imaging tomographic system", Meas. Sci. Techn., vol. 7, n. 3, p. 297-307, 1996. [KAF 96] KAFTANDJIAN V., PEDC G., BABOT D., PEYRIN F., "High resolution X-ray computed tomography using a solid-state linear detector", Journal of X-ray Science and Technology, vol. 6, p. 94-106, 1996. [KAK 87] KAK A.C., SLANEY M., Principles of Computerized Tomographic Imaging, IEEE Press, 1987. [PIN 99] PlNHEIRO P.A.T. et al., "Developments of 3-D Reconstruction Algorithms for ERT", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 563-570, 1999. [SCH 90], SCHNEBERK D.J., AZEVEDO S.G., MARTZ H.E., SKEATE M.F., "Sources of error in industrial tomographic reconstruction", Materials Evaluation, vol. 48, p. 609-617, 1990. [THI 99] THIERRY R. et al., "Simultaneous Compensation for Attenuation, Scatter and Detector Response for 2D-Emission Tomography on Nuclear Waste within Reduced Data", Proceedings of the 1st World Congress on Industrial Process Tomography, Umist Univ. (U.K.), p. 542-551, 1999. [ZHU 95] ZHU P., PEIX G., BABOT D., MULLER J., "In-line density measurement system using X-ray Compton scattering", NDT & E International, vol. 28, n. 1, p. 3-7, 1995.

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Chapitre 2

Phase contrast tomography

Hard X-ray radiography and tomography are common techniques for medical and industrial imaging. They normally rely on absorption contrast. However, the refractive index for X-rays is slightly different from unity and an X-ray beam is modulated in its optical phase after passing through a sample. The coherence of third generation synchrotron radiation beams makes a simple form of phase-contrast imaging, based on simple propagation, possible. Phase imaging can be used either in a qualitative way, mainly useful for edge-detection, or in a quantitative way, involving numerical retrieval of the phase from images recorded at different distances from the sample.

2.1. Introduction The phase of an X-ray beam transmitted by an object is shifted due to the interaction with the electrons in the material. Imaging using phase contrast as opposed to attenuation contrast is a powerful method for the investigation of light materials but also to distinguish, in absorbing samples, phases with very similar X-ray attenuation but different electron densities. Phase contrast imaging was pioneered in the early seventies by Ando and Hosoya [AND 72], who obtained images of bone tissues and of a slice of granite using a Bonse-Hart type interferometer [BON 65]. This technique developed into a quantitative three-dimensional imaging technique. Because of the limited quality of available lenses, elaborate forms of phase contrast imaging such as Zernike phase-contrast [ZER 35] or off-axis holography [LEI 62] are presently ruled out for hard X-rays. Three methods of phase sensitive imaging exist: the interferometric technique [MOM 95, BEC 97], the Schlieren technique [FOR 80, ING 95] and the propagation technique [SNI 95, CLO 96]. They are compared in section 3. The main advantages of the method used in this work, the propagation technique, are the extreme simplicity of the set-up and the better spatial resolution.

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This technique was mostly used up to now in the so-called 'edge-detection regime' to image directly the discontinuities in refractive index in the object. It is however possible to fully exploit the quantitative information entangled in the Fresnel diffraction patterns towards high resolution quantitative phase tomography. The 'holotomographic' reconstruction is performed in two steps: first the optical phase of the wave exiting the sample is retrieved numerically from images recorded at different distances from the sample. The refractive index distribution is then reconstructed from a large number of phase maps using a classical tomographic algorithm. Results of quantitative phase tomography on samples of interest to materials science are discussed.

2.2. X-ray phase modulation The interaction of a wave with matter affects its amplitude and phase. This can formally be described by the complex refractive index n of the medium. Because its value is nearly unity, it is usually written for X-rays as n = l - < J + i/?

[1]

A plane monochromatic wave propagating along the z-axis in vacuum is of the form exp(i^ L z) with A the X-ray wavelength. In a material with refractive index n this becomes exp(m^ L z). The refractive index decrement 6 results in a phase variation compared to propagation in vacuum. The imaginary part J3 determines the attenuation of the wave. The X-ray intensity is the squared modulus of the wave and the absorption index (3 is simply proportional to the linear absorption coefficient p.

,=

f>

[2]

The absorption index has a complex energy and composition dependence. It varies abruptly near the characteristic edges of the elements. The refractive index decrement 6 on the other hand is primarily due to Thomson scattering and has a much simpler dependency on the energy and the material characteristics. S is essentially proportional to the electron density in the material. Generally, it can be expressed as

where the sum extends over all atoms p, with atomic number Zp, in the volume V, rc = 2.8 fm is the classical electron radius, and f'p is the real part of the wavelengthdependent dispersion correction, significant near absorption edges, to the atomic scattering factor. If the composition of the material is known in terms of mass fractions qp, the following equivalent expressions can be used

Phase contrast tomography

31

[5]

with NA Avogadro's number and Ap the mass number. 6P and pp are respectively the refractive index decrement and mass density of the pure species. If the dispersion correction fp can be neglected, 6 is proportional to the electron density pe, i.e. S = r c A 2 p e /(27r). The ratios ZP/AP appearing in Equation 4 are similar for many atomic species (« 1/2), and 6 is thus to a good approximation determined by the mass density p of the material [GUI 94]

[6] Both 6 and ft are small, typically 10~5 - 10~6 and 10~8 - 10~9 respectively for light materials, indicating the power of phase sensitive imaging compared to the absorption. Figure 1 shows the ratio S /ft, a figure of merit for phase effects compared to attenuation effects, as a function of the X-ray energy E for aluminium. The energy range includes soft X-rays and hard X-rays. In the soft X-ray range, more precisely in the 'water window' where soft X-ray microscopes usually operate, a gain exists but it is relatively modest. On the other hand in the hard X-ray range (energies above 6 keV) this ratio increases with energy to huge values (up to 1000). Practically, if one selects for example an X-ray energy of 25 keV to be able to cross a thick aluminium sample, a hole in this metal should have a diameter of at least 20 /zm to produce 1 %

Figure 2.1. Ratio S//3 of the refractive ondex decrement and the absorption index as a function of the X-ray energy for the element aluminiu. This is a figure of merit for phase effects compared to attenuation effects

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X-ray tomography in material science

absorption contrast. Using the effect on the phase, the minimum detectable hole is reduced to about 0.05 yum. X-rays are adapted for imaging of thick samples thanks to their low absorption at high energies. If it is possible to visualise the phase of the transmitted wave, the sensitivity and spatial resolution remain good. For inhomogeneous samples the wave at the exit of the sample will be modulated in both phase and attenuation. Propagation inside the sample itself can usually be neglected and it is possible to project the object onto a single plane perpendicular to the propagation direction. The transmission function T(x, y) gives the ratio of the transmitted and the incident amplitudes. It can be compared to exp(— f n(x, y, z)dz) that gives the ratio of the transmitted and the incident intensities according to LambertBeer's law. This transmission function corresponds to the projection of the refractive index distribution through T(x,y) = A(x,y)eirtx>ri

[7]

with the amplitude A(x,y) = e-W*'*)

and

B(x,y} - y j 0(x,y,z)dz

[8]

(p(x, y) = Y / [1 - <5(z, y, z)]dz = (?0 - -^ / 6(x, y, z}dz .

[9]

and the phase modulation

(p0 is the phase modulation that would occur in the absence of the object. In classical absorption tomography the projection of n is determined for a large number of angular positions of the sample. The three-dimensional (3D) distribution of n(x, y, z) or equivalently of / 3 ( x , y , z} is then reconstructed from the set of projections using a tomographic reconstruction algorithm. Similarly if the phase map (p(x, y) is known for a large enough number of angular positions of the object, it is straightforward to reconstruct the distribution of the refractive index decrement 8(x, y, z).

2.3. Phase sensitive imaging methods There are three methods of phase sensitive imaging: the interferometric technique [MOM 95, EEC 97], the Schlieren technique [FOR 80, ING 95] and the propagation technique [SNI 95, CLO 96]. The co-existence of the different methods shows that they all have their advantages and disadvantages with respect to the accessible phaseinformation, the complexity of the set-up, the requirements on the beam or the spatial frequency range covered.

Phase contrast tomography

33

2.3.1. The interferometric technique Here contrast is due to interference of the beam transmitted through the object with a reference beam. If the beams are coherent with each other, the intensity will be directly affected by the local phase shift. Bragg-diffraction by perfect crystal slices cut out from a large, almost perfect monolithic silicon crystal is used to split, deviate and recombine the two beams. A possible configuration [HAR 75] is shown in Figure 2a. The recorded interference pattern cannot be exploited as it is because the interference fringes cannot be directly linked to a projection of the object and because an intrinsic fringe pattern is always present. The image treatment to quantitatively reconstruct the phase modulation introduced by the sample is however rather straightforward. Several images for different external phase shifts, typically 8 (including flatfield images), must be recorded to reconstruct a single phase-map. The possibility to perform phase tomography and to reconstruct the local distribution of the refractive index decrement with an X-ray interferometer was demonstrated by Momose et al and Beckmann et al in 1995 [MOM 95, BEC 95]. The interest of phase imaging compared to absorption imaging was frequently illustrated [MOM 96]. The complexity and stability requirements of this technique are however serious drawbacks. The sample must be immersed in a liquid that matches the refractive index of the sample. Otherwise large phase jumps at air-sample boundaries perturb the interference fringes and the large deflection in the sample reduces the visibility of the fringes. Some blurring is necessarily associated to the passage of the beam through the analyser crystal. This limits the resolution to about 15 /^m in the best case [BEC 97]. On the other hand the frequency range covered is not limited towards the low frequencies and a spatially homogeneous phase shift can be measured with respect to the reference beam.

2.3.2. Schlieren technique This differential phase contrast method is sensitive to the angular deviations of the X-ray beam. Phase gradients present in the object locally deviate the beam by an angle

Forster et al [FOR 80] used a double crystal arrangement similar to the one shown in Figure 2b. The first crystal acts as a collimator in limiting the angular and spectral range. The angular deviations introduced by the sample change the incidence angle with respect to the analyser that acts as an angular filter. The variety in the nomenclature for this approach can be noted: Schlieren-imaging [FOR 80, CLO 96], refraction contrast [SOM 91], phase dispersive imaging [ING 95, ING 96], phase contrast imaging [DAV 95] and diffraction enhanced imaging [CHA 98] are the most com-

34

X-ray tomography in material science

Figure 2.2. Set-up for phase sensitive methods: (a) Interferometric technique, (b) Schlieren technique and (c) Propagation technique

mon names. The possibility to visualise phase gradients (occuring for example at edges) was shown by many groups, but no reconstructed (differential) phase map was presented and the method was not extended to 3D imaging through tomographic techniques. Compared to the interferometric technique, the experimental set-up is simplified and the stability requirements are less stringent. To obtain a good sensitivity to phase gradients, the width of the rocking curve for one of the crystals relative to the other should be small, typically 2-10 yurad, and the angular stability should be about 0.2 yurad [ING 96]. As the alignment is less critical, the collimator and analyser crystal do not need to be part of a monolithic block, and the space available for the sample and its environment increases. The samples are in general not immersed in a liquid. The spatial resolution is again affected by the passage of the wave through the analyser crystal. This method is less adapted than the previous one to covering the low spatial frequency range, and very smooth variations of the phase may introduce a phase gradient that is too small to be detected. This imaging scheme can be used on a laboratory X-ray source. Most of the published results were obtained under these conditions, resulting in long exposure times of 15-30 minutes [ING 96] for a radiograph. This technique corresponds to Schlieren imaging in classical optics [HEC 98].

Phase contrast tomography

35

2.3.3. The propagation technique The spatial redistribution of the photons due to deflections or more generally Fresnel diffraction is considered a nuisance in absorption contact and projection radiography and in interferometric and Schlieren phase imaging. It is however also a unique contrast mechanism for phase sensitive imaging, with advantages in the simplicity of the set-up and the achievable resolution. In this case there is no distinct reference beam as in the interferometric technique, and the beam transmitted through the object plays this role itself. The occurrence of contrast can be understood as due to interference between parts of the wavefront that have suffered slightly different angular deviations associated to different phase gradients. The overlap between parts of the wavefront is only possible after propagation over a certain distance. As in previous case, this is a differential phase imaging technique. A homogeneous phase gradient cannot be detected because it corresponds to an overall deflection of the beam; detectable contrast requires the second derivative of the phase to be non-zero. When the direction of the X-ray beam is tangential to the edge of structures in the sample, such a perturbation of the wavefront is expected and contrast will appear. Possible internal structures are holes and cracks, inclusions, reinforcing particles or fibers in a composite material. Experimentally the sample is set in a (partially) coherent beam and the transmitted beam is recorded at a given distance d with respect to the sample [SNI 95, CLO 96]. The experimental set-up shown in Figure 2c is thus essentially the same as for absorption radiography except for the increased sample to detector distance. The crystal system upstream of the sample selects a narrow spectral range, delivering a quasi-monochromatic beam to the sample. The image contrast changes tremendously with the sample detector distance d. The latter determines the defocusing distance D through [BOR 80]

with / the source sample distance. In the case of the long ESRF beamline ID 19 (d
When it is small compared to the typical transverse dimension a of the features in the sample, a separate fringe pattern shows up for every border in the sample, and the images are characteristic of the 'edge-detection regime' (rp <§; a). Three-dimensional reconstruction of the boundaries inside the volume is feasible with the algorithm for absorption tomography (cf. section 4). At larger distance (rp w a) several interference fringes show up in the radiographs. These deformed images, corresponding to

36

X-ray tomography in material science

the 'holographic regime', give little direct information on the sample. However, combining such images recorded at different distances with a suitable numerical algorithm gives access to the phase modulation (cf. section 5). For the largest distances, rarely accessible with X-rays, one reaches the Fraunhofer limit (rp » a). Figure 3 shows as an example four radiographs of a 0.5 mm thick piece of polystyrene foam at increasing distances D. The beam is monochromatised to 18 keV. As the distance increases, the contrast and width of the Fresnel fringes both increase. The radiographs are recorded with a CCD based detector involving X-ray / visible light conversion in a transparent YAG:Ce screen [KOC 98], with an effective pixel size of 0.95 //m. The most striking advantage of this method is the extreme simplicity of the setup. It is essentially the same as for absorption radiography. The transition between absorption and phase radiography or between the different regimes of phase imaging is simply obtained by changing the sample detector distance. The stability requirements on the (few) elements downstream of the monochromator, i.e. the sample and the detector, are easily met. The monochromator can be well upstream of the sample and the sample detector distance can often be chosen quite large. The free space around the sample can be used for all kinds of devices for in-situ and real-time observations. However the optical elements of the beamline have to be carefully prepared to avoid spurious phase images. It can be shown that for a given defocusing distance the image is most sensitive to a specific frequency range. The optimum distance to be sensitive to phase features with spatial frequency / is such that

This frequency selectivity will intrinsically limit the accessibility to the low frequency range, i.e. the smooth variations in the object's phase. The optimum distance, increasing as the square of the object size, will not be reached for these frequencies due to physical limitations (size of the experimental hutch) or the coherence conditions. The image is not spoiled by the passage of the modulated wave through a crystal as is the case in the interferometric and Schlieren techniques. The resolution in the propagation technique depends on the image processing after recording. For untreated images recorded in the edge-detection regime the resolution is limited by the fringe spacing to about 2rp. When the fringes are disentangled in a holographic reconstruction, the spatial resolution is limited essentially by the detector. The stringent requirements on the beam incident on the sample explain why this technique emerged only recently [HAR 94, SNI 95, CLO 96, NUG 96] with the appearance of partially coherent X-ray beams delivered by third generation synchrotron radiation sources. The geometrical resolution in the Fresnel diffraction pattern is equal to Dsa with sa — f the angular source size and s the source size. The condition for observation of the spatial frequency / is

Phase contrast tomography

37

Figure 2.3. Phase sensitive radiographs of a 0.5 mm thick piece of polystyrene foam, with the detector at various distances dfrom the sample. X-ray energy 18 keV. (a) d= 0.03 m, (b) d = 0.2 m, (c) d = 0.5 m and (d) d = 0.9 m. The contrast and the width of the interference fringes increase trough the series

The blurring due to source size and detector resolution explains why no interference fringes are observed with classical laboratory sources although the propagation distances are also non-zero in projection radiography. Alternatively, the interference pattern involves the coherent superposition of laterally separated portions of the incident beam. The interfering waves must originate from points that are mutually coherent, and thus laterally separated by a distance smaller than the transverse coherence length /t that can be defined as l[ = A/(2s a ). The incident beam must be coherent over the first Fresnel zone for optimum conditions. The conditions on the monochromaticity are less stringent. The beam is usually monochromatised using a monochromator based on perfect silicon crystals. The en-

38

X-ray tomography in material science

ergy spread AA/A « 10~4 is thus very small. The thickness of the samples that can be investigated is not limited to the longitudinal coherence length (= A 2 /AA) [CLO 96]. An increase in energy spread by one or even two orders of magnitude can still correspond to quasi-monochromatic conditions. This allows to increase considerably the flux using a multi-layer monochromator with a very high substrate quality. Using the same propagation principle but working with the polychromatic radiation delivered by a laboratory X-ray microsource to retain some flux, deflection sensitive images were obtained [WIL 96, POG 97]. This seems promising for work in the edge detection regime as the main contrast, a white and black fringe, is unchanged over a large spectral range.

2.4. Direct imaging Most of the tomographic work performed until now using the propagation technique is based on the usual algorithm for absorption tomography. This is a workable solution especially when the defocusing distance D is small and the sample is made up of different (metallographic) phases with different densities [CLO 97]. This qualitative approach allows to visualise, in 3D, density discontinuities, such as reinforcing SiC particles in an aluminium matrix composite [BUF 99]. Density jumps appear as dark / light fringes. Another advantage is the possibility to detect and localise features that are actually smaller than the pixel size as the interference fringes produced can be larger than the feature itself (regime with rp > a). This makes it possible to detect cracks with sub-micron opening. An example of a tomographic slice of an aluminiumsilicon alloy obtained using this direct approach based on a single distance is shown on Figure 6b in the next section. However, with this approach the spatial resolution is limited by the Fresnel fringe distribution and artefacts occur in some cases due to the ill-suited algorithm. Binarisation of the edge-regime images is extremely tedious and was done essentially manually. It must also be noted that the quantitative information on density and composition is lost. For these reasons there is a need for more adapted algorithms preserving resolution and quantitative information, at the expense of an increased data volume and computational effort.

2.5. Quantitative imaging A new approach, holotomography, has now been implemented to extract the quantitative distribution of the phase (and attenuation) in two-dimensional projection images, then to turn it into 3D reconstructions. It is based on images obtained at several values of D for each angular position of the sample, in analogy with a technique developed for electron microscopy. The reconstruction of the 3D distribution of the refractive index is performed in two steps. In a first step (holographic reconstruction), the phase map and if relevant the attenuation map is restored numerically for every

Phase contrast tomography

39

projection from a set of in-line holograms. In a second step (toraographic reconstruction), the Radon transform is inversed using a classical tomographic reconstruction method, such as the filtered backprojection method. Details on the phase retrieval method are given elsewhere [CLO 99a, CLO 99b]. Here we only give two examples: the first is a non-absorbing polymer foam and the second an absorbing metal alloy that introduces very large phase shifts.

2.5.1. A polymer foam

Figure 2.4. Phase map retrieved with an algorithm that combines images recorded at several (here four) distances. The sample is a 0.5 mm thick piece of polystyrene. The X-ray energy is 18 keV.

A piece of polystyrene foam with a rather complex 3D structure was used to test the method. It had a cross-section of 0.5 x 0.7 mm2 and negligible attenuation. Four tomographic scans of 700 views each were automatically recorded at defocusing distances D of 0.03, 0.21, 0.51 and 0.9 m. The radiographs for a given angular position are those of Figure 3. The corresponding reconstructed phase map is shown in Figure 4. The phase retrieval included correction for the detector response and the partial coherence. The phase varies between about 0 and -7 rad. This phase map has a straightforward interpretation: it is a projection along the X-ray path of the electron density in the sample. Cells with sizes on the order of 100 ^m are clearly revealed. The holographic reconstruction was repeated for the 700 angular positions of the sample. The phase maps, projections of the refractive index decrement, were used to determine the 3D distribution of 6 in the sample with a filtered backprojection algorithm. Figure 5a shows a slice of the reconstructed volume. The gray scale is linear with respect to the index decrement 8(x, y, z), darker corresponding to a higher electron and mass density. The interpretation of the reconstructed slices is thus straightforward. The contrast of the polymer foam is excellent. This would certainly not be

40

X-ray tomography in material science

Figure 2.5. (a) A slice of the reconstructed distribution of the refractive index decrement 8 of a polystyrene foam. S is essentially proportional to the electron and mass density in the polymer. A phase map was determined for each of 700 angular positions using images at four distances. From the projections the 3D distribution was determined with a filtered backprojection algorithm. (b) Magnified portion of the reconstructed slice shown in a (a). (c) Profile along the arrow shown in (a). E = 18 keV.

the case in absorption tomography, the ratio of the refractive index decrement and the absorption index S/j3 being 2500. Figure 5c shows a profile of the index decrement along a line segment shown in Figure 5b. The air/polymer transition is abrupt and from this profile the spatial resolution is estimated to be about 2.5 jum (fwhm). In regions that apparently contain only polymer the refractive index decrement is 6.6 10~7, corresponding to a mass density of 0.96 g/cm3, in agreement with the expected density value for polystyrene of about 1 g/cm3. The tomographic slice of Figure 5a intersects a cell of the foam. It is completely enclosed by a polymer wall. The thin walls were not evident in the projection images, but they appear correctly after tomographic reconstruction. The shape of the cell is irregular and very distorted. This is probably due to a crushing process. The magnified portion of the slice shown in Figure 5b, indicates an excellent 3D isotropic spatial resolution that can be obtained thanks to the detector resolution, but also because the holographic reconstruction disentangled the object information from the defocused images.

Phase contrast tomography

41

Figure 2.6. Three tomographic slices of an aluminium-silicon alloy quenched from the semisolid state, obtained (a) using absorption contrast, (b) using phase contrast and a single propagation distance, (c) using phase contrast and holotomography based on four distances. E = 18 keV.

42

X-ray tomography in material science

2.5.2. A metal alloy The second sample, an aluminium-silicon alloy quenched from the semi-solid state, represents a more tedious problem. The phase modulation must be determined in the presence of attenuation by the matrix. The difference in attenuation between the two metallurgic phases is however small. A severe problem is due to the large phase modulation introduced by the cylindrical shape of the sample with a diameter of about 1.5 mm. This thickness of aluminium introduces at an X-ray energy of 18keV a phase shift of more than 200 radians and would be very difficult to reconstruct. This large shift was not present in the previous case because most of the foam is actually air. It was therefore decided to reconstruct only the phase variations with respect to the phase introduced by a homogeneous matrix. Figure 6a is a tomographic slice recorded at D = 1 mm, sensitive only to variations in absorption. It is impossible to distinguish the two phases, some bright spots appear corresponding to iron-rich inclusions. Figure 6b is a tomographic slice obtained for a single distance D = 0.6 m, revealing density jumps as dark / light fringes. Binarising such an image toward quantitative metallographic evaluation is extremely tedious. Figure 6c is a reconstructed map of the variations in refractive index decrement 6(x, y, z), clearly showing the slight difference in density of the two (metallurgical) phases (Ap w 0.05 g/cm3). The grey phase was the liquid in the semi-solid state and it consists of an aluminium-silicon eutectic. The dark phase was the solid in the semi-solid phase and it is essentially pure aluminium with substitutional silicon (see L. Salvo et al, these proceedings). The data set consisted of 4 times 800 images recorded at distances of 0.007, 0.2, 0.6 and 0.9 m from the sample. The beam was monochromatised to 18 keV by a Ru/B4C multilayer.

2.6. Conclusion Phase sensitive images are experimentally simple to obtain. Third generation synchrotron radiation facilities, such as ESRF, can easily produce the necessary coherent beam. A first approach is to use the phase radiographs directly or in combination with the algorithm for absorption tomography. In many cases this gives useful results, for example the detection of cracks or reinforcing particles in composite materials. A new approach, called 'holotomography' is now operational. It allows to perform quantitative phase contrast tomography with a simple set-up. We used for the holographic reconstruction, in analogy with electron microscopy, a numerical procedure based on images at different distances from the sample. The feasibility of this method was demonstrated on a polystyrene foam. The results are in quantitative agreement with the known sample composition. The study of a metal alloy, with metallographic phases that are not distinguishable in the absorption mode, demonstrates the usefulness of this technique for materials sciences. A drastic improvement in sensitivity and resolution is obtained compared to usual X-ray tomography.

Phase contrast tomography

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2.7. References [AND 72] ANDO M., HOSOYA S., "An attempt at X-ray phase-contrast microscopy", SHINODA G., KOHRA K., ICHINOKAWA T., Eds., Proc. 6th Intern. Conf. on X-ray Optics and Microanalysis, p. 63-68, Univ. of Tokyo Press, Tokyo, 1972.

[BEC 95] BECKMANN R, BONSE U., BUSCH F., GUNNEWIG O., BIERMANN T., "A novel system for X-ray phase-contrast microtomography", HASYLAB Annual Report II, p. 691692, 1995. [BEC 97] B E C K M A N N F., BONSE U., BUSCH F., GUNNEWIG O., "X-ray microtomography (//CT) using phase contrast for the investigation of organic matter", J. Computer Assist. Tomography, vol. 21, 1997, p. 539. [BON 65] BONSE U., HART M., "An X-ray interferometer", Appl. Phys. Lett., vol. 6, 1965, p. 155-156. [BOR 80] BORN M., WOLF E., Principle of Optics, 6th ed., Pergamon Press, Oxford, New York, 1980. [BUF 99] BUFFIERE J. Y., MAIRE E., CLOETENS P., LORMAND G., FOUGERES R., "Characterisation of internal damage in a MMCp using X-Ray synchrotron phase contrast microtomography", Acta Mater., vol. 47, 1999, p. 1613-1625. [CHA98] CHAPMAN D., THOMLINSON W., ZHONG Z., JOHNSTON R. E., PISANO E., WASHBURN D., SAVERS D., SEGRE C., "Diffraction Enhanced Imaging Applied to Materials Science and Medicine", Synchrotron Radiation News, vol. 11, 1998, p. 4-11. [CLO96] CLOETENS P., BARRETT R., BARUCHEL J., GUIGAY J. P., SCHLENKER M., "Phase objects in synchrotron radiation hard x-ray imaging", J. Phys. D, vol. 29, 1996, p. 133-146. [CLO 97] CLOETENS P., PATEYRON-SALOME M., BUFFIERE J. Y., PEIX G., BARUCHEL J., PEYRIN F., SCHLENKER M., "Observation of microstructure and damage in materials by phase sensitive radiography and tomography", J. Appl. Phys., vol. 81, 1997, p. 5878-5886. [CLO 99a] CLOETENS P., LUDWIG W., BARUCHEL J., VAN DYCK D., VAN LANDUYT J., GUIGAY J. P., SCHLENKER M., "Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays", Appl. Phys. Lett., vol. 75, 1999, p. 2912-2914. [CLO99b] CLOETENS P., LUDWIG W., VAN DYCK D., GUIGAY J. P., SCHLENKER M., BARUCHEL J., "Quantitative phase tomography by holographic reconstruction", BONSE U., Ed., Developments in X-Ray Tomography II, vol. 3772, 1999, p. 279-290. [DAV 95] DAVIS T. J., GAO D., GUREYEV T. E., STEVENSON A. W., WILKINS W., "Phasecontrast imaging of weakly absorbing materials using hard X-rays", Nature, vol. 373, 1995, p. 595-598. [FOR 80] FORSTER E., GOETZ K., ZAUMSEIL P., "Double Crystal Diffractometry for the Characterization of Targets for Laser Fusion Experiments", Kristall und Technik, vol. 1, 1980, p. 937-945. [GUI 94] GUINIER A., X-Ray Diffraction In Crystals, Imperfect Crystals, and Amorphous Bodies, Dover Publications Inc., New-York, 1994. [HAR 75] HART M., 'Ten years of X-ray interferometry", Proc. R. Soc. Lond. A, vol. 346, 1975, p. 1.

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[HAR94] HARTMAN Y. M., SNIGIREV A., "Some examples of high energy X-rays phase contrast", ARISTOV V. V., ERKO A. I., Eds., X-ray Microscopy IV, p. 429-432, Bogorodskii Pechatnik Publishing Company, Moscow, 1994. [HEC 98] HECHT E., Optics, Srded., Addison-Wesley, 1998. [ING 95] INGAL V. N., BELIAEVSKAYA E. A., "X-ray plane wave topography observation of the phase contrast from a non-crystalline object", J. Phys. D: Appl. Phys., vol. 28, 1995, p. 2314-2317. [ING 96] INGAL V. N., BELIAEVSKAYA E. A., "Phase dispersion radiography of biological objects", Physica Medico, vol. 12, 1996, p. 75-81. . [KOC 98] KOCH A., RAVEN C., SPANNE P., SNIGIREV A., "X-ray imaging with submicrometer resolution employing transparent luminescent screens", / Opt. Soc. Am. A, vol. 15, 1998, p. 1940-1951. [LEI 62] LEITH E. N., UPATNIEKS J., "Reconstructed Wavefronts and Communication Theory", J. Opt. Soc. Am., vol. 52, 1962, p. 1123. [MOM 95] MOMOSE A., "Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer", Nucl. Inst. Meth. A, vol. 352, 1995, p. 622-628. [MOM 96] MOMOSE A., TAKEDA T., ITAI Y., HIRANO K., "Phase-contrast X-ray computed tomography for observing biological soft tissues", Nature Medicine, vol. 2, 1996, p. 473475. [NUG96] NUGENT K. A., GUREYEV T. E., COOKSON D. F., PAGANIN D., BARNEA Z., "Quantitative Phase Imaging Using Hard X Rays", Phys. Rev. Lett., vol. 77, 1996, p. 29612964. [POG 97] POGANY A., GAO D., WILKINS S. W., "Contrast and resolution in imaging with a microfocus x-ray source", Rev. Sci. Instrum., vol. 68, 1997, p. 2774-2782. [SNI95] SNIGIREV A., SNIGIREVA I., KOHN V., KUZNETSOV S., SCHELOKOV I., "On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation", Rev. Sci. Instrum., vol. 66, 1995, p. 5486-5492. [SOM 91] SOMENKOV V. A., TKALICH A. K., SHIL'SHTEIN S. S., "Refraction contrast in x-ray introscopy", Zh. Tekh. Fiz., vol. 61, 1991, p. 197-201. [WIL96] WILKINS S. W., GUREYEV T. E., GAO D., POGANY A., STEVENSON A. W., "Phase-contrast imaging using polychromatic hard X-rays", Nature, vol. 384, 1996, p. 335338. [ZER 35] ZERNIKE F., Z. Tech. Phys., vol. 16, 1935, p. 454.

Chapitre 3

Microtomography at a third generation synchrotron radiation facility

The use of the modern synchrotron radiation sources for monochromatic beam microtomography provides several new possibilities. They include, in addition to reduced signal-to-noise ratio and enhanced spatial resolution, the easy setup of sample environment for in-situ experiments, the quantitative measurements, and the rapidly increasing field of phase imaging. Other topics, such as the present state of 'local' microtomography, and some new approaches like diffraction and fluorescence microtomography, are also briefly described. 3.1. Introduction Since their discovery X-rays have been used to image the bulk of materials which are non transparent for visible light, by taking advantage of their inhomogeneous absorption. A tremendous progress was obtained, 25 years ago, when three-dimensional (3D) visualization became available. This computerassisted tomographic approach provides the 3D information, from the many twodimensional (2D) images (radiographs) recorded at various angular positions of the object, and via appropriate algorithms ("filtered back-projection", described for instance in references [KAK88] and [PEY96]) and software. From the reconstructed 3D data cuts, projections, or perspective renditions of the object can be obtained at will. Microtomography, with spatial resolution better than 20 um, recent emerged in years [FLA87]. Several laboratory instruments have been developed [SAS98, RUE96], are commercially available, and produce good results. However, the best quality images, in terms of signal-to-noise ratio and spatial resolution, are obtained

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on instruments located in synchrotron radiation facilities, which use a parallel and monochromatic beam. The number of 2D images necessary is approximately equal to the number of pixel columns used by the image on the detector, typically between 100 and 1000. The time needed for recording the 2D images is a function of the source and the experimental setup. This chapitre reports the new possibilities associated with the availability of third generation sources of synchrotron radiation, such as the European Synchrotron Radiation Facility (ESRF), with examples which illustrate these possibilities. Section 3.2 comprises a brief presentation of the aspects of synchrotron radiation which are relevant to microtomography. Section 3.3 shows how a substantial improvement in the signal-to noise ratio is obtained when using these modern sources. Section 3.4 shows examples where the improvement in the spatial resolution is crucial for the observation of features of interest. Section 3.5 considers the quantitative measurements in the absorption case. Section 3.6 shows the present state of 'local' microtomography. Section 3.7 deals with the sample environment in microtomography. Section 3.8 gives a brief report on all that concerns phase imaging, which is treated extensively in a companion paper. Finally, section 3.9 introduces some new approaches such as diffraction and fluorescence microtomography. 3.2. Synchrotron radiation and microtomography Synchrotron radiation [RAO93] is the electromagnetic radiation produced by ultra-relativistic electrons (energies of several GeV) in a storage ring when they are

Figure 3.1. Schematic representation of a synchrotron radiation facility, showing the linear accelerator (LA) and the booster (B). In these two elements the electron beam energy is raised to the wished value (6 GEV at the ESRF). Then the electron beam is introduced in the storage ring (SR), where it produces, when accelerated by the magnetic fields which modify its trajectory, the X-ray beams. The X-ray beams are used at the level of the beamlines (EL), tangent to the storage ring

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accelerated by a magnetic field (figure 3.1). This field is uniform over a portion of the trajectory in the bending magnets. It oscillates spatially in the insertion devices (wigglers and undulators) which can be set on the straight sections between two bending magnets.

D

Figure 3.2. Relative brilliance (arbitrary units, logarithmic scale) of the X-ray sources from the discovery of X-rays to 3rd generation SR facilities

The X-ray beam produced is tangent to the curved trajectory of the electrons in the storage ring: the beamlines are thus located all around the storage ring, as indicated on figure 3.1. Along with all synchrotron radiation sources, the ESRF features very high intensity of the emitted beam (figure 3.2) and, when the source is a bending magnet or a wiggler, a continuous spectrum, spanning the whole range from the infra-red to X-rays. The original features of third generation synchrotron radiation facilities, and more particularly the ESRF, when considering the microtomographic applications, are: a) the very high intensity of the X-ray beam (fig. 3.2) b) the high energy (6 GeV) of the electrons producing the radiation, which implies the availability of high energy photons (beyond 100 keV) c) the policy of providing each beam-line, through appropriate choice of the insertion device, with the spectrum best suited to the experiment it is dedicated to, and d) the small size of the electron beam cross-section (< 100 um). This leads to high brilliance, but also to a very small angular extension of the source as seen from a point in the specimen, hence to a sizeable lateral coherence of the X-ray beam.

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These beam features make it possible to perform microtomographic experiments that are improved and/or radically new, through the use 1) of a beam that is very intense, homogeneous, parallel, and, after passage in a monochromator (either a perfect crystal or a multilayer), monochromatic. This is to be compared with the relatively weak, polychromatic and divergent beam used in laboratory tomography 2) of the coherence properties of the beam, making it possible to obtain phase images by simply adjusting the sample-detector distance ("propagation" technique) Two other points are not directly related to the source, but are nevertheless crucial for this type of experiments: 3) the availability of a suitable detector, displaying at the same time a large dynamic range, a low noise, and a transfer time shorter than the typical exposure time. No such a detector is presently available on the market, and we use the Fast REadout LOw Noise (FRELON) camera, developed at ESRF [LAB96], fitted with an optical system which leads to effective pixel sizes between one and several micrometers 4) the improvement of the reconstruction and image processing procedures, leading to software suited to the problem, associated with the required computing memory and calculation power. Figure 3.3 is a scheme of the experimental setup used at the beamline ID 19 of the ESRF, where most of the images presented as examples in the present paper were obtained. It shows that the incoming beam can be considered as parallel, the source being situated at 150 metres from the sample. This monochromatic parallel beam setup exhibits many advantages. Among them let us mention that the reconstruction algorithms are exact, free from approximations, and that quantitative measurements are possible, avoiding all the beam hardening effects. It also shows the main drawback of this approach: no magnification is obtained, and the spatial resolution mainly results from the effective pixel size of the detector.

Figure 3.3. Principle of the parallel and monochromatic beam microtomography

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When using the tomographic setup of the ID 19 beamline the total acquisition time is of the order of half an hour for recording 900 projections images, and around one hundred reference images for flat-field correction. Data acquisition for one sample typically represents 2 Gigabytes, and enables the reconstruction of a (1024)3 tomographic image. Due to the parallel geometry, the 3D reconstruction problem may be solved using a sequence of Filtered Back Projection algorithms run on each slice of the volume, after rearranging the data in a set of corrected sinograms. To avoid the duplication of the 2 Gigabytes of data, and save some computations, a reconstruction program directly handling the radiographic images acquired during experiment was developed. The program includes both pre-processing, and a 3D version of the FBP algorithm adapted to 2D parallel projections. This algorithm was parallelized on the workstation cluster of the ESRF, to speed up the reconstruction. Using 4 machines, the mean reconstruction time obtained for (512) 3 volumes, is approximately 2 hours, and depends on the workload of the system, since the system is multi-user. Reduced reconstruction times are presently reached by devoting an optimized computer to the reconstruction task.

3.3. Improvement in the signal to noise ratio in the 3D images The synchrotron radiation source allows to dramatically reduce the exposure times and improve the signal-to-noise ratio compared to a standard x-ray tube. Figure 3.4 illustrates the differences of signal-to-noise ratio on the images of a vertebra sample acquired once using the ID 19 Synchrotron microtomography setup, and once a standard x-ray tube microtomography setup at IBT [RUE 96]. Figures 3.4a) and b) represent a slice through both 3D images, whose pixel sizes are respectively 6.65 um, and 14 um. The image obtained at the ESRF is clearly less noisy. To give an order of magnitude, the signal to noise ratio estimated under the assumptions that the noise is stationary and uncorrelated, was found to be 70.8 on the ESRF image and 5.8 on the IBT image. The peak signal-to-noise ratio expressing the ratio of the image dynamic to the noise, are respectively 32.3 dB and 21 dB for these two images. These results come from the fact that the signal-tonoise ratio is proportional to the detected number of photons. The improvement of the signal-to-noise ratio on the IBT image would be possible if the acquisition time was increased, which may not be compatible with a 3D acquisition. The differences are smaller on surface rendering displays of the 3D images as illustrated on figures 3.4c) and d). However, this display is obtained after segmentation of bone from background, which is more difficult on a noisiest image. A detailed comparison of these two techniques for the quantitative analysis of three-dimensional bone microarchitecture is reported in [PEY98] and [SAL98].

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Figure 3.4. Images of a vertebra sample obtained using the ESRF Micro-CT and the IBT Micro-CT: a)-b): 2D slices extracted from the 3D images, c) d) 3D displays

3.4. Improvement in the spatial resolution The availability of intense, parallel and monochromatic beams made it possible to obtain images from very diverse materials, and to reconstruct the volume with a resolution of the order of one u,m, much better than provided by laboratory generators (limited in spatial resolution to about 20 urn). As already pointed out, this spatial resolution is mainly determined by that of the detector, a CCD camera, with suitable optical setups, specially developed for these experiments. Increasing spatial resolution while keeping the same signal to noise ratio requires to dramatically increase the number of photons. Under some estimations of the signal to noise ratio in the reconstructed image, it may be considered that the

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required number of photons grows as the third power of the number of pixels in the image [PEY96]. As an illustration of the capabilities of micrometer synchrotron micro tomography, figure 3.5 represents a reconstructed slice of a three dimensional image of a fetal mouse bone, recorded with a voxel size of 1.8 um. An inferior spatial resolution would not allow to observe micro-structures within this bone.

Figure 3.5. Reconstructed slice of a fetal mouse bone, pixel size : 1.8 \jan

Examples were an improved spatial resolution is a crucial point can be found in the literature: this is the case of building materials (brick, concrete), where these techniques allow to visualize the various components, as well as the pores. Models for fluid diffusion (air, vapors) using as parameters the values deduced from the images are in good agreement with experimental data [QUE98]. A synchrotron radiation microtomography investigation of porous reservoir rocks, where both oil and brine are visible on the reconstructions, was also published [COL98], and this type of investigation is of growing importance. Other examples, as the penetration of liquid Ga in Al-based alloys, can be found in the present volume (paper by Ludwig and Bellet).

3.5. Quantitative measurements (absorption case) A clear-cut tendency of absorption microtomography using synchrotron radiation is the quantitative evaluation of the images. Thus the interest shifts from just the architecture and porosity of the object to the densitometry of the solid parts. The monochromaticity of the beam is the key point since this condition is a basic assumption for tomographic reconstruction. This condition is not fulfilled when using conventional polychromatic x-ray sources, and this may result in beam hardening artifacts in the reconstructed images, due to the more important attenuation of soft x-rays in the sample. Monochromaticity of the beam thus enables

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quantitative measurements of the absorption coefficient u (x,y,z), and consequently quantitative density measurements. Figure 3.6 is the illustration that it is now possible to access not only the architecture and porosity of the object but also the densitometry of the solid parts: it represents a reconstructed slice through a bone biopsy, where different gray levels may be observed within the cortical region. This indicates the feasibility of mapping 3D bone mineralization, using a suitable calibration. The agreement between the theoretical and reconstructed linear attenuation coefficients of hydroxyapatite phantoms with different concentrations is shown on figure 3.7 [SAL99] .

Figure 3.6. Reconstructed slice through a human bone biopsy, showing different gray levels within the cortical region, indicating differents levels of mineralization (voxel size: 10 /jm)

Figure 3.7. Linear attenuation coefficient of hydroxyapatite phantoms of various concentrations

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3.6. Present state of "local" tomography Another emerging trend is the implementation of "local" or "zoom" microtomography, i.e. the high-resolution reconstruction of an interesting region within a matrix which gets only low-resolution reconstruction (or even no reconstruction at all). This approach is essential for applications where it is not possible or desirable to extract from the matrix a sample small enough to be entirely illuminated by the beam.

Figure 3.8. Two reconstructed slices of the same sample a) with a 2 jam voxel size (local tomography, region of 2mm in diameter), b) with a 6.6 jum voxel size ( 6 mm diameter sample completely immersed into the beam)

Since the 2D detector is based on a 1024x1024 CCD, the size of the field of view is determined by the pixel size in the image. Tomographic acquisition theoretically requires that the field of view encompasses the entire transaxial extent of the object at every angle. When the sample is only partly illuminated by the beam, the projections are truncated, and the reconstruction problem is known as 'Local Tomography'. Many efforts are devoted to obtain reliable Local Tomography images, in spite of the missing information. Among them, let us mention a wavelet-based algorithm for multiresolution tomographic reconstructions [PEY99]. However if the region of interest is centered, and if the sample is globally homogeneous, a good approximation in the center of the structure of the sample can be obtained. Figure 3.8 shows that in this case the reconstructed image from truncated projections is close to the image reconstructed from the complete data. The projections from a 'composite with fibers' sample were recorded both with a 6.65 urn, and 2 urn, pixel size on the detector. In the second case the projections are

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truncated, which generates the important circular artifact at the border of the image. However we may notice that even if the values in the center of the image may not be used quantitatively, they provide an accurate information on the structure of the sample.

3.7. Sample environment in microtomography

Figure 3.9. Snow samples resulting from the evolution of fresh snow, collected on the field a) obtained by immersion in water at 0°C, well-rounded grains, b) displaying faceted crystallites, obtained by transformation under the action of a temperature gradient (l°C/cm). 10 keV'X-rays.

Figuer 3.9 corresponds to a quite common porous medium, snow, which requires to remain at low temperature to be investigated. The sample was located in a cryostat especially designed for microtomography, featuring X-ray absorption that is weak and independent of the angular position of the sample during its rotation. The snow was maintained at -60°C, in a regulated temperature nitrogen flow. It sitted within a cylindrical enclosure with polished double plexiglass walls, 0.5 mm thick . This device allows to establish a catalog of snows: fig. 3.8 shows that tomographic methods provide data on the three-dimensional microstructure of snow that is statistically significant because it represents a large number of grains, and at a high resolution compared to the grain's scale [BRZ99]. Indeed, snow is a mixture of ice particles, air and occasionally liquid water which can take different aspects. In most cases, recent snow is a very loose powder, which can transform into hard, crusted or pasty material, according to weather conditions and exposure. The growth of ice particles is caused by vapor diffusion in dry snow and melt-freeze exchanges in wet snow. Normally, dry snow covers are warmer at the bottom than on the top. The value of the temperature gradient determines whether rounded (small gradient) or faceted (large gradient) crystals will grow. These transformations have huge consequences: changes over several order of magnitude in the physical and

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mechanical properties of the different snow layers are commonplace. In some cases the snow crystals are able to stick on vertical rocks, whereas under some conditions a single skier can release a slab avalanche. The shape (specific area, local grain curvature) and arrangement (grain connections) of the grains, and the quality of the ice bonds will govern the snow properties. These parameters cannot be directly derived from classical two-dimensional observations. Other examples where the sample environment is a crucial part of the scientific case are 1) the investigation of fatigue failure in silicon carbide particle reinforced aluminum based composites, ([BUF97] and present volume), which employs a traction machine specially designed for microtomographic experiments, or 2) the 'local tomography' in-situ investigation of an open-cell polyurethane foam at several levels of compressive strain. This work correlates the macroscopic behaviour (stress/strain curve) with the local structure modifications. It shows, during the initial phase of compression (linear elastic response) the struts bending, and, on a further compression stage (plateau in the stress/strain curve) the collapse a whole band of cells [WINOO]. 3.8. Phase Imaging It is shown, in another paper of the present volume [Cloetens et al.], that the Xray beams produced at third generation synchrotron radiation facilities exhibit a high degree of coherence. This allows to use them to record "phase images" by just varying the sample-to-detector distance ("propagation technique"). The great advantage of this new type of imagery is the increased sensitivity it provides, in particular for light materials such as polymers, or for composites made up of materials with neighboring densities (for example Al and SiC). This implies, of course, that the spurious contrast due to inhomogeneities of the beamline components has been eliminated [ESP 98] Phase microtomography based on the visualisation of the edges was used, for instance, to understand the mechanisms of degradation in aluminum-SiC composites. It is possible not only to easily visualize the SiC reinforcing particles, but also to observe the nucleation and propagation of cracks when the material is submitted, in situ, to tensile stress. The cracks appear first in the elongated particles, and this imaging technique has shown that their number is 50% more than suggested by surface investigations [PEI 97, BUF 97]. Another example of application is the work on quasicrystals. These materials are non-periodic solids with long-range order, discovered in 1984. Their growth mechanism and stability is still a controversial topic. While they are not usually considered as porous materials, the phase images show all the investigated high quality icosahedral Al-Pd-Mn quasicrystals to contain internal holes, with dodecahedral shape and orientation reflecting the icosahedral point symmetry of the host [MAN98]. Their sizes exhibit a discrete distribution. The jump from one size to

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the next is consistent with the factor t3, where T is the golden mean, a basic ingredient in all theoretical approaches of quasicrystals. The 3D reconstruction resulting from phase microtomography shows that the average distance between neighbouring holes is, again, about T3 times the hole size. The observed features are in fair agreement with a geometrical approach which describes the quasicrystalline structure in terms of a hierarchical self-similar packing of overlapping atomic clusters, such that an inflation scale factor x3 preserves long range order but generates a hierarchy of holes [JAN 99]. This geometrical approach appears to be controversial among the concerned scientific community, and further experiments are being performed to better understand the physical implications of these observations. Phase imaging based on the visualisation of the edges is not a quantitative technique, and its spatial resolution is limited by the occurrence of the fringes used to visualize the borders. A more quantitative approach of phase imaging and tomography was recently developed. It is based on the combination of several images recorded at different distances. An algorithm, initially developed for electron microscopy by the Antwerp group, was successfully adapted to the X-ray case, and allows the "holographic" reconstruction of the local phase, well beyond the images of edges [CLO99a]. Once the phase maps are obtained through holographic reconstruction, there is no conceptual difficulty in bringing together many maps corresponding to different orientations of the sample, and in producing the tomographic, three-dimensional, reconstruction. This combined procedure was applied to a polystyrene foam: for each of 700 angular positions of the sample, the phase map was retrieved using images recorded at four distances. The highest accessible spatial frequency is determined by the resolution of the detector (~2 um our case). Quantitative phase mapping and tomography ("holotomography") are now operational, and provide a new approach to the characterization of materials on the micrometer scale [CL099b]. 3.9. Other new approaches in microtomography The high intensity beams available at third generation synchrotron radiation facilities are very favorable for X-ray scanning microscopies. In this approach, one or more selected features of the sample's scattering diagram (diffraction, fluorescence...) is plotted as a function of the position of the specimen as the latter is scanned across the beam, to produce a mapping sensitive to crystalline phases, impurities... Resolution in the micron range is obtained by focusing the beam. Various focusing elements are currently used (curved mirrors or crystals, Fresnel zone plates, refractive lenses, tapered capillaries...) to produce very small beams. This kind of experiments is being extended to three-dimensional diffraction tomography [KLE98] as well as to fluorescence microtomography, in which trace

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elements can be mapped inside a sample [SIM99]. These techniques, and their applications, will surely exhibit a rapid expansion in the next years, in connection with the enhanced availability of the modern synchrotron sources.

3.10. Conclusion X-ray microtomography is an invaluable tool to obtain 3D data on a large variety of materials. The use of a third generation synchrotron radiation source, such as ESRF, opens up new possibilities. The most important features, as far as absorption microtomography is concerned, are the very broad choice available of photon energy (typically between 6 and 120 keV), the quantitative evaluation of the experimental data made possible by the beam being monochromatic and parallel, and the improved spatial resolution. Obviously, some of the applications require an even better spatial resolution. The detection of X-rays is performed through visible light scintillators. The system is then diffraction limited. A possible way to overcome this limitation is to use a lens to magnify the image before the scintillator. Promising attempts have been performed using asymetrically cut crystals, or parabolic refractive lenses [LEN99]. These techniques are well adapted to in-situ experiments, where the material, in an adequate sample environment, is imaged as a function of an external parameter (temperature, stress...). The very small angular size of the source provides, in an instrumentally simple way, phase images which reveal phenomena hardly visible by other means. Other approaches, using focus beams, and where diffraction or fluorescence are measured, are also being developed. 3.11. References [BRZ99] BRZOSKA J.B., et al. 3D Visualization of snow samples by microtomography at low temperature. ESRF Newsletter 32, 22-23 (April 99) [BUF97] BUFFIERE J.Y., et al.. Damage assessment in an Al/SiC composite during monotonic tensile tests using synchrotron X-ray microtomography Mat. Science and Engineering A234-236, 633-635 (1997) [CLO97] CLOETENS P., et al. Observation of microstructure and damage in materials by phase radiography and tomography J. Appl. Phys. 81, 5878-5886 (1997) [CLO99a] CLOETENS P., et al. Hard X-ray phase imaging using simple propagation of a coherent synchrotron radiation beam J. Phys. D: Appl. Phys.32, A145-A151 (1999) [CLO99b] CLOETENS P., et al. M Holotomography: quantitative phase tomography with micrometer resolution using hard synchrotron radiation X-rays Appl. Phys. Lett., 75, 2912-2914 (1999) [COL98] COLES M.E., et al. Pore level imaging of fluid transport using synchrotron X-ray microtomography J. Petroleum Science and Engineering 19, 55-63 (1998)

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[ESP98] ESPESO J.I., et al Conserving the coherence and uniformity of third generation synchrotron radiation beams: the case of ID 19, a 'long'beamline at the ESRF Journal of Synchrotron Radiation, 5, 1243-1249 (1998) [FLA87] FLANNERY B.P., et al. Three dimensional X-ray Microtomography Science 237, 1439-1444(1987) [JAN99] JANOT C., et al.. Self-similar porosity in quasicrystals Mat. Res. Soc. Symp. Proc. 553, 55-66 (1999). [KAK88] Kak A.C., SLANEY M. Principles of computerized tomographic imaging. IEEE Press, New York (1988) [KLE98] KLEUKER U. et al. Feasibility study of X-ray diffraction computed tomography for medical imaging Phys. Medicine and Biology 43, 2911-2923 (1998) [LAB96] LABICHE J.C., et al. FRELON Camera: Fast REadout LOw Noise. ESRF Newsletter, 25, 41-42 (1996) [LEN99] LENGELER B., et al. Imaging by parabolic refractive lenses in the hard Xray range Synchrotron. Rad., 6, 1153-1167 (1999) [MAN98] MANCINI L., et al. Investigation of defects in icosahedral quasicrystals by combined synchrotron X-ray topography and phase radiography Philosophical Magazine A, 78, 1175-1194 (1998) [PEI97] PEIX G., et al. Hard X-ray phase tomographic investigation of materials using Fresnel diffraction of synchrotron radiation SPIE 3149, p.149-157 (1997). [PEY96] PEYRIN F., et al. Introduction to 2D and 3D tomographic methods based on straight line propagation: X-ray, emission and ultrasonic tomography (text in French) Traitementdu Signal 13, 381-411 (1996). [PEY97] PEYRIN F., et al. Quantification of the trabecular structure from 3D synchrotron radiation microtomography: comparison to histology, Osteoporosis Int., 7, 268 (1997) [PEY98] PEYRIN F., et al. Micro-CT examinations of trabecular bone samples at different resolutions: 14, 7 and 2 micron level, Technology and Health Care, IOS Press, 1998, vol 6(5-6), p. 391-401. [PEY99] PEYRIN F., et al. Local tomography in 3D SR CMT based on a nonseparable wavelet approach SPIE 44th Ann. Meeting, Developments In XRay Tomography II, Denver USA, (1999) [QUE98] QUENARD D., et al. Micro structure and transport properties of porous building materials: 3D modeling from 2D-SEM images and X-ray microtomography Workshop on modeling of deterioration in composite building components due to heat and mass transfer, 22-23/1/98, Tsukuba, Japan (1998) [RUE96] P. RUEGSEGGER, et al. A microtomographic system for the non destructive evaluation of bone architecture, Calcif. Tiss. Int., vol 58, p 24-29, 1996. [SAL98] SALOME-PATEYRON M. Acquisition et quantification d'images du reseau trabeculaire osseux en microtomographie tridimensionnelle utilisant le rayonnement synchrotron, PhD thesis (Genie Biologique et Medical) INSA Lyon, 1998.

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[SAL99] SALOME-PATEYRON M. et al., Description of a synchrotron radiation microtomography device for 3D trabecular bone imaging, Med. Phys., 26, n° 10, p. 2194-2204(1999) [RAO93] RAOUX D Introduction to synchrotron radiation and to the physics of storage rings Neutron and Synchrotron Radiation for Condensed Matter studies, HERCULES, edited by Baruchel J. et al., Ed. de Physique and Springer-Verlag (1993) [SAS98J SASOV A. et al. Desk-top microtomography: gateway to the 3D world European Microscopy and Analysis, 17-19 ( March 1998) [SIM99] SIMIONOVICI A. et al. X-ray fluorescence microtomography: experiment and reconstruction SPIE 3772, 304-310 (1999) [WINOO] WINDLE A. et al. In-situ microtomographic investigation of the compression of a polyurethane foam in press

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Chapitre 4

Introduction to reconstruction methods

This paper introduces reconstruction methods for X-ray transmission computed tomography (CT) applied to materials science. Basic principles of analytic reconstruction methods such as filtered back-projection (FBP) are recalled and illustrated in 2D. Then we introduce algebraic reconstruction techniques (ART) that can be implemented when conventional, complete data sampling cannot be achieved.

4.1. Introduction The purpose of a computed tomography (CT) system is to build a 2D or 3D representation of the inner structures of an object, from a set of projection measurements, acquired from a number of points of view. In the case of transmission CT with X-rays or gamma-rays, the parameter that is reconstructed in order to represent these structures is the linear attenuation coefficient u of the object. It is assumed to be proportional, in most non-destructive testing (NDT) applications, to the mass density (p in g/cm3) regardless of the chemical nature of the object. Calibration procedures are usually necessary to transform the raw measurements into quantitative projection data and to correct for X-ray polychromaticity effects, but they are beyond the scope of this paper. In this paper we intend to illustrate the basic concepts of reconstruction methods in parallel-beam geometry, through a simulated example, rather than going into the details of mathematical developments. References for this paper have been found in [AMA85], [LEW 83], [CEN 83] and [HER80]. The reader can find a description of recent developments including cone-beam or spiral CT in, for instance, [HIR97], [GRA99] or [NAT99].

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4.2. Description of projection measurements Let us assume that the acquisition system is able to provide a set of projections of /j. along straight lines, or source-to-detector "rays", in a so-called "parallelbeam" geometry. The object is supposed to be fixed, while a measurement system, composed of one source and one collimated detector, translates and rotates around the object, as shown in figure 4.1.

Figure 4.1. Acquisition geometry for parallel-beam tomography

In most NOT systems the source-detector system is fixed and the object moves, but in order to explain reconstruction methods it is easier to place oneself in a reference system attached to the object. Let (x,y) be the coordinates of a point M in the reference system attached to the object. The origin O of this reference system is at the axis-of-rotation. A source-to-detector ray is defined by parameters (r, 0) where 9 is the angle of the x axis with the perpendicular to the ray, and r is the algebraic distance from the axis-of-rotation to the ray. If $) is the number of photons delivered by the source, and is the number of photons after attenuation by the object, the Beer-Lambert law states that: [1]

If one assumes that both

and

are measured by the acquisition system, then

the projection of u along a ray (r, 0) is expressed as follows : [2]

Note that (r, o) do not correspond exactly to polar coordinates since generally :

Introduction to reconstruction methods p(0,

p(o,

for

63

[3]

and r may be negative. We refer to as "a projection" the set of measurements acquired for the same rotational angle. The set of projections over 180° (or 360°) is the Radon transform of function u. In parallel-beam geometry, 180° are sufficient to represent the Radon transform since :

p(r,o,) = p(-r,0 + x)

[4]

The linear and angular sampling must be fine enough so that the projection data can be mathematically considered as a satisfactory sampled version of the continuous Radon transform, and function u, must be null outside the circle defined by the acquisition system when it rotates. Obviously, real measurements are not acquired with infinitely thin rays, but each ray has a width that is determined by the focal spot size of the X-ray source, the detector width, the source-to-object distance and the source-to-detector distance. Accordingly, the translation step must be small enough to be consistent with this ray width and with the size of the structures in the object. The number of angular projections must be between and 1 4 times the number of pixels per projection. The simulated phantom is made of a disk with /j = 1 including two elliptic inserts with // = 2 . The simulation code is based on an analytic computation of the lengths crossed by the rays in each component of the object. It does not include any beam-hardening effect or other physical considerations such as scattering or energydependent detector efficiency. Such a realistic simulation code is available in our laboratory [GLI98], but for the purpose of reconstruction, possible distortions due to physical effects are supposed to be corrected for by adequate pre-processing of the projections. We simulated 180 projections of 128 rays per projection, equally spaced over 180°. The reconstructed image is made of 128 x 128 pixels. Figure 4.2 shows the reconstructed image, which can be considered as the reference representation of the object.

Figure 4.2. Reconstructed image of the object

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If the projections are displayed in grey levels, one below the other, then the 2D representation which is obtained (figure 4.3) is called a "sinogram" because a point of the object describes a sinusoid in this representation, in effect, from figure 4.1 it can be deduced that:

Figure 4.3. Right: sinogram or set of projections over 180° - Left: three projections

It can be seen on the sinogram that the long objects inside the main cylinder are better detected when they are seen through their longest direction. Let us assume now that projection data are available, and that the image whose Radon transform corresponds to these measurement data is unkown.

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4.3. Backprojection The first intuitive operation which can be implemented is backprojection. It consists in assigning to each point of the object the average value of all the projections at the corresponding location, as illustrated in figure 4.4. The result of backprojection of the sinogram of figure 4.3 into a 128 x 128 pixels matrix is shown in figure 4.5.

Figure 4.4. Backprojection in a point of the object

Figure 4.5. Backprojection of the sinogram The backprojected image, when compared with the "perfect" object, is highly blurred. As a result of the "projection-then-backprojection" process, each pixel contains information about what the object really contains at the pixel location, but this information is added to a blurred version of the rest of the object. An exact mathematical correction of the "projection-then-backprojection" smoothing effect can be done by an appropriate pre-filtering of the projections, as in the Filtered Backprojection (FBP) algorithm. This can be demonstrated based on Fourier considerations.

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4.4. Projection-slice theorem The "projection-slice theorem" is the basis of many reconstruction methods. It states that the 1D Fourier transform p(R,0) of projection p(r,0) in the 0 direction is equal to the cross-section, in the same direction 9, of the 2D Fourier transform fr( vi, v2) of the original function //(*, y):

Figure 4.6. Magnitude of the Fourier transform of projection for 0 = 0. Left: linear representation. Right: log representation

Figure 4.7. Magnitude of the Fourier transform of the object. Left: linear representation. Right: log representation

This is illustrated by figure 4.6 and 4.7: figure 4.6 is a horizontal profile of figure 4.7. Note that for this object the 2D Fourier transform contains 2 principal components (one vertical and one horizontal) which correspond to the direction the two ellipses inside the object. The representations of figure 4.6 and 4.7 are discrete Fourier transforms; they are a valid representation of the continuous Fourier

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transform only if the Nyquist sampling conditions are satisfied. Let us recall that, if Ar is the sampling step on the projections, and Nr the number of pixels per projection, then the discrete Fourier transform of the projection is made of Nr points and the maximum frequency which can be represented is :

4.5. Fourier reconstruction methods From the projection-slice theorem can be built a reconstruction method: if equally sampled projections are acquired over 180°, then the set of their ID Fourier transforms constitute a representation, on a polar grid, of the 2D Fourier transform of u(x, y). If a polar-to-cartesian resampling is done, followed by a reverse 2D Fourier transform, then the reconstruction is done, we will name this method the "Fourier method". Another method, referred to as the FBP method, can be deduced from the Fourier method. The reverse 2D transform of /}( Vj , v2 ) is written : dvldv2

[8]

If ( R ,6} are the polar coordinates in the ( v1 , v2 ) Fourier space it comes :

RdR dO

[9]

The change of variables may seem a little tricky since R varies over ]-oo,oo[ in the ID Fourier transform of the projection p(R,0} , while R > 0 in the expression of u(v 1 ,V2)in polar coordinates. Taking into account symmetry properties, eventually, it comes :

which is the formula of the FBP reconstruction method. The first step of this methods is :

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\R\dR

[12]

This is a filtering operation applied to the projection p(r, 6}, the filter being represented in the frequency domain by /(/?) = R\ . This filter is noted HD because it is the Hilbert transform of the first derivative. It is also known as the "ramp filter".

Figure 4.8. Filtered projection for 0 = 0.

Figure 4.9. Backprojection of filtered projections for 1, 2, 4 and 8 directions

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This filtering operation has the effect of creating negative components on the filtered projection (see figure 4.8), which will compensate for the contribution of other projections in the backprojection step (see figure 4.9). The second step of the FBP reconstruction is :

which is the backprojection of the filtered projections. This operation is illustrated in figure 4.9. It is important to understand that the averaging of all the projections in the image pixels as explained in figure 4.3 is equivalent with the successive spreading of the projections over the image as in figure 4.9.

4.6. Filtering in Fourier methods In the presence of noise, the "ramp" filter tends to multiplied by an apodisation high frequencies to decrease the filter becomes :

is it even more important to filter the projections that increase the high frequencies. Usually the filter is function, such as a Manning window, which forces the smoothly down to 0. If Rc is the cut-off frequency,

Figure 4.10. Reconstruction filters: ramp, ramp*Hanning (e=2), ramp*Hanning (e=6)

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Apodisation tends to reduce noise but also to blur the edges of the structures. There is a trade-off to find between noise and spatial resolution as shown in figure 4.11 (a gaussian random noise has been added to the projection data before reconstruction)

Figure 4.11. FBP reconstruction of noisy data (all three images are displayed with the same gray scale)

4.7. ART-type methods The methods presented above are based on an analytical expression of the inversion of the Radon transform. Discretization is introduced only at the time of implementation. These methods assume that the measurement data constitute a discrete representation of the Radon transform, which means that they must be complete and equally spaced. When it is not the case, the problem may be written directly in the form of a discrete linear system :

[15] where p is the vector made of the measurement data, x is a vector made of all the pixels of the image, M is the projection matrix. This can be written :

where i is an index of source-to-detector ray, j is an index of image pixel and mij is a coefficient representing the contribution of pixel j to measurement i. There are several ways of modelizing the projection matrix; usually mij coefficients are

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computed as the length crossed by ray j to measurement through pixel j. Several other models exist in the literature, for instance the Xj may be considered as basis functions with bell-shaped profiles on a circular, overlapping support [LEW 92] in order to regularize and accelerate the reconstruction. In this paper we restrict ourselves to conventional square pixels. Since data are perturbed by noise, the inversion of the system is done in the least square sense. The quadratic error to be minimized is :

The gradient of the quadratic error is :

The minimum of the quadratic error is found for x such that :

Even though many m^ coefficients are null, the size of the system is generally huge, and the system is not inverted directly, but using a method iterative by blocks. A block is a set of rays (or a set of rows in the matrix). Very often, a block will be equivalent with a "projection", that is to say all the rays for the same rotational angle (then the method is called the SART technique for "simultaneous algebraic reconstruction technique"). At iteration q, one block of rays is taken into account, and the current estimate x^' of x is refined in order to minimize the quadratic error for the rays of this block :

The difference between the projection

M block . x (q) of the current image

estimate and the actual measurement Phiock for this block is backprojected in order to update the image estimate. This can be written :

where

Pi

is the actual measurement across ray i,

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is a relaxation factor, which prevents from going too fast towards the solution corresponding to the current block, regardless of the other blocks, mi is the ilh row of matrix M, it corresponds to the projection along ray i. This algorithm is made of two basis modules which are projection and backprojection. Projection : pi (q) = <mi,x ( q ) > = mij xj(q} [22] j is the computed projection of the estimate X along ray / (at the iteration q)

Backprojection :

[23]

is the backprojection into the image of Spi = pi - pi which is the error on measurement i (at iteration q). The value which is backprojected into pixel j for ray sum / is :

[24] k on ray i

The weighting factor

gives the redistribution of the projection kon rayi

error to the pixels crossed by ray i. If mij is homogeneous to a length and is expressed in cm, then this weighting operation is equivalent with dividing the error by the length crossed by the ray in the object. If the projection data are equivalent to an attenuation ( u • / ) then the reconstructed value will be u ( x , y } in attenuation per cm. This is the "basic" ART method. In order to regularize the problem in the presence of incomplete or noisy data, a priori constraints are often taken into account. Among them are found: positivity constraint (ju cannot physically be negative), or support constraints (function u is known to be null outside a given area). Filtering of the image or of the projection data may also be introduced. If precautions are taken in order to ensure data consistency, ART methods can even be used for "local" tomography :

Introduction to reconstruction methods

Figure 4.12. Local tomography is possible with SART.

We performed a simple backprojection and a SART reconstruction of the phantom using only 3 projections at -30° (150°), 0° and 30°.

Figure 4.13. Reconstruction of 3 projections Left: Backprojection, Right: SART reconstruction (30 iterations and A. = 0.1)

Figure 4.14. Horizontal profiles across the images reconstructed from 3 projections

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The vertical ellipse, which is in the direction of the projections used for the reconstruction, can be seen on projection 9 = 0 (see figure 4.3), on the SART reconstructed image (profile 1), and hardly on the backprojected image (profile 1). The backprojected image is not quantitative, it has been divided arbitrarily by 50 in order to be comparable to the other profiles. The SART reconstruction provides a value of // which is at least of the good order of magnitude. The backprojected image, and in a better way the SART reconstruction, also provide an information about of the depth of this ellipse within the object. The quality of this information is very dependent of the maximum angle between the projections (here, 60°). The horizontal ellipse cannot be located nor quantified since measurement data along its main axis have not been used for this reconstruction. This example is of course very simple, but it shows the potentialities of reconstruction from few view angles, for instance for the detection of defects in homogeneous media. 4.8. Conclusion This paper has introduced basic FBP and ART methods in parallel beam in 2D for the reconstruction of an image from its projections. Adaptations of these algorithms are necessary in most applications in order to use ID or 2D detectors. Simulated results for conventional CT and for very few (3) projections have been shown. 4.9. References [AMA 85] AMANS J.L., CAMPAGNOLO R.E., GARDERET P., Imagerie medicale : methodes de reconstruction et techniques instrumentales, Note technique CEA/LETI/MCTE N°1511 du 10 dec 85. [CEN 83] CENSOR Y., "Finite Series-Expansion Reconstruction Methods", Proceedings of the IEEE Vol 71 N°3 March 1983 p 409-419 [GLI 98] GLIERE A., "Sindbad: From CAD model to synthetic radiographs", Rev. Of Progress in Quantitative Nondestructive Evaluation, Vol. 17A, 1998, pp 387394. [GRA 99] GRANGEAT P., "Fully three-Dimensional Image Reconstruction in Radiology and Nuclear Medicine", to be published in Encyclopaedia of Computer Science and Technology. [HER 80] HERMAN G.T., Image reconstruction from projections: the fundamentals of computerized tomography, Academic Press, New York, NY 1980. [HIR 97] HIRIYANNAIAH H.P., "X-ray Computed Tomography for Medical Imaging", IEEE signal Processing magazine, p 42-59, march 1997.

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[LEW 83] LEWITT R.M., "Reconstruction algorithms : transform methods", Proceedings of the lEEEVol 71 N°3 March 1983 p 390-408 [LEW 92] LEWITT R.M., "Alternatives to voxels for image representation in iterative reconstruction algorithms", Phys. Med. Biol., 1992, N°3, 705-716 [NAT 99] NATTERER F., Numerical methods in tomography, Acta-Numerica. vol.8; 1999;p.l07-41 , 1999

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Chapitre 5

Study of materials in the semi-solid state

A synchrotron X-ray source (ESRF, FRANCE), has been used to investigate the microstructure of semi-solid materials. Two kinds of materials have been investigated, Al-Cu alloys with a high volume fraction of solid to study hot tearing and an industrial 356 alloy (Al-Si) which is the common material used for semisolid forming. Two imaging modes have been used, the absorption mode for the AlCu material and the phase contrast mode for the Al-Si. Discussion is presented on the interest of such a technique to characterise semi-solid microstructures.

5.1. Introduction The semi-solid state is a specific state in which the solid and the liquid phases coexist. This state is obtained for every alloy with a solidification interval when it is solidified from the liquid state (such as in the continuous cooling of metals), or when it is reheated from the solid state (such as in semi-solid forming). In the first case, during the solidification of the material, cracks can be generated in some alloys (Al-Cu alloys in particular) owing to the deformation of the solid phase [FRE 79] which leads to hot tearing. This phenomenon is not fully understood from a theroretical point of view. The second case (reheating from the solid phase) is related to the forming in the semi-solid state which is a competitive forming process compared with liquid injection [Gffl 96]. This process involves the forming when the material is reheated between the solidus and the liquidus with a solid fraction close to 0.5. For the moment the alloys used in this technique are aluminum-silicon alloys like 356 and 357 which are hypoeutectic alloys. The main advantage of this process is to provide laminar filling of the mould during injection which avoids any

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porosity formation and therefore allows to produce thin parts which can be heat treated in addition to increase their mechanical properties. Figure 5.la presents the microstructure of an Al-Cu alloy quenched from the semi-solid state (T=570°C). At this temperature the volume fraction of the solid phase was 0.8. The solid phase is in grey and the black phase is the eutectic mixture which was liquid when the material was in the semi-solid state. In the following the eutectic will often be designated by the liquid. In this material, the solid phase can be described as globules surrounded by the liquid. Figure 5.1b presents the microstructure of an Al-Si alloy also quenched from the semi-solid state. The solid phase is also surrounded by the fine eutectic mixture (which was liquid in the semisolid state). As previously, the solid phase can be described as globules embedded in the eutectic liquid. Some liquid droplets are also observed in the solid phase.

Figure 5.1. Typical microstructure of (a) an Al-Cu alloy (b) an Al-Si alloy

However, 2D observations of such materials are not sufficient to describe the microstructure. Some attempts have been made to get 3D images of the solid in semi-solid materials. [ITO 92] studied an Al-Si alloy with a very low solid fraction (0.2). The employed technique was to take micrographs of the sample, to polish it, to take again micrographs and this procedure was repeated every 40 microns. Then the volume was reconstructed. These images of the solid in a semi-solid alloy obtained for the first time in 3D, even though at a very low solid fraction, indicate that clusters of solid can not be properly characterised using 2D sections only. More recently [VOL 97] have used the same technique on a small volume of a model SnPb alloy with a very high solid fraction (0.8). They took micrographs every 21 microns and reconstructed the volume. Their observations again clearly demonstrate the limitation of 2D analysis for such a material. They found a complete connectivity of the solid particles within the volume. The size of the solid particles (around 0.35 mm) is large in the volume investigated (1.3 x 1.7 x 1.7 mm) and the authors agree on the fact that the precision (21 microns) may create artificial

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connections between particles. However, it must be noticed that this paper presents the very first 3D images of a reasonable volume. Finally the last study published on 3D images of semi-solid materials is by [NlR 98]. This paper shows 3D images of some solid particles obtained from an 2xxx alloy. The technique employed is the same as that of the other papers except that the precision is higher, around 10 microns for the section spacing. However, there is no attempt in this paper to study the connectivity of the solid phase and only few solid particles have been investigated. Therefore, 3D characterisation of solid phase in semi-solid materials is now becoming necessary to better understand microstructural evolutions, like Ostwald ripening and coalescence processes, and deformation mechanisms. The aim of this paper is to present a new technique which allows to get 3D images of any semi-solid materials with a high resolution. This technique is high resolution X-ray tomography carried out at ESRF. This paper will first present the experimental device and then the materials under study. Finally 3D characterisations of these semi solid materials will be presented.

5.2. Experimental device and procedure The experimental device is available on line ID 19 at the ESRF and since three years several studies were carried out with the X-ray tomography device. This is at the moment the only way to get 3D images with a high level of resolution (of about one micron). A X-ray beam with a high coherency and a high energy (up to 60 keV) hits the sample and a CCD camera records the transmission image. The sample is placed on a high precision rotating table and 600-900 transmission images are taken while the table is rotating by 180°. The CCD camera has 1024 x 1024 pixels and the pixel size can range from 15 to 0.9 microns. Therefore the higher the resolution is, the smaller the volume analysed is. In the investigated materials, two scales are present in the microstructure, the globule size of the solid phase (around 100 microns) and the size of the liquid film between the globules (a few microns). In order to image enough globules in the investigated volume and obtain a good resolution to observe the liquid film, we have decided to work with the 2 microns resolution. The samples were machined as cylinders of 1.6 mm in diameter and 3 mm in height. X-ray tomography can be performed in several ways. The most classical is the absorption mode, when the CCD camera is placed just behind the sample (3 mm for example). This mode will provide images for which the contrast is obtained owing to the difference in X-ray absorption of elements in the sample. This mode has been used by [BUF 98] to study the spatial distribution of Al2Cu nodules in an Al-Cu alloy. This mode was used also in this study for the Al-Cu alloys. However, this mode does not allow to study multiphase materials when the absorption coefficients of the various phases are not different enough. The contrast is then too low, which is the case of the Al-Si alloy. It is then possible to use the phase contrast : this particular mode requires the CCD camera to be placed far from the sample (up to

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900 mm). Therefore the contrast is due to constructive interferences of X-rays diffracted at the interface between phases. This special mode has been for example used to study damage in metal matrix composites based on the Al/SiC system [BUF 99]. The phase contrast is enhanced when the distance between the sample and the camera is increased, but the resolution of the interface between the phases becomes lower. In both Al-Cu and Al-Si alloys experiments, the energy of the X-ray beam was fixed to 17.5 keV. A complete scan (800 projections) including references images every 100 projections lasts about 10-15 minutes. It is important to notice that it is not possible to do "in situ" scan on semi-solid alloys since the microstructure changes when the material is held in the semi-solid state. Consequently, all the scans were performed at ambient temperature on quenched samples from the semi-solid state. The quenching must be very fast in order to get a very fine eutectic mixture in between the solid phase, which will then appear homogeneous in contrast. This is very important since, if the eutectic mixture is not fine enough, the high resolution will allow to distinguish the two phases of the eutectic and therefore will cause trouble to identify the solid globules.

5.3. Results on Al-Si alloys In this section we will first present the best experimental conditions for this material, then we will present several 2D sections of a 3D reconstructed volume and finally we will study the influence of the remelting time on the microstructure of an Al-Si alloy. The 356 material has been provided by Pechiney and it is mainly Al7wt% Si-0.5wt%Mg. It has been electromagnetically stirred during solidification in order to get a non dendritic structure. This material was partially remelted in the semi-solid state in an induction furnace at 583°C (7°C above the eutectic temperature) and maintained at this temperature during 5 minutes. This time is usually required before the forming of this material. Several X-ray tomography experiments have been carried out for various distances between the sample and the CCD camera and figure 5.2 presents 2D sections of the reconstructed volume of the Al-Si alloy for 4 distances. As expected in absorption mode (distance=3mm), there is no contrast between phases, whereas a good contrast is observed when the distance is 200 mm or larger : indeed it is possible to distinguish all the phases (liquid, solid and droplets of liquid). In order to keep good resolution of interfaces between solid and liquid and to have enough contrast, the best distance seems to be in between 200 and 600 mm. We have used 500 mm for the subsequent experiments.

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Figure 5.2. Influence of the sample / camera distance on the contrast in Al-Si alloys

Figure 5.3 presents 2D sections every 4 microns. The microstructure is very different between figure 5.3a and 5.3d although there is only a 12 microns distance between the two sections. These changes concern the solid and the liquid phases. Most of the droplets are effectively entrapped in the liquid when considering several 2D sections. However, some droplets of liquid which seem to be entrapped in the solid phase are in fact connected to the interdendritic liquid (square box in figure 5.3). Until now, these droplets were assumed to be isolated in the solid phase owing to morphological and metallurgical assumptions but it was really difficult to distinguish between entrapped and connected liquid [SAL 96]. Concerning the solid phase, within 12 microns, new solid globules can appear as illustrated in the dotted

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box in figure 5.3 and connection between globules can appear also as illustrated by the arrow. Therefore a resolution of 2 microns is at least necessary to get complete information on the connectivity of the solid phase of industrial semi-solid materials.

Figure 5.3. Serial section on an Al-Si These results therefore demonstrate that it is very important to make a real 3D characterisation of such materials. The problem with the phase contrast technique is that segmentation of the volume is not easy to perform. Apparently the edges of each phase seem to be well defined (see figure 5.2) but this is not always the case. The origin of this problem can arise from the experimental device but in our case it is mainly due to the sample : as explained before quenching is sometimes not fast enough so that the eutectic does not appear as homogeneous but as a mixture of two phases : silicon and aluminium, which can not be distinguished from solid globules. Therefore, it becomes delicate to get information on the connectivity of the solid phase. As an automatic segmentation does not permit for the moment to know

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precisely its influence on the connectivity, we decided to perform manual segmentation on 2D sections every 2 microns and reconstruct part of the volume, this procedure is long since it requires to draw by hand the globules and the liquid phase using painting software. This technique allows also 3D information about entrapped liquid and connectivity and shape of the solid, since segmentation is easy. If information on the connectivity of the solid is only required, there is a quicker way : the principle is to draw a line between connected globules in 2D in order to get the skeleton. Then this 2D skeleton is placed on the next section and modification of the connection can be added or suppressed. This faster technique, however, must only be used to count solid phase globules in the volume and not to get information about the morphology. Figure 5.4 presents a 3D reconstruction of the material presented in figure 5.3 . Figure 5.4a illustrates a fully connected solid phase, since all the globules have been found to be connected by 3D analysis of the volume. This observation is very interesting since the 2D photographs of figure 5.3 do not suggest at all this high level of connectivity of the solid phase. This feature indicates that partial remelting of a semi-solid Al-Si alloy during 5 minutes leads to a highly connected solid phase. This explains why the material can be handle in this state and does not collapse even if it contains 50% of liquid. Figure 5.4b presents the entrapped liquid, which is not connected to the interdendritic liquid. As suggested by the 2D sections presented in figure 5.3, these droplets are mainly spherical.

Figure 5.4. a) solid phase, b) entrapped liquid

The last experiment which was conducted was designed to study the evolution of the microstructure during partial remelting with the same sample. As explained earlier, it is not possible to perform in situ scan of these materials since the evolution of the microstructure is very fast. Therefore, we decided to do what we call

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interrupted in situ scans. A sample is scanned, then partially remelted during five minutes, quenched and scanned again, remelted again for some time and so on. The important point here is to put marks on the sample to be able to image the same part of the sample after the remelting experiments. The sample was studied in the as received conditions, after 5 minutes remelting at 587°C and after 10 minutes. Figure 5.5 presents 2D sections of reconstructed volumes of the same region in the specimen. It is therefore possible to see the evolution of the microstructure within 10 minutes remelting at 587°C. It indicates that the evolution of the microstructure during the time required for a complete scan is too important to perform real in situ measurements. The microstructure evolves very rapidly during the 5 first minutes so that it is difficult to follow the same region. Indeed coalescence of dendrite arms occurs as generally observed which leads to the entrapped liquid [SEC 84]. Between 5 and 10 minutes, the evolution becomes slower and it is then easy to follow some groups of globules as those underlined in black in figure 5.5.

Figure 5.5. Influence of remelting time on the microstructure of an Al-Si alloy

A 3D representation of a group of solid globules are shown in figure 5.6 for 5 and 10 minutes remelting. The morphology of the solid phase changes considerably, since a small globule seems to disappear and the curvature of the solid phase decreases.

Figure 5.6. Solid phase evolution during partial remelting

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The main mechanism of this evolution is the well known reduction of the liquid/solid interface area [VOO 84] and this figure is a real 3D representation of this mechanism.

5.4. Results on Al-Cu alloys A grain refined Al-8wt%Cu alloy was studied by X-ray microtomography in the absorption mode. Figure 5.7 shows successive sections along the z-axis. Considering the first and the last images (z = 0 and z = +10|jm) indicates that the two liquid films present in the circle are not connected. The other sections of figure 5.7 conversely show that these two liquid films are connected. This kind of information about the connectivity of the liquid phase is very important in the study of hot tearing phenomena. Indeed, if the liquid phase is totally connected, the cracks will easily propagate into the liquid [FRE 79]. Sections closer than 10 microns are thus required and it seems in this case that sectioning for every 4 microns gives a good information about the microstructure. 3D analysis of such a volume indicates that there is no entrapped liquid in this volume.

Figure 5.7. Successive sections on the Al-Cu alloy

We also performed an interrupted in situ test for the Al-Cu alloy. For this alloy, the air furnace temperature was 555°C (the eutectic temperature is 548°C) and four

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cumulated times were analyzed: 5, 15, 30 and 60 min. Figure 5.8 presents a 2D section of the same sample after 30 and 60 min of partial remelting. As already mentioned for the Al-Si material, there is an evolution of the microstructure which obeys the Ostwald ripening process [Voo 84] but the evolution is lower since the solid fraction is higher. During the partial remelting treatment, the liquid fraction decreases from the Scheil equation value (0.19) to the equilibrium value given by the phase diagram (0.1).

Figure 5.8. 2D sections at constant z during partial remelting experiments

5.5. Conclusion and perspectives The microtomography is a very powerful techniques which allows to study the microstructure of semi-solid materials. The resolution of 2 microns that is possible to use at ESRF allows to study the connectivity of the solid phase in such materials in 3D. We have shown that for the Al-Si materials the scan has to be made in the phase contrast mode with a distance of 500mm and that the resolution of 2 microns is a good trade off. In the case of Al-Cu materials the absorption contrast gives good images and it seems that owing to the large solid fraction a resolution of 4 microns might be sufficient. The first 3D analysis of the Al-Si shows that the solid phase is entirely connected and lots of entrapped liquid are present. In the case of Al-Cu no entrapped liquid has been found. We performed in both alloy interrupted in situ tests in order to follow the evolution of the microstructure of the same sample. This gives very interesting results concerning the solid phase morphology and entrapped liquid in the case of Al-Si alloy. Future efforts will be concentrated on the evolution of the microstrutructure during deformation and during partial remelting. This needs 3D analysis of the volumes and therefore a good segmentation technique, which is possible for Al-Cu but at the moment difficult for Al-Si.

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5.6. References [FRE 79] H.FREDRIKSSON and B.LEHTINEN, Solidification and Casting of Metals, 1979, p. 260-267 [GHI 96] G. GHIARMETTA , Conference Proceedings of the 4th International Conference on Semi-Solid Processing of Alloys and Composites, 19-21 June 1996 Eds D.H. Kirkwoodand P. Kapranos, p204-208 [ITO 92] Y. ITO, M.C. FLEMINGS, J. A., Cornie Nature and Properties of Semi-Solid Materials, eds J.A. Sekhar and J.A. Dantzig, 1992, p3-17. [VOL 97] T.L. VOFLSDORF, W.H. BENDER, P.W. VOOHREES, Acta Mater. vo145, n°6, 1997, p 2279-2295 [Nm. 98] B. NlROUMAND , K. XlA, Conference Proceedings of the 4th International Conference on Semi-Solid Processing of Alloys and Composites, June 1998 p637-644 [BUF 98] J.-Y. BUFFIERE, E. MAIRE, P. CLOETENS, J. BARUCHEL, R. FOUGERES,

Poceedings of ICAA 6 , Aluminium Alloys vol 1, 1998, p529-534. [BUF 99] J.-Y. BUFFIERE, E. MAIRE, P. CLOETENS, G. LORMAND, R. FOUGERES, Acta Mater vol 47 n°5, 1999, p!613-1625. [SAL 96] L. SALVO, M. SUERY, Y. DE CHARENTENAY, W. LOUE, Conference Proceedings of the 4th International Conference on Semi-Solid Processing of Alloys and Composites, 19-21june 1996 Eds D.H. Kirkwoodand P. Kapranos, plO-16 [Voo 84] P.W. VOORHEES, M.E. GLICKSMAN, Met Trans A, vol 15A, 1984, pi0811089 [SEC 96] J.-F. SECONDE, M. SUERY , Journal of Materials Science, vol 19, 1984, p3995-4006

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Chapitre 6

Characterisation of void and reinforcement distributions by edge contrast

We have studied the inter-relation between reinforcing particle distribution and void distribution prior to ductile failure in a model, ZrC>2 reinforced Al matrix, metal matrix composite. Dual energy X-ray microtomography measurements above and below a strong Zr absorption edge have been use to make reconstructions of ZrC»2 and void distributions in failed metal matrix composite tensile test specimens. The distribution of ZrO2 particles in the composite prior to testing is shown to be spatially uncorrelated. There is a correlation between the local density of voids nucleated during tensile straining and the density of ZrO2 reinforcement. However, there is no evidence for preferential nucleation or growth of voids formed prior to fracture in regions of higher particle density.

6.1. Introduction In this study we have investigated the evolution of void volume fraction during the plastic tensile straining of a particle reinforced metal matrix composite (MMC) using X-ray microtomography (XMT). Previous studies of void evolution in MMCs using conventional absorption contrast XMT have been unable to distinguish between the contribution of void and reinforcement populations [MUM 93a, 95]. Recent work using coherent synchrotron X-radiation has been able to use phase contrast to image individual reinforcement particles in model MMCs [BUF99]. However, this was only achieved with a model MMC containing very large SiC reinforcement of diameter > 200 am.

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Here we will describe a different approach used to characterise reinforcement distributions tomographically. This is dual energy XMT, where the X-ray absorption is measured at two X-ray energies either side of a critical X-ray absorption edge of an element in the reinforcement material. This limits our selection of reinforcements to those which possess a suitable absorption edge. Zr has an absorption edge at 18.2 keV and so model MMCs of ZrC>2 particles in an Al alloy matrix were used in these studies. Combining the information from the two signals measured above and below the absorption edge allows us to distinguish between the contribution of reinforcement and voids to the X-ray absorption density. This enables the tomographic reconstruction of separate images of the void distribution and reinforcement distribution, allowing the relation between void evolution and local reinforcement density to be studied.

6.2. Dual energy X-ray microtomography XMT measurements were carried out on a "first generation" microtomography apparatus. The details of this apparatus have been described elsewhere [ELL 90, 94] and similar apparatus was used in our earlier studies of MMC fracture [MUM 93a, 93b, 95]. This apparatus uses a collimated, pencil beam of X-rays defined by a lOum aperture. X-rays are detected by a single, energy dispersive, high purity Ge detector. This is connected to a spectroscopy amplifier and then a calibrated 2048 channel multi-channel analyser (MCA). A laboratory microfocus X-ray source is used with a Mo target run at typically 40 kV with a current of 2 mA. The resulting beam is polychromatic and of 15um diameter at the specimen; this defines the reconstruction resolution (voxel size). The specimen is translated through the beam by a stage which positions to an accuracy of better than 0.5 (am. To provide the data to reconstruct a single 2-dimensional slice, the specimen is stepped to provide a line projection of 128 measurements. It is then rotated through equal angular intervals of 1.434° to obtain 251 line projections of X-ray absorption over 360°. A standard filtered back projection algorithm is used to reconstruct the data from each slice onto a 256 x 256 pixel array. Three-dimensional reconstructions are obtained by collecting data from a series of slices. In dual energy XMT, two independent sets of transmitted X-ray intensity are made at two different energies along identical beam paths through the specimen. In order to achieve this, the MCA is used to distribute the X-rays detected by the Ge detector into different energy bins of 0.025 keV width. In order to get independent data sets for this technqiue, the ratio of matrix to reinforcement X-ray absorption coefficient should be as different as possible at the two X-ray energies chosen. This can be most easily achieved if the reinforcement has an X-ray absorption edge in the energy range available from the X-ray source. In this case a suitable model reinforcement is ZrO2 with the Zr absorption edge at 18.2 keV.

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In conventional XMT, the reconstruction provides a spatial distribution of linear X-ray absorption coefficient //. In each reconstructed voxel the absorption coefficient is made up of three components: u voxel = where V is the volume fraction of each component matrix, particles and void, defined by subscripts m, p and v respectively. The general method of reconstructing elemental quantified XMT data using multi-energy X-ray absorptiometry has been described elsewhere [KOZ 99]. In dual energy XMT, the absorption coefficients of each phase are not separately reconstructed; instead, the individual contributions of the matrix and reinforcement to the total absorption are decoupled. This requires the use of the mass absorption coefficient, which is the linear X-ray absorption coefficient of fully dense material divided by its theoretical density. We use the notation mc,? to represent the mass absorption coefficient of component c at energy E2. At energy E\ the absorption measured is thus:

where p is the density of the component in the voxel, r is the specimen thickness and I and I, are the intensities of the incident and transmitted beam respectively. Note that because a void has a mass absorption coefficient taken to be zero, it is not included in the above calculation. Equation 2 can be rewritten for the second X-ray energy E2 and combined to give a solution for the local particle component density:

This calculation is carried out on the X-ray intensity values collected at each specimen position. This results in "intensity of density" values on which reconstruction is carried out directly yielding voxel particle density: Ip=exp-[Joppdx|

[4]

The above equations 2-4 can also be solved for pm. Once the values of pm and pp are known for an individual voxel, the void content of that voxel can also be calculated by

[5] Pp Pm where pp' and pm' are the theoretical densities of the particles and reinforcement.

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Dual energy XMT reconstruction was carried out on data collected at X-ray energies of 17.4 keV and 19.6 keV. These energies correspond to the MoKa and MoKp characteristic X-ray energies and were chosen because the characteristic count rates are high. The strong X-ray absorption of ZrO2 restricts the maximum diameter and reinforcement volume fraction of the specimens that can be examined. Thus, a maximum reinforcement volume fraction of 7.5% was used which allows a maximum specimen thickness of about 1mm. The typical data collection time on these specimens was 14 hours per slice; a minimum of 20 slices with 50 urn separation between each slice were reconstructed for each specimen examined.

6.3. Experimental materials All materials were formed using a standard powder metallurgical fabrication route of vacuum hot pressing followed by extrusion. The matrix Al alloys chosen for this study were commercially pure Al (1100) and a Mg and Si containing precipitation hardening alloy (6061). Argon gas atomised powders were supplied by Alpoco. The Al 1100 and Al 6061 powders had quoted mean particle diameters of 7 fim and 45 u,m respectively. The morphology of both powders was spherical. The reinforcing particles used were cubic Mg stabilised ZrO2 (Johnson Matthey) of sieved particle range 28 - 45 urn.

Figure 6.1. Variance of local area fraction of ZrO2 particles in a 7.5% ZrO2 reinforced Al 6061 matrix, plotted against the size of the area used for measurement (logarithmic scale for both axes)

Quantitative measurements of the microstructure were made using a Kontron IBAS V2.0 image analysis system connected to an optical microscope. The spatial

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distribution of the reinforcements was determined by measuring the local volume fraction of reinforcements in sub-areas within 4 fields. An important question is whether this distribution in local area volume fraction reflects a truly random distribution of particles or whether some clustering or ordering occurs. If a distribution of points is truly random in space, there should be no influence of the scale of the measurement unit on results. For particle clustering, this can be determined by plotting the logarithm of the variance of the measured local area fraction of particles against the log of the measured area [MIL 61, LI 92, WOR 96]. This is similar to a Richardson plot in fractal analysis. If the distribution is random over all the length scales of the measurement, this plot should be linear with a gradient of -1. Gradients between -1 and 0 indicate a fractal dimension < 3, characteristic of the spatial distribution of points. For example, if the points are distributed on randomly distributed planes (perhaps on grain boundaries), at certain length scales, a fractal dimension would be measured between 2 and 3, and in our plot a gradient > -1 would be measured. When a deviation from linearity occurs on the plot, this indicates a change in the characteristics of the particle distribution, e.g. clustering, at a scale characteristic of that measured area. Figure 6.1 shows such a variance plot for an Al 6061 matrix MMC on measurements parallel and perpendicular to the extrusion direction. The gradient is closer to -1 on measurements made perpendicular to the extrusion direction, but both directions show no change in slope characteristic of clustering at a particular areal length scale.

Figure 6.2. Schematic illustrating the location of XMT specimens machined from failed tensile specimens

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For XMT characterisation of undeformed MMCs, a 1 mm diameter cylinder of a fully dense Al 1100 containing 7.5 vol% ZrO2 was machined from an extruded rod. Specimens were also extracted from failed tensile specimens just below the fracture surface by electro-discharge machining (figure 6.2). They were of square section, approximately 1 mm x 1 mm. These specimens were used in a parallel study on acoustic emission prior to fracture]. Three specimens were used for fully quantitative XMT studies containing nominally 7.5% by volume of ZrO2 in annealed Al 1100, peak aged Al 6061 and over aged Al 6061.

6.4. Results and discussion Figure 6.3a shows conventional XMT reconstructions of a single slice from the fully dense 7.5% ZrO2 Al 1100 MMC measured below and above the Zr absorption edge at 17.4 and 19.6 keV respectively. The same features can be identified below and above the absorption edge at 18.2 keV. The apparent low density halo seen at the specimen edge in the 19.6 keV reconstruction is an artefact generated by the edge pixels being incompletely occupied by material. Figure 6.3b shows a separate

Figure 6.3. Reconstruction of a single slice obtained from dual energy XMT measurements on a fully dense, extruded bar of 7.5vol% ZrO2 reinforced Al 6061 MMC. a) Single energy reconstruction of absorption density from 17.4 keV and 19.6 keVX-rays either side of the Zr K-edge. b) Reconstruction of Al and ZrO2 density computed from the data in (a)

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reconstruction for each of the Al and ZrO2 distribution data sets obtained from the data of figure 6.3a. It is clear that regions of high ZrO2 density correspond with regions of low Al density. The reinforcement volume fraction averaged across this slice was measured at 4.5%. The maximum reinforcement volume fraction of a single voxel was 56%, which is attributed to a combination of the interpolation carried out during back-projection reconstruction and the similarity between reinforcement particle size and voxel linear dimension. The void content averaged over this slice was measured as 0.4%. Figure 6.4 shows images of ZrO2 density distribution and total material density distribution. The void density distribution is the inverse of the total relative density. These have been calculated from two slices of a dual energy XMT reconstruction from beneath the fracture surface in the peak aged 7.5% ZrO2 Al 6061 MMC. Slice numbering is arbitrary with increasing number towards the fracture surface. Both images in slice 16 show a common feature corresponding to a section of the fracture surface intersecting the slice. Regions of high void density correlate with regions of low ZrO2 density as expected.

Figure 6.4. Dual energy XMT reconstructions from a peak aged 7.5vol% ZrO2 reinforced Al 6061 MMC showing ZrO2 particle distribution and totsl density distribution. Two slices are shown, slice 12 is from the bulk of the specimen while slice 16 intersects the fracture surface: a) total density distribution (light regions are voids), b) ZrOz density distribution

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It is possible to analyse the quantitative XMT data in a number of ways [MUM 93a, 93b, 95]. Changes in sample external dimension can be used to measure local strain. The void content can be averaged over complete slices of material which have undergone a known plastic strain to determine the local void content as a function of strain [MUM 93a]. The reinforcement volume fraction can also be averaged over a complete slice of material. It is also possible to determine the reinforcement and void content in each individual voxel and compute the probability distributions of these measures. It is thus possible to investigate whether there is any correlation between the particle and void distributions over a range of length scales. Finally, variance analysis can be carried out on both the reinforcement and void density distributions in any single slice of material to determine the extent of any void clustering as a function of plastic strain. Measurements of ZrO2 volume fraction from each complete slice in the three fractured specimens were in the range 5.7 - 7.4%. This is lower than the nominal volume fraction of ZrO2 present (7.5%) and is also lower than that measured by optical metallography in this study (8.3% for the Al 1100 MMC and 8.2% for the Al 6061 MMC). The precise reasons for this divergence are unclear but are probably artefacts of both the tomographic reconstruction algorithms and the determination of 3-dimensional information from the 2-dimensional data of quantitative optical metallography. The maximum reinforcement content measured in any reconstructed voxel from the fractured specimens was 18.5% (figure 6.5). This is significantly lower than that found with the as extruded specimen (56%). This may indicate an accumulation of damage in the regions with the greatest local reinforcement volume fraction. Figure 6.5 also shows no systematic variation in maximum reinforcement content with distance up to 1mm from the fracture surface, this suggests that all high reinforcement content regions have associated voids throughout this region near the fracture surface.

Figure 6.5. Maximum ZrO2 volume fraction measured in an individual voxel on each reconstructed slice as a function of distance from the fracture surface

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Figure 6.6 shows the results of variance analysis on both reinforcement and void distribution within each slice of an XMT reconstruction. The resulting straight lines have gradients between ~ -0.8 and - 0.5. These are similar, if a little further from the ideal case of -1, to those found for the distribution of particles on optical crosssections. Thus, we might infer there to be no significant clustering of voids. However, the algorithm used during filtered back projection reconstruction of an XMT slice introduces a correlation between adjacent voxels, hence leading to a reduction in variance at small sampled areas, as can be clearly seen from the gradual change in slope as the sampled area decreases. This reduces the utility of variance analysis with this form of tomographic reconstruction algorithm, however, the absence of any abrupt change in slope suggests that there is no clustering within the length scales sampled.

Figure 6.6. Variance analysis with gradient of plot, b, shown of a) Zr02 distribution and b) void distribution, determined from dual energy XMT slice reconstruction data

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Figure 6.7. Local void content as a function of local reinforcement volume fraction measured from a dual energy XMT reconstructed slice 300 fjmfrom the fracture surface of a peak aged 7.5vol% ZrO2 reinforced Al 6061 MMC. Linear regression plots are shown for two different sampled areas

Finally we can consider representing a direct relationship between local reinforcement content and local void content in any reconstructed slice, i.e. at constant distance from fracture surface and hence, constant nominal strain. The measure of a local value of any parameter with a degree of inherent randomness depends on the measured area. Figure 6.7 shows the relation between local void content and reinforcement volume fraction for the peak aged Al 6061 MMC measured over 12 x 12 pixel (180 x 180 um) and 16 x!6 pixel (240 x 240 um) areas. There is considerable scatter but in both cases a straight line can be fitted by regression analysis with a reasonable correlation coefficient. Table 6.1 shows the correlation coefficient between local particle density and void fraction as a function of distance from the fracture surface and sampling area for the annealed Al 1100 and peak aged Al 6061 MMCs. With the Al 1100 MMC there are only significant void volume fractions within about 250 urn of the fracture surface (figure 5a). In this region the correlation coefficient is in the range of 0.3 - 0.4. The peak aged Al 6061 MMC has a significant void fraction throughout the volume sampled and a higher fraction than the Al 1100 MMC at most sections (figure 5a). This shows a stronger correlation with a coefficient typically in the range of 0.3 - 0.8. In this material there are a few "rogue" slices (e.g. at 300um and 600 um) with a systematic anomaly in the correlation. The 24 x 24 sampling is more susceptible to rogue results, presumably because its size limits the number of data points. In both MMCs there is no systematic change in correlation coefficient with sampling size which is consistent with there being no clustering of voids.

Characterisation of void and reinforcement distributions

Distance 4x4 6x6 Annealed 7.5% ZrO2 Al 1100 0.25 0.18 50 0.38 0.33 100 0.34 150 0.35 0.29 0.39 200 250 0.35 0.33 0.12 300 0.18 350 0.16 0.27 0.24 0.32 400 0.04 450 0.05 0.21 500 0.05 0.14 550 0.00 0.05 600 0.01 650 0.20 0.09 0.34 0.20 700 0.04 750 0.01 Peak Aged 7.5% ZrO2 Al 6061 50 0.45 0.50 100 0.34 0.35 150 0.42 0.41 200 0.44 0.38 250 0.31 0.30 300 0.60 0.65 350 0.44 0.42 400 0.46 0.43 450 0.37 0.27 500 0.51 0.53 550 0.41 0.46 600 0.01 0.05 650 0.33 0.32 700 0.33 0.29 750 0.33 0.20

99

Sampling Area (Voxels) 8x8

12x12

16x16

24x24

0.22 0.29 0.41 0.40 0.34 0.09 0.22 0.46 0.08 0.19 0.18 0.02 0.29 0.44 0.07

0.29 0.28 0.21 0.44 0.34 0.19 0.12 0.40 0.18 0.07 0.16 0.18 0.48 0.54 0.06

0.46 0.29 0.25 0.39 0.03 0.22 0.11 0.50 0.08 0.07 0.22 0.10 0.45 0.40 0.23

0.52 0.09 0.04 0.38 0.58 0.43 0.39 0.52 0.38 0.74 0.02 0.71 0.97 0.47 0.79

0.59 0.44 0.43 0.35 0.37 0.70 0.50 0.43 0.33 0.58 0.51 0.13 0.32 0.47 0.20

0.57 0.33 0.18 0.41 0.36 0.85 0.24 0.61 0.15 0.65 0.76 0.30 0.06 0.33 0.88

0.88 0.45 0.60 0.40 0.27 0.89 0.43 0.43 0.08 0.73 0.74 0.32 0.30 0.57 0.24

0.81 0.38 0.83 0.66 0.89 0.95 0.34 0.99 0.24 0.74 0.90 0.37 0.89 0.32 0.62

Table 6.1. Linear correlation coefficient between local reinforcement volume fraction and local void fraction as a function of distance from the fracture surface and different sampling areas

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Let us now consider the evidence for any relationship between local particle volume fraction and local void volume fraction in these materials after straining. These materials were produced by powder metallurgy and our analysis of the particle distribution using a Richardson (variance) plot indicates a self similar distribution with fractal dimension between 2 and 3. There is no apparent change in slope seen on the plot over the range of sampled areas selected and thus no significant change in distribution character. Similar random distributions are seen when the XMT data of void and particle distribution are analysed, with the caveat concerning possible autocorrelation introduced during reconstruction. Linear regression analysis indicates a significant correlation between void fraction and reinforcement content. However, this correlation is independent of plastic strain and sampled area. Thus there is no evidence for either an increased void nucleation rate or growth rate in regions of higher local reinforcement content. Our results are consistent with a model of random void nucleation at particles and subsequent growth independent of local environment; i.e. there is greater void volume in regions of high reinforcement content purely because there are more nucleation sites. These results only apply to our materials with random particulate distributions and may not apply to highly clustered MMCs obtained by casting routes [LLO 91]. There is some evidence from acoustic emission experiments that artificially constructed MMC microstructures with reinforcement clusters nucleate damage at lower strains than random microstructures [BUR 97]. Indeed, there is some very limited evidence from our results that might support this hypothesis. Highly clustered regions of reinforcement are rare in our material, but their presence was identified in the undeformed material where some voxels in the tomographic reconstruction had reinforcement contents in excess of 50%. Beneath the fracture surface, we never found any voxels with reinforcement contents greater than 20%. Thus, the albeit rare regions of high reinforcement disappear after straining. Clearly more work is required to address the fracture behaviour of more clustered microstructures during tensile straining.

6.5. Conclusions Dual energy XMT has been used successfully to determine the spatial distribution of voids and reinforcement in model ZrO2 reinforced Al alloy MMCs. It has not been possible to demonstrate the equivalence of reconstructed XMT sections with coincident metallographic sections, because the system resolution was below that required to image individual ZrO2 reinforcement particles. However, the ability to measure local reinforcement density in each reconstructed voxel allowed the analysis of local particle clustering. The initial ZrO2 reinforcement distribution appears to be random and with no significant clustering as determined by variance analysis. Void distributions after straining had similar properties. Unfortunately because of correlations introduced by

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the tomographic reconstruction algorithms we cannot be certain that this analysis is valid for the XMT data. There appears to be a weak but consistent correlation between void density and particle density. This supports the hypothesis that in powder route fabricated MMCs, void density during plastic straining is directly related to particle density and that no enhanced void nucleation or growth occurs in regions of high reinforcement concentration. Acknowledgements We would like to acknowledge the EPSRC for the provision of funding to support this project through its composite materials programme, reference GR/H33817 and GR/J25628. IJJ would also like to acknowledge the EPSRC for the provision of a postgraduate research studentship. 6.6. References

[BUF 99] BUFFERS J.-Y., MAIRE, E., CLOETENS P., LORMAND G. and FOUGERC R., Acta Mater, vol. 47, p. 1613 (1999). [BUR 97] BURDEKIN N.A., STONE I.C. AND MUMMERY P.M., presented at 4th Inter. Conf. Compos. Eng. (1997), unpublished. [ELL 90] ELLIOT J.C., ANDERSON P., DAVIS G.R., DOVER S.D., STOCK S.R., BREUNIG T.M., GUVENILIR A. AND ANTOLOVICH S.D., /. X-Ray Sci. Tech. vol. 2, p. 249(1990). [ELL 94] ELLIOTT J.C., ANDERSON P., DAVIS G.R., WONG F.S.L., and DOVER S.D., JOM-J. Min. Met. Mater. Soc. vol.46, p. 11 (1994). [HIL 61] BILLIARD J.E., CAHN J.W., Trans. Metall Soc. AIME vol. 221 p. 344, 1961. [KOZ 99] KOZUL N., DAVIS G.R., ANDERSON P. AND ELLIOT J.C., Meas. Sci. Technol. vol. 10, p. 252 (1999). [LI 92] LlQ.F., SMITH R., McCARTNEY D.G., Mater. Char. vol. 28 p.189 1992. [LLO 91] LLOYD D.J., in Metal Matrix Composites - Processing, Microstructure and Properties, Proceedings of the. 12th Ris0 Inter. Symp. Mater. Sci., p. 81 1991 Ris0 National Lab., Roskilde, Denmark. [MUM 93a] MUMMERY P.M., ANDERSON P., DAVIS G.R., DERBY B. and ELLIOTT J.C., Scripta Metall. Mater, vol. 29, p. 1457 (1993). [MUM 93b] MUMMERY P.M. and DERBY B., in Prcoeedings of the 9th International Conference on Composite Materials. , Vol. 1, Metal Matrix Composites, p. 424, 1993, Woodhead Publishing, Cambridge, UK. [MUM 95] MUMMERY P.M., ANDERSON P., DAVIS G.R., DERBY B. and ELLIOTT J.C., J. Micros, vol. 177, p. 399 (1995). [WOR 96] WORRALL C.M., WELLS G.M. Proceedings of the European Conference on Composite Materials, London, p. 247 1996.

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Chapitre 7

Characterisation of MMCp and cast Aluminium alloys

This paper describes recent results on the application of high resolution X-ray tomography to the study of metal matrix composites and cast aluminium alloys. The advantage of phase contrast X-ray tomography over classical attenuation X-ray tomography is shown in the case of an Al/SiC metal matrix composite. The possibility of studying in situ damage initiation and development within a material under stress is also illustrated. Finally, examples of quantitative results extracted from reconstructed three dimensional images are given with respect to microstructure and damage characterisation of different structural materials.

7.1. Introduction Almost all structural materials contain micro-heterogeneities such as precipitates, reinforcing particles, holes, etc. Generally the mechanical properties of those heterogeneities are different from those of the base material (the matrix). Therefore, those heterogeneities form a preferential site for the nucleation and development of damage inside structural materials under stress. Ceramic reinforcements can be added, for instance, to a metal matrix in order to obtain a Metal Matrix Composite (MMC) with a high specific elastic modulus. However, the co-existence of a brittle material with a very ductile matrix induces some elastic and plastic incompatibilities which result in poor fracture properties. A lot of attention has been paid to the investigation of the damage mechanisms of Metal Matrix Composites (MMC) under stress. However, most of the work carried out so far relies on surface observations of samples mainly because experimental

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evidences of the damage mechanisms occurring in the bulk of the samples are rather difficult to obtain. Another example of micro-heterogeneous structural materials are cast aluminum alloys. These materials offer various advantages over wrought aluminium alloys when one is concerned with the manufacturing of an object with a very intricate shape, like, for example, hollow shapes. However, aluminium, like other metals, contracts when it solidifies and besides, hydrogen dissolves very easily in the molten metal. Therefore, unless special (expensive) processing methods are used, aluminium cast objects always contain a certain amount of porosity. These pores are a major problem because of their detrimental effect on the mechanical properties of the finished object. For this reason, a lot of research is being carried out on the study of porosity in cast aluminium alloys and on its influence on mechanical properties. However, like in the case of MMCs, the characterisation of porosity in cast alloys is quite difficult to achieve experimentally through classical metallographic examination. For the moment, X-ray Computed Tomography (XRCT) is the only non destructive characterisation technique which provides direct images of the bulk of heterogeneous materials. As a matter of fact, this method has been used extensively in medicine for years, but the low resolution of the images obtained (around 100 um) impeded its use in materials science. In the last ten years, however, the use of synchrotron X-ray radiation coupled with new detectors has opened new possibilities in this field. 3D images of the interior of micro-heterogeneous materials have been obtained with a resolution of a few microns (see for example [CLO 97] for a review). In this paper we present some recent results in the study by XRCT of damage initiation and development inside a notched sample of MMC. The aim of the experiment is to evaluate the influence of the non uniform stress state on damage development within the material. Moreover, the same technique has also been used for the characterisation of model cast alloys. Some quantitative results on the description of the pore population in these alloys are shown. 7.2. Experimental methods 7.2.1. Materials. 7.2.1.1. Al/SiC composite The composite studied in this work was a 6061 Al alloy reinforced by a 10 % volume fraction of SiC particles. The average size of the reinforcement, 120 fam, was large compared to industrial material. This size had been chosen in relation with the resolution of the 3D imaging technique (see below). The average aspect ratio of the particles was 1.6. The material was processed through a rheocasting route under

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nitrogen and was subsequently extruded at 813 K with an extrusion ratio of 16 and a ram displacement rate of 100 um.s"1. The extruded bars were solutionised at 803 K for 2 hours, quenched in water and matured at room temperature for 2 weeks (T4 heat treatment) before testing. 7.2.1.2. Al cast alloys Three different model Al cast alloys have also been studied and characterised by X-ray tomography. The nominal composition of the three alloys is given in table 7.1. Those alloys were chill mould cast by Pechiney. Different amounts of hydrogen (H2) and Argon (Ar) gases were introduced in the melt thanks to a propeller. At the beginning of solidification, part of the gas content was rejected and formed artificial pores. By varying the H2/Ar ratio (2%, 7% and 10%) three different average sizes of pores were obtained for each alloy (alloy A, B and C, respectively). Thin cylinders (diameter 3mm) were machined from the ingots for XRCT characterisation. The volume of material investigated during each scan was around 170 mm3.

Si

Fe

Mg

Ti

Sb

6.6-7.0

0.09-0.14

0.29-0.34

0.11-0.14

0.12-0.14

Cr <0.01

Others <0.01

Table 7.1. Chemical composition (wt%) of the studied castAl alloys

7.2.2. Attenuation vs. phase contrast tomography High resolution XRCT experiments have been carried out at the European Synchrotron Radiation Facility (ESRF) in Grenoble on line ID 19. A schematic view of the experimental set up used can be seen on figure 7.1. The X-ray beam coming out of the ring first encounters a double silicon monochromator before hitting the

Figure 7.1. Schematic view of the experimental set up used for the tomography experiments on line ID 19 at ESRF. Note the unusual large distance between source and detector which results in a large coherence of the X ray beam.

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studied specimen which stands on a rotating stage. The transmitted X-ray beam, is converted into visible light before being recorded by a 1024*1024 CCD detector specially designed at ESRF. A tomographic scan comprises 600 2D projections of such type. They are obtained by a 180 ° rotation of the sample around its vertical axis with a 0.3° step. Those projections are used to reconstruct a 3D numerical image of the sample through a classical filtered back projection algorithm [ROB 99]. The voxel size in the reconstructed volume, fixed by the experimental setting of the detector, was about 6.5*6.5*6.5 um3 for the materials studied in this work. In classical tomography, the contrast obtained on the reconstructed images comes from differential X-ray attenuation between the constituents of the studied material. The pores in the studied cast al alloys show an X-ray attenuation very different from the matrix. Therefore, they give a good contrast on the 2D projections and in the reconstructed images. The case of the Al/SiC composite is slightly more problematic as the X ray attenuation of the SiC particles is very close to that of the matrix. Thus, the contrast of the SiC particles in the reconstructed images is very faint. This is illustrated on figure 7.2a. To circumvent this problem, the so called «phase contrast» tomography has been used. This technique, which is based on the high coherence of the X-ray beam on line ID 19, greatly improves the detection of SiC particles (see figure 7.2b). It has been shown, besides, that the phase contrast also enhances the detection of damage (cracks) within reinforcements in MMCs [BUF99].

Figure 7.2. Reconstructed images of two different cross sections of the studied Al/SiC composite obtained by a) attenuation tomography and b) phase contrast tomography. On the left, the contrast given by the SiC particles is very faint because of the small difference in Xray attenuation between SiC and Al. Reconstruction rings are also visible because the contrast of the image has been enhanced to visualise the SiC particles. On the right, thanks to the phase contrast technique available on ID 19 beamline at ESRF, the contrast between matrix and reinforcement is enhanced by the presence of bright/dark fringes at each particle/matrix interface. Those fringes are obtained by simply setting the CCD detector at 83 cm behind the sample

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7.2.3. Mechanical tests The mechanical tests have only been performed on the Al/SiC composite. A double shouldered tensile sample cut along the extrusion direction was used. The cross-section of this sample was 2*2 mm2 and its gage length was 4 mm. A notch was cut in the central part of the sample using a diamond saw. Its height and depth were 365 urn and 500 urn respectively. The radius of curvature at the root of the notch was around 130 urn. Before testing, the flat faces of the sample were mechanically grinded using SiC paper down to 1200 grit. The tensile tests were conducted at room temperature using a specially designed tensile testing device which was set directly on the goniometer and which experienced the same rotation as the sample during the scans. In order to avoid the frame of the machine to hide the beam during the 180 degree rotation, a PMMA tube was used to transmit the load between the upper mobile grip and the lower grip fixed on the goniometer. This tube was carefully polished and gave a weak attenuation (which was corrected in the reconstructed images) on the 2D radiographs. The machine could be used in tension or in compression with a maximum load of 2500 N. The force and the crosshead displacement were recorded on a computer and monitored during the test. A constant crosshead displacement rate of 150 um.min'1 was used. This corresponded to an average strain rate of 6 10"4 s"1 in the sample. Five tomographic scans were performed corresponding to the initial undeformed state and to four consecutive steps at increasing values of the applied stress (40, 95, 130, 140 MPa). These steps will subsequently be referred to as steps 0, 1, 2, 3 and 4. For each scan, the sample was maintained with a constant crosshead displacement on the tensile testing device. All observations were consequently made in the loaded state. For the initial state and also for each stress step, one 3D volume of the sample was reconstructed. Only a restricted zone (2*2*1.7 mm3) around the notch has been considered in the reconstruction in order to reduce the volume of data to handle. 7.3. Results and discussion 7.3.1. Al/SiC material Figure 7.3 shows three reconstructed images of the same internal section of the AlSiC sample at step 0, 1 and 2. The analysis of this figure shows, first, that some voids induced by the extrusion process were present at some particle/matrix interface at the initial state. A few cracked particles were also observed. For the experimental conditions investigated, the voids induced by the extrusion process did not tend to propagate under stress and no clear evidence of particle matrix de-cohesion could be seen. The main damage mechanism was the mode I cracking of the SiC particles as

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already observed for the same material during tensile test on a smooth sample [BUF 99]. The evolution of the cracked particles spatial distribution within the sample, all along the test, is shown on figure 7.4.

Figure 7.3. Reconstructed images of the same internal section of the AlSiC sample at step 0, 1 and 2. The arrows indicates the presence of a crack in a SiC particle at the root of the notch where the stress level is high

It can be seen, on that figure, that the particles tend to break first near the root of notch where a higher stress level is obtained. When the stress is increased, however, the size of the volume containing the cracked particles steadily increases. The presence of a tri axial state of stress in the studied sample is generally believed to alter the mode of damage, by promoting particle matrix decohesions [CLY93]. In our case, however, the mode of damage ( mode I particle cracking) seems to be the same as that observed in a smooth sample in the same deformation conditions. At least two effects can account for this observation. One is related to the phase contrast technique. A thin interfacial crack between matrix and reinforcement would give an extra fringe in the reconstructed image which would probably be quite difficult to detect with a 6.65 urn resolution. The second reason is related to the material itself. In MMCs indeed, a large size of the reinforcements is known to promote particle cracking at the expense of other damage modes. Thus, the large size of the SiC particles studied here are likely to favour only the particle cracking as damage mechanisms in the early stages of straining. Further work with a material

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containing smaller particles (around 20 urn) and with a higher (2 urn) resolution should help to clarify this point.

Figure 7.4. 2D projection (as indicated by the arrows on the 3D sketch on the right) of the SIC particle centroids in the reconstructed volume of the material. The different symbols used on the figure indicate at which step a particle broke. The tensile stress was applied vertically. The superimposed square indicates approximately the zone where a higher number of broken particles was found at the first stages of deformation

In order to study quantitatively by 3D image analysis the evolution of damage in the material all along the test, the reconstructed images were binarised. The average grey level in the SiC particles was quite close to that of the matrix [CLO97], impeding the use of simple well known threshold-based methods. Instead, the binarisation process was carried out on the 2D projections with an edge detection algorithm. The resulting binary images were analysed with a 3D software OTIP 3D specially designed for the classification of 3D images [MIC 98]. For each particle, of the 3D volume, a set of parameters was determined by the software (volume of the particle, centroid coordinates, size of the surrounding box ...). In this work, 260 particles were analysed individually around the notch. For each of them, the state of deformation (broken or not broken ) was studied manually on the 1250 sections of the different 3D volumes. From those results, a Weibull analysis of damage was undertaken. The Weibull probabilty Pr(i) for the rupturebof a particle i is given by the following formula:

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V0 is a reference volume which is a constant, V, stands for the volume of particle i, oi is the stress in the particle, m and a0 are the Weibull distribution coefficients which are to be determined. The volume Vi can be obtained directly from the results of image analysis on the 3D images. The stress cri arises from elastic and plastic incompatibilities between matrix and reinforcements. Its value, is obtained through quite complex analytical calculations which take into account the local stress value. This local stress being obtained via a 3D FEM model of the sample. By coupling this calculation and the result of the analysis of the 3D reconstructed volumes, the rupture stress of each particle was determined. From this data a plot of the particle rupture probability as a function of the stress in the particles was obtained. The shape of this curve suggests that the particle failure probability does follow a Weibull law as already observed in MMCs [LEW 95]. The corresponding Weibull parameters m and a0 were found equal to 3.2 and 1000 MPa, respectively. The value of the Weibull modulus m calculated in the present study is relatively low, suggesting that the rupture of the SiC particles is spread over a large stress range. As a matter of fact, the analytical calculation of oi does not take into account several parameters such as the introduction of damage in the particles during specimen preparation or the degree of clustering of the particles (a high local volume fraction should enhance the stress level in the particles). Such factors could probably account for a large distribution of the rupture stresses. Thus, the Weibull parameters obtained in this study should be regarded as those of the studied SiC particles embedded in a deforming matrix, rather than the inherent fracture behaviour of the particles themselves. 7.3.1. Cast aluminium alloys Figure 7.5 shows a volume rendering of the internal pores inside alloy A and B as observed by X-ray micro-tomography. It can be seen, from this figure that the size and the volume fraction of the pores increases when the amount of hydrogen introduced in the melt increases. Besides the shape of the pores appear to vary a lot, from rather round shapes to more tortuous ones. By comparing reconstructed sections of the material with the corresponding 2D optical micrographs, it was shown that the reconstructed images of the pores with a 6.65 urn resolution do correspond (both in shape and size) to the actual pores in the material [SAV 98]. On that basis, a quantitative study of the pores was carried out from the 3D reconstructed images. As already mentioned, the differential X-ray attenuation between a gas pore and the aluminium matrix being large, the pore grey level in the reconstructed images is very different from the matrix grey level and thus, binary images of the pores could be obtained by simple thresholding of the

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images. The binary images have been studied by automatic 3D image analysis. Two stereological parameters have been defined to try and characterise the shape and size distributions of the pores. The first one, called the equivalent size, is defined as the diameter of a sphere which would have the same volume as the considered pore. The second one, called the sphericity s, is defined from the ratio of the volume of the pore to its surface. A perfect sphere would have a sphericity parameter of 1 while a pore with a very tortuous shape would have a sphericity much smaller than 1 [SAV 98].

Figure 7.5. Reconstructed images of internal pores in two model cast aluminum alloys. Artificial pores have been obtained by introducing H2 and Ar gases in the melt metal. As the solubility of hydrogen decreases with temperature, when the metal solidifies, H2 is rejected and forms gas pores which are easily visualised using X-ray tomography. Two different H/Ar ratio have been used: 2% (left) and 7% (right). The edge size of both cubic boxes is 1.33 mm Figure 7.6 shows a plot of those two parameters for the three investigated alloys. It can be seen from this figure that at least two population of pores (cooresponding to two clouds of points on the plot) can be found in the material. The first one, in the lower part of the plot (equivalent size lower than 50 um), shows a weak correlation between shape and sphericity and is represented for the three alloys. The second one on the contrary can be found for sizes larger than 50 um and shows a strong correlation between size and shape. The volume fraction of the pores belonging to this last population can be seen to increase when the amount of hydrogen introduced in the liquid metal is increased. The two populations have different origins. One corresponds to microshrinkages when the metal solidifies while the other corresponds to artificial gas pores. The latter appears during the whole solidification process while the former appear at the end of it. Both types of pores have to form between the growing dendrite arms network. Thus the growth of large pores is geometrically hindered by

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the dendrites and the resulting shape is tortuous (low sphericity). By analysing carefully the data it is possible to show [SAV 00] that the volume fraction of microshrinkage pores remains roughly constant for the three investigated alloys while the volume fraction of gas pores increases exponentially with the H2 content as already observed in the literature.

Figure 7.6. Plot of the equivalent size of the pores inside alloys J, K and L, as a function of their sphericity. Two populations of pores can be seen respectively below and above an equivalent size of about 50 jam. Those two populations correspond to artificial gas pores (equivalent size larger than 50 pm) and to micro-shrinkages (below 50 /jm)

7.4. Conclusion We have shown in this paper how X-ray tomography can be used to study quantitatively structural micro heterogeneous materials. Two examples have been given corresponding to the case on an Al/SiC particulate metal matrix composite and to model cast aluminium alloys. Although 3D X-ray images of internal pores in cast alloys are quite easy to obtain thanks to the large X-ray attenuation difference between gas pore and aluminium matrix, the visualisation of the SiC reinforcements in an aluminium matrix is not possible with classical tomography. In that case, it is necessary to use phase contrast tomography which can be obtained with a coherent X ray beam like the one delivered on the ID 19 beamline at ESRF. This technique also improves the detection of damage in reinforcements. With regard to the composite material, it was possible to follow the initiation and development of damage inside a notched sample during in situ tensile deformation.

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No influence of the tri axial stress state could be found on the mode of damage (mode I cracking of the particles) which appears to be localised near the notch root at the begining of the deformation process. By coupling the 3D data on the SiC particles and analytical calculation a Weibull analysis of the particle fracture was carried out, leading to Weibull modulus of 3.2 and a a0 constant of 1000 MPa. For the model cast aluminium alloy, the 3D data obtained by tomography has been used to characterise the pore population for three different processing conditions. From this characterisation it is possible to distinguish between degassing pores and micro-shrinkage pores. 7.5. References [CLO 97] CLOETENS P., et al. Observation of microstructure and damage in materials by phase sensitive radiography. Journal. Appl. Phys. 81,9 p.5878, 1997. [ROB 99] ROBERT-COUTANT C, et al. Introduction to reconstruction methods This Workshop. [BUF 99] BUFFIERE J.-Y., et al. Characterisation of internal damage in a MMCp using X-Ray synchrotron phase contrast microtomography Acta Mater. 47, 5, pp.1613-1625, 1999. [CLY 93] CLYNE T.W., WITHERS P.J., An Introduction to Metal Matrix Composites Cambridge University Press, 1993. [MIC 98] MICHOUD P., DARSONVILLE F., Segmentation et visualisation d'images 3D application par croissance de region a I'imagerie medicate dentaire PhD thesis Saint EtienneUniversity June 1998. [LEW 95] LEWIS C.A., WITHERS P.J., Weibull modelling of particle cracking in metal matrix composites Acta metall. mater., 43, 10 1995. [SAV 98] SAVELLI S., et al. Characterisation by synchrotron X-ray microtomography of internal features and their detrimental effects with respect to the fatigue properties in aluminium cast alloys Proc. of ICAA 6 Toyobashi Japan p.529 1998 . [SAV 00] SAVELLI S., PhD thesis INS A Lyon 2000 to be published.

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Chapitre 8

X-ray tomography of Aluminum foams and Ti/SiC composites

The High resolution X ray tomography technique has been used to investigate the internal structure during the deformation of two very different materials : an Al foam and a SiC fibre reinforced titanium matrix composite. The observations show that the metallic foam deforms via local buckling events. This collapse mechanism is also studied by means of a modelling technique based on the finite element method. The microstructure of the metal matrix composite is also easily revealed with a sifficient resolution despite the high attenuation of the matrix. Cracks can be observed during the straining on a fibre touching the surface of the sample. It was not possible to show if this preferencial rupture was due to the weakness of this fibre or to the stress field which is different close to the surface.

8.1. General introduction This paper presents the investigation, with a single technique, of two completely different materials. As a consequence, it is divided into two parts. The first part is devoted to the characterisation of the microstructure of an aluminium foam and its evolution during a compression loading. The second part is devoted to an investigation of the microstructure and damage of a Ti/SiC composite during in situ tensile straining.

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8.2. Aluminium foams The newer low density cellular materials such as aluminium foams are being produced at low cost by several companies thanks to recent developments in manufacturing methods [MCC 99]. These materials show potential for use in lightweight structures. One of their interesting properties is the amount of energy absorbed during the deformation which is directly related to the way the material collapses in compression. We will focus our study on the compression behaviour of these aluminium foams. We performed 3D inspections by means of X ray computed microtomography (XRCMT) [KAK 88] on a commercial foam at different deformation stages in compression. XRCMT has recently emerged as a powerful technique capable to give a non destructive picture of the interior of the structural materials including foams. We also present a method to transform the images of the actual microstructure into a mesh which can be used to model the behaviour by finite element calculations. We finally use these calculations to assess the stress distribution inside the compressed walls.

8.2.1. Experimental set-up and results &.2.I.I. Set-up The tomograph used is based on the principle of « fan-beam » X-ray computed tomography scanners. A schematic view is shown in Fig. 1 and a more complete description can be found in [KAF 96]. An X-ray tube (Pantak HF 100) is used to irradiate a thin slice of the scanned object. The transmitted intensities are measured by means of an X-ray linear detector constituted by a unique row of 1024 sensitive elements with a pitch of 0.225 mm. The height of the sensitive row is 0.5 mm. Within each sensitive diode, a scintillator converts X-ray photons into visible light which is detected by a photodiode. A charge-coupled device allows multiplexing of the individual charges accumulated in each photodiode and delivers, after offset and gain calibration, a one-dimensional signal of 1024 points encoded on 8 bits. The line integration time is 8 ms. The scanned sample is set between the focus and the detector, at a distance allowing a geometrical magnification of 1.5. As the pixel size, measured along the linear detector, is 0.225 mm, the corresponding elementary feature imaged at the level of the sample is therefore 150 urn wide and 300 urn high. A first stepping motor allows to rotate the sample by steps of 360/N degrees. The rotation axis must be set perpendicular to the array and to the beam axis. The acquisition of N successive lines delivers a complete « sinogram » which is then processed by computer (computed tomography) to reconstruct the map of the

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slice. For the present application, N was set to 900. A second stepping motor then generates a translation of the sample along a direction parallel to the rotation axis and a new adjacent slice can thus be imaged. The acquisition of the data corresponding to a single slice lasts about 3 min.

Figure 8.1. Schematic description of the experimental setup for the tomograph used

Reconstruction is achieved using a C-language program written by the CEALETI (Commissariat a 1'Energie Atomique - France) and based on the Feldkamp algorithm [FEL 84]. A ramp filter is selected, thus preserving the spatial resolution. A DEC 500 MHz workstation is used to reconstruct the entire volume, slice by slice. Reconstruction of the complete volume (350 x 340 x 128 voxels) lasts approximately 1 hour. We have translated the sample of 300 (jm between each slice in z, a value which corresponds to the height of the diodes. In order to get an isotropic 3D representation of the sample (i.e. reduce the resolution in the z direction to 150 |jm ), we have numerically added extra slices in between the slices in z. We calculated the grey level in these extra slices by linearly interpolating the grey levels in the corresponding x, y voxels of the two adjacent slices. 8.2.1.2. Results The sample was scanned for tomographic analysis in its initial state. It was then compressed at three increasing values of remnant strain (true strain measured after unloading). The internal microstructure was imaged using tomography at these three steps after removal of the compression load. We then have a picture of the interior

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of the sample at four values of the remnant true strain in compression: 0, 0.065, 0.2 and 0.6. A 3D view of the analysed sample is shown in figure 8.2. The cell walls are clearly imaged and the 3D structure can be analysed. However, it is quite difficult to visualise the deformations with this kind of representation especially in the of closed cells for which the outer walls hide the inner ones. In what follows the qualitative results will be shown under the form of 2D reconstructed slices which are easier to analyse. Figure 8.3 (a and b) show a set of 2 D images of the same zone of the sample numerically extracted from the volume at 0 to 6.5 % of remnant strain respectively. These slices are parallel to the compression axis which is vertical on the figure. One can clearly observe the deformation mechanism of the studied foam in compression by comparing these two figures. The large value of the plastic strain at this stage is obviously not due to a homogeneous plastic straining of the whole sample, but to the local buckling of several walls like these surrounded by circles on the picture. From the observation of the entire population of buckled walls, we have observed that they are located in a band perpendicular to the loading axis (see figure 8.3b) i.e. the centre of gravity of these buckled walls is located in a narrow range in the z dimension (roughly between z=60 and 100 pixel compared to a total height of 255 pixel for the sample). Between the second and the third step, the number of new buckled walls is small. The deformation process leads to the complete closing of the already collapsed cells instead of the appearance of new ones. Some new collapsed cells can be seen in step 3, but even at this very large plastic strain, some regions of the sample are not deformed at all.

Figure 8.2. 3D view of the sample before compression

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Figure 8.3. Part of a 2D reconstructed slice of the same zone at two deformation steps initial state (left) and 6 % (right)

8.2.2. Finite element modelling The internal structure of the studied foam is extremely complex: the cell size and the length of the walls are parameters which are widely spread. The non periodic character of the structure leads also to a very difficult calculation of the actual force applied to each wall compared to the case of honeycombs which is another class of cellular materials. The complexity of this entanglement lead us to analyse the distribution of the stress and deformation in each wall using an appropriate tool: the finite element analysis. In what follows, we present an easy way to use this 3D picture to generate meshed models of the actual microstructure. These models are readable by the ABAQUS commercial code which we have used to perform the calculations in 3D. 8.2.2.1. Direct meshing of the actual microstructure The result of a tomographic inspection (after reconstruction) is a 3D table of gray levels. Each point of this table (voxel) defined by its coordinates x, y and z, represents the linear attenuation coefficient of the X-rays within the elementary volume of the material at this location. Given the experimental resolution of the setup used, the volume of the element exhibiting this transmission coefficient is 150um x ISO^m x 150um in our study. The grey level is high if the voxel is situated inside a cell and it is low if the voxel is situated inside an aluminium wall. The method described below is based on a simple idea: each voxel in the 3D image can be represented by a cubic element in a finite element mesh. We developed a software to produce models meshed according to the following procedure : - a 3D parallelepipedic set of nodes is generated with a regular and identical spacing in the three directions. This spacing is equal to 150 jam. The number of nodes is governed by the number of voxels in the actual sample in the three directions (number of nodes = number of voxels +1);

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- this set of nodes is used to define cubic-eight-nodes-brick-elements according to the standards used in the ABAQUS commercial code (element type C3D8). - the table of the values of the local transmission coefficient in the sample is read in parallel with the 3D mesh. The elements located in the cells (high grey level) are removed from the mesh and the elements located in the walls (low grey level) are kept in the model. We considered that the boundary between the steel plates of the rig and the aluminium foam was perfect. The displacement of the lower nodes were fixed in the three directions. We generated a steel plate perfectly bounded to the upper nodes and applied a displacement along the compression direction to the upper nodes of this steel plate. 8.2.2.2. Results Given the number of degrees of freedom of the problem to be solved, it was not possible to treat the complete foam in 3D. We firstly grouped 8 voxels into one voxel which lead to a resolution of the mesh twice lower than this in the initial picture. We calculated the deformation of a 503 voxels block which represents a 1003 voxel block in the initial foam i.e. approximately one eight"1 of the entire volume of the studied foam. The calculation was performed in the elastic regime. The deformed and undeformed models are compared in figure 8.4 a and b respectively. One can see the deformed shape of the walls.

Figure 8.4a. FEM undeformed model

Figure 8.4b. FEM deformed mode

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8.3. Titanium composites 8.3.1. Experimental procedure 8.3.1.1. Imaging setup This part of the experiment has been carried out at ESRF on beam line ID 19. The electron beam energy was 6 GeV. The white beam was restricted by slits and monochromated by a set of two parallel silicon single crystals selecting the photons exhibiting the chosen energy (33 keV for the present application). The unusually large distance of 140 m between the source and the experimental hutch on ID 19 leads to a high lateral coherence of the photons. The distance between the sample and the detector (a cooled CCD camera developed at ESRF) was set to about 70 mm. This particularity in the setup combined with the high lateral coherence of the photons has been shown [CLO 97] to lead to an improvement in the detection of phase features (like cracks) thanks to a so called 'phase contrast' which is added to the regular attenuation contrast. More details about this technique can be found in the article by Cloetens et al. which will be published in the present proceedings. In addition to simple x-ray radiography, the ID 19 beam line allows tomographic observations to be performed on structural materials. Tomography is a new non destructive technique which allows the 3D internal structure and damage of a sample to be imaged [BUF 99] with a resolution down to 1 (jm. The sample can be fixed on a turntable and rotate in the beam to provide a set of 900 radiographs which are used by a reconstruction software to give a 3D numerical image of the studied material. 8.3.1.2. Sample

Figure 8.5. Schematic description of the sample used

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Figure 8.6. Tomographic reconstruction of the sample in its initial state. View along the z axis. The three plys can be observed

Figure 8.7. Tomographic reconstruction of the sample in its initial state. View along the y (right) and the x (left) axis

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The sample used for this study was made of three layers of monofilamenatry SiC fibers embedded into a titanium matrix. The schematic geometry of the sample is shown in figure 8.5. The initial state was characterised using the tomographic setup described above. The results are shown in figures 8.6 and 8.7 under the form of 2D reconstructed tomographic slices along the 3 principal directions of the sample according to the geometry defined in figure 8.5. The internal structure of these composites can be clearly imaged at the resolution of 6.65 um used for the present study. 8.3.1.3. Tensile rig and testing procedure A tensile rig has been especially designed to allow the observation of damage by tomography during the deformation of materials [BUF 99]. This rig can be set on the turntable. In order to avoid the frame of the machine to hide the beam when rotation is necessary (for the tomographic inspection) a PMMA tube is used to transmit the load between the upper mobile grip and the lower fixed grip. This tube was carefully polished and gave negligible attenuation nor phase contrast on the 2D radiographs. The force and the crosshead displacement were recorded on a computer and monitored during the test. A crosshead displacement rate of 150 um.mn 1 was used for the tests corresponding to an average strain rate of 5.10 s'1. Once sample 1 was set in the machine, a first radiograph was recorded in order to characterise the initial state. The sample was then loaded step by step in tension : during the test, the crosshead displacement was stopped at increasing values of the deformation and a radiograph was recorded during each of these steps.

8.3.2. Results Figure 8.8 shows the observation of the evolution of damage during the straining of the sample. After the first step, one fiber was broken in two locations. After the second step, the same fiber was broken in a third location. These were the only visible events detected during the straining which then had to be stopped because the sample was too strong compared to the limited capacity of the tensile rig. The fiber which broke was this at the bottom left corner of the sample in figure 8.6. This fiber was the only one cut by the free surface. It is not clear if the early damage of this fiber is due to the fact that its intrinsic resistance was reduced because of pre damage during the sample preparation or if the stress field inside this fiber was modified by the effect of the presence of the surface.

T

8.4. General conclusion The collapse mechanism of a closed cell foam has been studied during compression thanks to X ray tomography experiments. The resolution in this first case was 150 |im, a value which was chosen because of the big size of the sample. The deformation mechanism by buckling of the walls of the foam has been evidenced. These microbuckling events are located in a band perpendicular to the compression axis. A software has been developed to transform the actual voxel information into a meshed model. The deformation of a part of the foam has been calculated by the finite element method in the elastic regime. In future work, we will try to define a criterion for local buckling of the walls from the comparison of the actual buckling configuration observed and the calculation of the stresses inside the walls. The microstructure and damage of a titanium composite reinforced with SiC long fibres has also been clearly imaged with the same kind of technique but

Figure 8.8. 2D reconstructed tomographic slices showing the evolution of the microstructure of a same zone for the three steps of the tensile test: inital step, fisrt step (center) and second step (right)

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with a much higher resolution (6.65 urn in the resent case). This resolution was achieved using synchrotron radiation. The observation of damage was carried out in tension. Unfortunately, the sample was too strong for the tensile rig, but the observations made during this primarily experiment are promising. We were able to visualise several ruptures in a same fibre at different steps during the test.

Acknowledgements The tomographic scans on the Al foam were performed with the help of G. Peix using the equipment of the CNDRI laboratory in Villeurbanne. The Ti/SiC samples were provided by Alex Madgewick and Phil Withers from the UMIST in Manchester and the tomographic scans were made at ESRF in collaboration with J. Baruchel and P. Cloetens.

8.5. References [BAR 98] H. BART-SMITH, A.F. BASTAWROS, D.R. MUMM, A.G. EVANS, DJ. SYPECK, and H.N.G. WADLEY. Acta mater. (1998). 46, N° 10, PP 35823592. [BUF 99] J.-Y. BUFFIERE, E. MAIRE, P. CLOETENS, G. LORMAND, R. FOUGERES. Characterisation of internal damage in a MMCp using X-ray synchrotron phase contrast microtomography. Acta Met. (1999) vol. 47, N°5 pp 1613-1625. [CEN 99] CENDRE E., DUVAUCHELLE P., PEIX G., BUFFIERE J.-Y. and BABOT D. Conception of a high resolution X-ray computed tomography device ; application to damage initiation imaging inside materials. Proceedings of the 1st World Congress on Industrial Process Tomography (1999) BUXTON (U.K.), 14-17avril. [CLO 97] P. CLOETENS, M. PATEYRON-SALOME, J.-Y. BUFFIERE, G. PEIX, J. BARUCHEL, F. PEYRIN, J. SCHLENKER. J. Appl. Phys. (1997), 81,9. [DAY 83] G.J. DAVIES and S.J. ZHEN. Mater sci. (1983), 18, 1899. [PEL 84] L.A. FELDKAMP, L.C. DAVIS, J.W. KRESS. Practical cone-beam algorithm. J. Opt. Soc. Am. (1984), 1 N° 6, pp. 612-619. [KAK 88] C. KAK, M. SLANEY. Principles of Computerized Tomographic Imaging. (1988). Cotellessa RF (Editor). New York : IEEE Press (Phs). [KAF 96] V. KAFTANDJIAN, G. PEIX, D. BABOT and F. PEYRIN. J. of X ray Science and Technology (1996). 6, 94. [MCC 99] K.Y.G. McCuLLOUGH, , N.A. FLECK and M.F ASHBY. Toughness of aluminium alloy foams. Acta Mater. (1999), 47 pp. 2331-2343.

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Chapitre 9

Simulation tool for X-ray imaging techniques

A computer code was developed to simulate the operation of radioscopic or tomographic devices. This code can be helpful to predict and optimise the performance of any imaging system. The use of computer aided drawing (CAD) models allows to carry out simulations with complex three-dimensional (3D) objects. The whole geometry of the system, from the source to the detector, can be defined and monochromatic or polychromatic beams can be chosen. Computed images present the advantage to contain no photon noise. Nevertheless, if necessary, noise can be added to the images afterwards, with an adjustable level. Movements of the object can be performed automatically, in order to simulate, for example, the acquisition of a sinogram. 2D tomographic images of a spatial resolution phantom, simulated with different sets of parameter values, have been reconstructed. A 3D tomography has also been carried out on a geometrically complex sample. The results prove overall coherence of our simulation tool and show rich possibilities.

9.1. Introduction X-ray imaging techniques, such as radiography, radioscopy and tomography, are used in a wider and wider range of specific applications, notably in the medical field and in materials science [PEL 89][PEI 97]. To develop new imaging systems, or optimise existing ones, it is generally necessary to carry out long and expensive series of tests and measurements. Proceeding in such a way, by trials and errors, often reveals itself unrewarding because it is practically difficult to study the influence of the many parameters (beam energy, focal spot size of the X-ray tube, detector type, geometric parameters, etc) that condition the final image quality.

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To go beyond those material limitations, simulation tools can be appealed to, with the aim to create a virtual imaging environment. Thus complex situations can be investigated, with important saving of time and low cost [INA 98]. When developing a new system, simulation offers powerful means to predict and optimise the future device performance, and to choose the most suitable components. Simulation can also be very helpful to interpret complex experimental data, by comparing them to the corresponding simulation results [KOE 98]. Finally, simulation can also be a tool to train operators. In that context, a four years research program was initiated 12 months ago in the laboratory "Controle Non Destmctif par Rayonnements lonisants", to model and simulate the functioning of any X- or y-ray imaging chain, from the photon emission phenomena, up to the detector behaviour, including the photon-matter interactions within any 3D object. During this first period, we started with photon attenuation in the object, and we have laid emphasis on 3D geometric aspects. This paper presents the basic principles we have used, the possibilities offered by the computer code we have developed and a few examples of simulated images, especially in tomography.

9.2. Background Our simulation software requires various input data, intended to describe an experimental situation with maximum accuracy. From the geometric point of view, we have to define the source (shape, size, position, orientation), the detector (position, orientation, pixel number and size), and the sample. The use of a computer aided drawing (CAD) model describing the object offers great advantages: complex 3D samples can be drawn in a short time with automatic surface mesh generation (triangular meshes), and with high accuracy. The object may consist of several parts, possibly of different materials. Ray-tracing techniques, together with the X-ray attenuation law, are the basis of our computer code. From each source point, a set of rays is emitted towards every pixel centre of the detector. Each ray may intersect a certain number of meshes on the object surface (fig. 9.1). The attenuation path length in every object phase is calculated by determining the coordinates of all those intersection points. The photon number N(E) that emerges from the sample and reaches the detector surface is given by the attenuation law: N(E) = N0(E) A0Q[])xp[-//,(£)*,.]

[1]

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where N0(E) refers to the number of photons with energy E, emitted by the source per solid angle unit, AQ is the solid angle corresponding to the pixel, observed from the source point, ui(E) designates the attenuation coefficient associated to the material i at energy E, and xi the total path length through the material i. The image computed by using equation [1] does not take into account the contribution of scattered photons. Moreover, the simulated image does not contain any photon noise. So far, we have used a perfect detector, in which every pixel is able to count all of the photons that hit its surface. The Compton and Rayleigh scattering contributions to the image and the photon interactions with a real detector will be investigated in a future work [DUV 98].

Figure 9.1. Principle of radiography simulation. A point source, a matrix detector, and a CAD model of the object with triangular meshes are used

9.3. Simulation possibilities 9.3.1. Source geometry As mentioned before, the simulation is based on ray-tracing from a source point. Consequently, if a unique source point is used, the simulated image does not take into account any image degradation due to geometric unsharpness. Accurate evaluation of geometric unsharpness is of great importance, e.g. in diagnostic radiology [SCH 98], but it is a very delicate task, because the unsharpness associated to a point in the object depends on several parameters:

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the position of the point in the object the shape, size, position and orientation of the gamma or X-ray source the position and orientation of the detector.

To carry out realistic simulations, we chose a straightforward method, which consists in cutting the source area into elementary source points, and repeating image computation with each of the latter (fig. 9.2). The source shape (rectangular or elliptic) can be chosen, and the number of source points as well as their spacing in the two directions are adjustable. In this way, the influence of geometric unsharpness can be evaluated, even in complex situations. This can be useful to determine the best geometric adjustment of an imaging system.

9.3.2. Beam spectrum The photon spectral distribution of the incident beam is another important aspect we had to give care to. To simulate an image, the photon attenuation calculation has to be repeated with a set of discrete energy values. So far, we have used the Birch and Marshall catalogue [BIR 79], in which semi-empirical spectral data, in the tube voltage range 30 to 140 kV, are tabulated. The corresponding attenuation

Figure 9.2. Simulation of geometric unsharpness. The sample is a thin square plate with a circular hole 1 mm in diameter. The detector is composed of 80 x 80 square pixels, 0.1 mm wide. Simulation A: point source (geometric unshrapness UA = 0 mm); B: 1 mm wide square source (UB = 1 mm); C: 2 mm wide square source (Uc = 2 mm). The grey level profile corresponding to the central column of the detector is presented in each case. The profile C reveals a degradation of the image contrast, due to geometric unsharpness

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coefficients, associated to any elementary or compound substance, can be obtained by means of available databases, such as XGAM [BER 88] or EPDL97[CUL97]. When performing a simulation, energy intervals (or channels) with adjustable width can be defined, and a set of images associated to each one is stored. A global image, obtained by adding all those partial images is computed too. It is therefore easy to monitor the spectrum evolution (beam hardening) when the beam goes through the object (fig. 9.3). The influence of a filter, wherever it is located in the imaging chain, can also be visualised. Simulation of dual energy imaging techniques can be carried out as well. By comparing simulation results obtained at different tube voltages, it is possible to predict the more favourable energy, leading to optimal image quality, in any specific application. For example, when the photon energy rises, the quantity of transmitted photons is more important and the relative photon noise is smaller. However, the image contrast is lower and therefore a compromise has to be found. To determine the latter, simulation can be a helpful tool. 9.3.3. Sample movements To simulate the functioning of radioscopic or tomographic systems, multiple simulation with automatic movements of the sample is necessary. Those movements play the same role as stepper motors in experimental setups and they can be a combination of translations and rotations. For example, radioscopic image acquisition can be performed by means of a linear detector, with a sample translation, perpendicular to the plane constituted by the X-ray focal spot and the detector [KAF 95]. Tomographic devices can be simulated as well, with automatic computation of all the projections that compose the sinogram. Depending on the simulation type (simple radiographic projection, 2D tomographic slice, 3D tomography with a matrix detector) and parameters (detector resolution, number of source points, mono or polychromatic beam, complexity of the sample), the computation time order of magnitude can vary from 0.1 second up to several hours on a PC, with a 400 MHz microprocessor.

9.3.4. Phone noise An interesting characteristic of our simulation code is its deterministic principle. The same experiment, if it is simulated several times, will always yield exactly the same results. Moreover, the synthetic images have the advantage not to contain any photon noise. If necessary, photon noise, i.e. Gaussian random fluctuations, can be added to the image, when the simulation is complete. The effect of different noise levels can thus be evaluated with a single simulation.

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Figure 9.3. Radiograph simulation of a gate, with a step wedge. A polychromatic beam is used and the detector is split into 100 energy channels with a 1 keV interval, (a): CAD model of the gate, (b): Simulated radiograph, (c): Incident beam spectrum, (d): The transmitted photon spectra associated to the points labelled a to c reveal a beam hardening phenomenon, stronger where the part is thicker, i.e. close to the edge

9.4. Simulation examples in tomography 9.4.1.2D tomography on a spatial resolution phantom As a first application, we decided to simulate 2D tomographic experiments, with a linear detector (fig; 9.4). Several sinograms were simulated with different source sizes to assess the geometric unsharpness influence on spatial resolution. We used a CAD model of an aluminium spatial resolution phantom, cylindershaped, with square holes from 0.4 mm to 2.5 mm in width. This phantom was defined in the frame of an industrial tomography workgroup, in order to quantify the performance of a tomographic system. We have also studied the effect of beam spectral distribution. When a monochromatic beam is used, the reconstructed numerical value of the linear attenuation coefficient is constant in the homogeneous aluminium phase, and

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Figure 9.4. Simulation of a tomographic slice on an aluminium spatial resolution phantom, (a): Simulated setup and CAD model of the phantom. The detector is linear, composed of 500 pixels (0.2 mm wide). 800 projections were computed, (b): Reconstructed slices. Left: point source; right: 4 mm wide source, cut into 15 elementary source points (geometric unsharpness associated to the centre of the phantom: U - 1.7 mm). In both simulations, the beam is polychromatic (100 kV tube voltage), (c): The grey level profiles, corresponding to the lines labelled A and B show the effect of geometric unsharpness. Computation time: 10 min (point source) and 2.5 h (large source)

corresponds, as expected, to the attenuation coefficient of aluminium at the beam energy (fig. 9.5). This quantative result proves that the whole simulation process is coherent. If the beam is polychromatic, it hardens itself when penetrating into the part, causing a "cupping effect"[CEN 98][KAK 88]. If necessary, it is also very simple to add photon noise, i.e. Gaussian fluctuations to the sinogram and to the reference image (obtained when the sample is removed from the beam) before reconstructing the tomographic slice. In this way, different noise levels can be visualised easily (fig. 9.6). Experimentally, photon noise is conditioned by the choice of tube voltage, current intensity, filters, exposure time and geometric parameters.

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Figure 9.5. Influence of the incident beam spectrum on a reconstructed slice (a): monochromatic 100 keV beam, (b): polychromatic beam with 100 kV tube voltage. In both cases, a point source was used, (c): (-n) reconstructed value profiles (n designates the linear attenuation coefficient). When a monochromatic beam is used (graph A), the baseline of the profile has a constant value, equal to the value of(-u) associated to aluminium at 100 keV. In contrast, when the source is polychromatic, the beam hardens itself in the object, causing a "cupping effect" (graph B) 9.4.2.3D tomography on a complex mechanical part Finally, we simulated a 3D tomographic system, with a matrix detector, consisting of 250 x 220 pixels. The whole volume of the object was reconstructed successfully. As the beam was polychromatic, cupping effect, as well as streaks, are visible in the reconstructed slices (fig. 9.7).

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Figure 9.6. Influence of Gaussian photon noise. Gaussian fluctuations have been added to the sinogram and to the reference image (obtained by performing an acquisition without sample) before reconstruction, (a): The photon number reaching the central pixel of the detector, during the first projection, is equal to 50. (b): The photon number is 10 times greater, (c): Grey level profiles corresponding to the lines labelled A and B

9.5. Conclusion and future directions This preliminary work phase allowed us to develop the framework of a computer code, able to produce within a short time realistic synthetic images, simulating the operation of radiographic, radioscopic or tomographic devices. The strong points of this tool are the use of CAD models to describe complex 3D objects, the ability to simulate geometric unsharpness and to deal with a polychromatic beam. The

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Figure 9.7. 3D computed tomography, (a): Schematic view of the simulated setup and CAD model of the sample. A matrix detector (250 x 220 pixels) and a point source were used. The dotted lines labelled A, B and C indicate schematically three slices presented on the right (b)

computed images present no photon noise, but Gaussian random fluctuations can be added afterwards, if needed. Several issues need further development and will be addressed in the next phase of our research program. As a priority, we envisage to enrich our simulation tool, by taking into account the scattering contribution. The 3D mapping of the absorbed dose in the sample will be broached simultaneously. Modelling and simulating the influence of the detector response on the image quality will also be an essential task. Finally, quantitative experimental validations will be necessary. Possible application areas are numerous. Simulation can be useful, when developing any specific application, to choose the best components, optimise the experimental parameters and save time by reducing the number of experimental tests. Simulation also presents an important potential, to test the performance of image processing procedures in a virtual environment, where all parameters are fully controlled.

9.6. References [BER 88] BERGER, M. J., "MIST Standard Reference Database 8, X-ray and Gamma-ray Attenuation Coefficients and Cross Sections, XGAM", distributed by Office of Standard

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Reference Data, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, ©1988. [BIR 79] BIRCH, R., MARSHALL, M., ARDRAN, G. M., Catalogue of Spectral Data For Diagnostic X-rays, The Hospital Physicists' Association, 1979. [CEN 98] CENDRE, E., Tomographie a haute resolution par rayons X, application a I'etude de la perte osseuse chez le sujet age, these de doctorat, Institut National des Sciences Appliquees de Lyon, 1998. [CUL 97] CULLEN, D. E., HUBBEL, J. H., KISSEL, L., "EPDL97: The Evaluated Data Library, '97 Version", UCRL-ID-50400, Vol. 6, Rev. 5, 1997. [DUV 98] DUVAUCHELLE, P., Tomographie par diffusion Rayleigh et Compton avec un rayonnement synchrotron: application a la pathologic cerebrale, these de doctorat, Universite Joseph Fourier - Grenoble 1, 1998. [PEL 89] FELDKAMP, L. A., GOLDSTEIN, S. A., PARFITT, A. M., JESION, G., KLEEREKOPER, M., "The direct examination of three dimensional bone architecture in vitro by computed tomography", Journal of Bone and Mineral Research, vol. 1, n° 4, p. 3-11, 1989. [INA 98] INANC, F., GRAY, J. N., JENSEN, T., Xu, J., "Human body radiography simulations: development of a virtual radiography environment", Part of the SPIE Conference on Physics of Medical Imaging, Vol. 3336, p. 830-837, 1998. [KAF 95] KAFTANDJIAN, V., Reconnaissance automatique de defauts dans les produits metalliques par radioscopie numerique, these de doctorat, Institut National des Sciences Appliquees de Lyon, 1995. [KAK 88] KAK, A. C., SLANEY, M., Principles of Computerized Tomographie Imaging, IEEE Press, 1988. [KOE 98] KOENIG, A., GLIERE, A., SAUZE, R., RJZO, P., "Radiograph Simulation to Enhance Defect Detection and Characterization", Proceedings of the 7tn European Conference on Non-Destructive Testing, Vol. 1, p. 444-451, 1998. [PEI 97] PEIX, G., BUFFIERE, J.Y., CARDINAL, S., CLOETENS, P., SALOME, M., PEYRIN, F., BABOT, D., "Caracterisation de rendommagement dans les materiaux de structure par tomographie haute resolution a rayons X", Revue des composites et des materiaux avances, Vol. 7, n° hors serie, p. 59-67, 1997. [SCH 98] SCHIABEL, H., SlLVA, M. A., C-LIVEIRA, H. J. Q., MARQUES, P. M. A., FRERE, A. F.,

"A computer simulation technique to preview the influence of the recording system on the image sharpness in mammography", Part of the SPIE Conference on Physics of Medical Imaging, Vol. 3336, p. 602-609, 1998.

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Chapitre 10

Micro focus computed tomography of Aluminium foams

X-ray microfocus computed tomography (CT) is a non-destructive inspection method that provides the true cross-sectional images of the internal details of a component. Due to their X-ray absorption properties, CT technique has been found to be very useful for the internal investigation and quality control of metal foams. Therefore, some non-destructive tests have been performed on closed cell aluminium foams produced by powder metallurgical and casting processes. This method provides a detailed microstructure characterisation of any cross-section of the sample as well as a three-dimensional (3D) visualisation of the specimen using backprojection algorithms. Moreover, digital imaging with CT technique enables the description of the deformation mechanisms within the foam at different strain values.

10.1. Introduction Metallic foams are porous metals with high porosity (from 50 to 99 %). In the past few years there has been a considerable increase in interest for metal foams, especially made of aluminium or aluminium alloys. The unique properties of aluminium foams like high stiffness to weight ratio, high energy dissipation, low density, reduced acoustic, thermal and electrical conductivity, chemical resistance, easy recycling etc..., make them potentially useful in many high technology industries such as automotive and aerospace. Therefore, they are currently used for energy impact absorbers [GIB 88, EVA 99], in lightweight structures (in the cores of sandwich panels) [DAY 83, GIB 88, KRE 99], heat exchangers [GIB 88, BAR 98, EVA 99], sound absorbers or silencers [DAV 83, GIB 88] and filters [GIB 88]. However, successful thermostructural implementation relies not just on their thermome-

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chanical properties, but on additional attributes such as manufacturing cost, environmental durability and fire retardancy [EVA 99]. Efficient use of foams requires detailed understanding of their mechanical behaviour. Therefore, simples equations related to their structural features and the properties of the material from which they are made have been derived [GIB 88, THO 75]. Thereby, computed tomography which is a non-destructive technique of characterisation, has been used for a detailed 3D description of the microstructure (which can not be achieved by a commonly used 2D characterisation technique) as well as for the investigation of deformation mechanisms within the foam at different strain values. Furthermore, input parameters for both the optimisation of production process and for the modelling can be expected as feedback.

10.2. Production process of Aluminium foams Aluminium foams can be produced by various methods [DAY 83, GIB 88]. For our tests two types of Al-foam samples with closed cell structure have been used. These specimens were made by a powder metallurgical process and by a continuous foam-casting route.

10.2.1. Powder metallurgical process Aluminium foam samples made by powder metallurgical route were produced by LK Ranshofen [KRE 99], a department of the Austrian Research Centres. The process, that is based on the Alulight®-technique, is described schematically in figure 10.1. Aluminium alloy is prepared by mixing metal powder in the appropriate relations. The foaming agent, titanium hydride or zirconium hydride is then added

Figure 10.1. Schematic illustration of the powder metallurgical process

for making metal foams

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to the mixture. The content of the foaming agent depends on the metal to be foamed and the desired density. The mixture, containing the agent, is then compacted by extrusion. As a result a semi finished product is obtained in which the foaming agent is homogeneously distributed within a dense, virtually non-porous metallic matrix. This foamable material is processed into pieces of the desired shape by rolling and cutting. Finally, foamed metal parts were obtained by heating the material to temperatures above the melting point of the matrix metal. The metal melts and the foaming agent releases gas in a controlled way, so that the metal transforms into foamy mass which expands slowly. The foaming takes place inside simple closed moulds which are completely filled by the foam. After the mould is filled, the process is stopped by simply allowing the mould to cool to a temperature below the melting point of the metal. The density of the metal foams is controlled by adjusting the content of foaming agent and by varying the heating conditions. The resulting foamed blocks have closed outer skins (figure 10.2) which can be removed by cutting them into samples of the desired dimensions.

Figure 10.2. Aluminium foams made by powder metallurgical process, the right picture is a sample with closed outer skins, and the left one is a sample without skins.

10.2.2. Continuous foam-casting process Another production method for aluminium foams is the continuous foam-casting route developed simultaneously and independently by Alcan and Norsk Hydro in the late 1980's [ASH 99, SAN 94]. A principle sketch of the process is given in figure 10.3. Gas is dispersed into small bubbles in an aluminium composite melt by rotor impellers. The walls of the created bubbles are stabilised by dispersed refractory particles avoiding coalescence between them. The bubbles rise to the surface where they accumulate. The accumulated foam on the melt surface is then transferred to a conveyor belt, where it solidifies and cools. The melt may constitute of different alloys and refractory particles. The most common alloys and particles are AlSiSMg (or equiv.), AlSiSMgCuNi with SiC particles and AA6061 (or equiv.) with A12O3

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particles. The particle size ranges from 10 to 30 um and their amount in the melt is comprised in the interval of 10 to 30 vol%.

Figure 10.3. principle sketch of the foam-casting route employed by Nosk Hydro

Figure 10.4. Aluminium foams made by casting process, the right picture is a sample with density 0.3 g/cm, and the left one is a sample with density 0.18 g/cm

Foams may be produced in densities from 0.1 to 0.5 g/cm3 by this method. The density is controlled by the process parameters, the most important being the rotor speed, the gas flow through the rotor and the amount of particles in the melt. At the present stage the production is directed towards slabs having typically 8-12 cm thick, 70 cm wide and 200 cm long; however samples of the desired dimensions (figure 10.4) can be cut out from these slabs.

10.3. Mechanics of foams A first impression of the mechanical behaviour of foams can be obtained by a uniaxial compression test, because in most applications, foams are loaded in compression. A typical compressive stress-strain curve is shown in figure 10.5. One

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can see an initial quasi-static increase of stress for small deformations followed by a long collapse plateau, truncated by a regime of densification where the stress steeply rises.

Figure 10.5. Schematic compressive stress-strain curve for an elastic solid and a foam made from the same solid, showing the three regimes of compressive behaviour and the dissipated energy at a given peak stress for both foam and the solid from which the foam is made

Gibson and Ashby [GIB 88] have developed models to describe the typical mechanisms responsible for the different mode of deformation of foams. The mechanical properties of foams can be described by equation [1] derived using dimensional analysis:

X

[1]

where X and p are the mechanical property and density of the foam, Xs and ps are those of the solid cell wall material and (|) is the volume fraction of solid contained in the edges, C and a are constants. Foams are especially good for energy absorption. Energy absorption in a foam is related to the area under its stress-strain curve in figure 10.5, energy absorption of a foam for a given peak stress is compared to that of the solid from which it is made. For the same energy absorption, the foam always generates a lower peak force, but the stress is limited by the long flat plateau of the stress-strain curve [GIB 88]. Thereby, by choosing the right cell wall material and relative density, best combination of the properties can be achieved for energy absorbers.

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10.4. Non-destructive investigation of Aluminium foams Previous models for mechanical performance of foams (equ.[l]) have focused on their dependence on the variation of the apparent density, the properties of the solid material from which the cell face are made, and the cell geometry. The geometrical feature include cell shape and size, distribution in cell size, defect and flaws in the cell structure as well as the microstructural parameter like cell wall and edges thickness and material distribution between cell face and cell edge. X-ray microfocus computed tomography (CT) is a non-destructive inspection method that provides the true cross-sectional images of the internal details of a component. Due to their X-ray absorption properties, CT technique has been found to be very useful for the internal investigation and quality control of metal foams. Unlike conventional microstructure characterisation methods [GRO 99], no previous preparation of the sample is required when using CT technique. This method delivers any cross-section of the sample as well as a three-dimensional (3D) visualisation of the specimen using back projection algorithms. Moreover, digital imaging with microtomography enables to the description of the deformation mechanisms within the foam at different strain values. 10.4.1. Principle of microfocus computed tomogrphy Microfocus computed tomography technique is based on X-ray attenuation inside an object [BRU 92 - MAI 89]. The attenuation depends on the material's atomic number, density and thickness of the sample and on the energy of X-ray beam. A diagram showing the basis of CT technique is given in figure 6. The object to be inspected is mounted on a turntable. Rotation and translations are controlled par stepping motors. A conical X-ray beam is generated from an X-ray source and focused on a detector located on the opposite side of the specimen. After traversing the sample, the beam is recorded by an X-ray CCD camera. Several projections are needed to fully construct the two-dimensional cross-section. There are obtained by rotating the object through 180 degrees. Conversion of these projections to the final image is conducted using a back projection algorithm [BRU 92].

Figure 10.6. Basis of method for X-ray microfocus computed tomography

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The CT images obtained by this method represent cross-sectional slices through the specimen in a horizontal plane perpendicular to the rotation axis of the turntable. Magnifications up to 100 are possible on small objects resulting in a high resolution (10 um) and therefore, very small details can become visible. 10.4..2. Results Samples of Al-foams made by both powder metallurgical (LKR) and casting (Norsk Hydro) processes were chosen for non-destructive testing. The typical dimensions of the specimens were approximately 50 x 50 x 50 mm3. The density of the Hydro samples were measured to be 0.31, 0.18, and 0.10 g/cm3 while the density of the LKR specimens were measured to be 0.35 and 0.22 g/cm3. The desktop X-ray microtomograph used for scanning contains a microfocus tube, an X-ray CCD camera and a Pentium computer with an integrated software package called TOMOHAWK running under Windows NT V4.0. The TOMOHAWK software developed and commercialised by AEK Technology Company in OXON (UK) provides a package of programs for calibration, data acquisition, processing, display and analysis of both computed tomography and digital radiography data. The samples were placed at such distances from the X-ray source and the imaging system to achieve a magnification of 2.3 and a resolution of 200 um. To acquire the information needed for tomography, the specimens are rotated for 180 degrees in steps of 0.5 degree. To increase the signal-to-noise (SNR) ratio, the image processing hardware is first used to digitise the real-time TV images into arrays of 512 x 512 pixels, each having 8 bits of intensity information. Integration of 256 TV frames is then used to increase the SNR by a factor of 16. The data collected as described above are processed to obtain the cross-sectional CT images using an appropriate filtered backprojection algorithm [BUR 92]. Several different sets of data at various heights can however be readily extracted simultaneously (up to 50 slices) from the integrated TV images. The total data acquisition time is about 70 minutes and the CT reconstruction lasts about 20 minutes. 10.4.2.1. Qualitative micro structure characterisation of Al-foams Figure 10.7 shows CT images of Al-foam samples made by Norsk Hydro casting process. The specimens have an average density of 0.10, 0.18 and 0.31 g/cm3. The pictures reveal a broad density distribution through the samples. Regions with a high density (figure 10.8) consist of spherical cavities embedded in the metallic matrix while for those with a lower density, individual cells can fell the presence of neighbouring cells during solidification [PRA 95].

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Figure 10.7. CT images of cross-sections of Norsk Hydro Al-foams; (a) sample with average density of 0.10 g/cm3, (b) sample with average density of 0.18 g/cm, (c) sample with average density of 0.31 g/cm

We can also distinguish from figure 10.7 that the cells are oriented in a preferred direction (perpendicular to the outer skins) and on the other hand that there is a broad cell size distribution. A qualitative description of the material distribution between cell walls and cell edges is shown in figure 10.9. Different situations that influence the mechanical properties of foams through the factor <j> in equation [1] are specified.

Figure 10.8. CT images of cross-sections of Norsk Hydro Al-foams showing regions of high density and others of lower density; (a) sample with average density of 0.18 g/cm , (b) sample with average density of 0.31 g/cm

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CT images of Al-foam specimens made by LKR powder metallurgical route are given in figure 10.10. The samples have an average density of resp; 0.22 and 0.35 g/cm3. Two slices at two different heights have been extracted for each sample.

Figure 10.9. CT images of cross-sections of Norsk Hydro Al-foams showing the material distribution between cell walls and cell edges: (a) concentration of the material in the cell walls, (b) concentration of the material in the cell edges, (c) equal distribution of the material between cell walls and cell edges

Cell size distribution seems to be homogeneous but the presence of big holes around the centre of the structure leads to a broad density distribution through the specimens. These holes might be caused by the foaming process itself because the latter is stopped by simply allowing the mould to cool. The temperature in the centre of this mould is still sufficiently high to enabling the foaming process to continue while it has stopped elsewhere. In figure 10.11 the image obtained by a zooming of the circled region (figure 10.11(a)) in the CT slice of the specimen, reveals a significant decrease of the resolution. To overcome this problem, a scanning of a smaller sample is done (figure 10.1 l(b)). Therefore, the size of the specimen has to be chosen carefully to achieve the required resolution.

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Figure 10.10. CT images of cross-sections of LKR Al-foams showing the presence of big holes in the foam structure: (a) sample with average density of 0.35 g/cm, (b) sample with average density of 0.22 g/cm3

Figure 10.11. C7 images of cross-sections of Norsk Hydro Al-foams: (a) picture obtained by a zooming of the circled region, (b) picture obtained by scanning a smaller specimen of 10 x 10 x 30 mm leading to a resolution of 90 [im

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10.4.2.2. Description of the deformation mechanisms CT technique has been found to be very useful for the investigation of deformation mechanisms of Al-foams at different strain values. Hydro samples of 20 x 20 x 30 mm3 have been strained up to 10% and scanned again to show the deformations that occur within the specimen, the CT slice of unloaded sample was taken as a reference.

Figure 10.12. CT images of cross-sections of Norsk Hydro Al-foams having typical dimensions of 20 x 20 x 30 mm3: (a) picture of the unloaded specimen, (b) picture after 10% straining showing the deformation mechanisms within the foam.

From figure 10.12 we can see that deformation was concentrated in a local area having the lowest density of the sample [KRI 99, BAR 98]. The main observed plastic deformation mechanisms were bending and buckling of cell walls, nodes which are centre of high concentration of metallic material were left unchanged in most cases, some of them seemed to undergo a slight rotation. Some cells were even found to be completely intact, they had only undergone a translation. 10% straining is perhaps too much to describe the onset of the deformation bands in the specimen, so in subsequent tests, samples will be strained at lower values (2 - 4 %) to identify the cells that initiate the bands and to provide a morphological explanation to the importance of band formation in plastic deformation. 10.4.2.3. Three-dimensional (3D) reconstruction of Al foam structure CT technique provides 3D information of the scanned specimen concerning cell arrangement like orientation, gradient, homogeneity, etc. Images of contiguous planes can be stacked by mean of a reconstruction algorithm, called T3D, to form 3D images of a section or if the entire part has been scanned, a full volumetric image of the specimen. T3D is a scientific visualisation application for graphically

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analysing volumetric data. It has been developed and commercialised by Fortner Software Company.

Figure 10.13. 3D reconstruction of Al-foam structure using T3D software: (a) 3D reconstruction of a Norsk Hydro Al-foam structure based on 50 CT slices, (b) numerical oblique slice through the 3D data volume

Figure 10.14. 3D reconstruction of Al-foam structure using T3D software: (a) 3D reconstruction of a Norsk Hydro Al-foam structure based on 50 CT slices with a numerical cut-out, (b) 3D reconstruction of an LKR Al-foam structure based on 50 CT slices with a numerical cut-out

TT

Once data is imported, the reconstruction algorithm can quickly generate slices in any orthogonal plane or at oblique angles through the 3D data volume (figure 10.13) as well as creating cut-outs in the reconstructed 3D foam structure (figures 10.14, 10.15). The presence of the big holes in the Al-foam specimens produced by LKR powder metallurgical process is clearly shown in figure 10.15

Figure 10.15. 3D reconstruction of an LKR Al-foam structure based on 50 CT slices, the presence of big holes in the structure is clearly shown

10.5. Conclusion Aluminium foams of similar relative density can exhibit a wide dispersion of the mechanical properties due to various effects, such as, gradient of density distribution, preferred pore orientation, cell size distribution and cell shape features [SIM 99, GRA 99, SIM 98, GIB 88]. Therefore a clever characterisation of these material is required. Besides the fact that microfocus computed tomography is a non-destructive technique of investigation, it brings a fundamental change to the microstructure characterisation by delivering 3D data as well as it provides a powerful tool to describe the deformation mechanisms within the specimen at different strain values.

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Future work will focus on the use of microfocus computed tomography for quantitative characterisation of Al-foams as well as the identification of the cells that initiate the deformation bands by straining the specimens at lower values (2-4 %). Furthermore, input parameters to support modelling work can be expected. Acknowledgements We wish to acknowledge the financial support of Renault S.A. Company. The authors would like to thank Claude Peytour-chansac and Christophe Grolleron from Renault S.A. for their support and useful discussion.

10.6. References [BUR 92] BURCH S.F., LAWRENCE P.P., "Recent advances in computerised X-ray tomography using real-time radiography equipment", British Journal of NOT, Vol. 34, n° 3, p. 129-133, 1992. [CEN 96] CENDRE E. et al., "High resolution X-ray computed tomography applied to bone structure characterization", Proceedings of the 14th Word Conference on Non Destructive Testing, p. 1211-1214, 1996. [SAS 98] SASSOV A., "Desktop X-ray microtomography", Proceedings of the 7h European Conference on Non Destructive Testing, p. 2837-2836, 1998. [MAI 89] MAISL M., REITER H., "Non destructive investigation of new materials and electronics by microfocal radiography and high resolution X-ray computed tomography", Proceedings of the 12th Word Conference on Non Destructive Testing, p. 1667-1672, 1989. [GRO 99] GROTE F., SCHIEVENBUSH A., "Characterization of cast and compressed foam structures by combined 2D-3D analysis", Proceedings of the International Conference on Metal Foam and Porous Metal structures, p.227-232, 1999. [GIB 88] GIBSON L.J., ASHBY M.F., Cellular Solids: Structure and Properties, Pergamon Press, 1988. [SUG 97] SUGIMURA Y. et al., "On the mechanical performance of closed cell Alalloy foams", Acta Mater., Vol. 45, n°12, p.5245-5259, 1997. [THO 75] THORNTON P.H., MAGEE C.L., " The deformation of Aluminium foams", Metallurgical Transactions A, Vol. 6A, p. 1253-1263, 1975. [BAR 98] BART-SMITH H. et al., "Compressive deformation and yielding mechanisms in cellular Al alloys determined using X-ray tomography and surface strain mapping", Acta Mater., Vol. 46, n°10, p. 3583-3592, 1998 [GRA 99] GRADINGER R., RAMMERSTORFER F.G., "On the influence of mesoinhomogeneities on the crush worthiness of metal foams", Acta Mater., Vol. 47, n ° l , p . 143-148, 1999. [SIM 98] SlMONE A.E., GIBSON L.J., "Effects of solid distribution on the stiffness and strength of metallic foams", Acta Mater., Vol. 46, n° 6, p. 2139-2150, 1998. [PRA 95] PRAKASH O. et al., "Structure and properties of Al-SiC foam", Materials Science and engineering A199, p. 195-203, 1995.

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[DAV 83] DAVIES G.J., ZHEN S., "Metallic foams: their production, properties and applications", J. Mater. Sci., Vol. 18, p. 1899-1911, 1983 [SAN 94] SANG et al., "Process for producing shaped slabs of particle stabilized foamed metal", US Patent 5334236, 1994. [EVA 99] EVANS A.G., HUTCHINSON J.W., "Multifunctionality of cellular metal systems", Proceedings of the International Conference on Metal Foam and Porous Metal structures, p.45-56, 1999. [KRE 99] KRETZ R. et al., "Manufacturing and testing of aluminium foam structural parts for passenger cars demonstrated by example of rear intermediate panel", Proceedings of the International Conference on Metal Foam and Porous Metal structures, p.23-27, 1999. [ASH 99] ASHOLT P., "Aluminium foam produced by the melt foaming route process, properties and applications", Proceedings of the International Conference on Metal Foam and Porous Metal structures, p. 133-140, 1999. [SIM 99] SlMANClK F., Reproducibility of aluminium foam properties", Proceedings of the International Conference on Metal Foam and Porous Metal structures, p.235-240, 1999. [KRI 99] KRISZT B. et al., "Deformation behaviour of aluminium foam under uniaxial compression (a case study)", Proceedings of the International Conference on Metal Foam and Porous Metal structures, p.241-246, 1999.

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Chapitre 11

3D observation of grain boundary penetration in Al alloys

Synchrotron radiation X-ray microtomography is used to investigate the penetration of liquid gallium into the grain boundaries of aluminium.Penetration and wetting processes play a key role in the understanding of liquid metal embrittlement.The large difference in absorption coefficients between Ga and Al allows micrometer and sub-micrometer thick liquid Ga films to be observed as they are wetting grain boundaries of Al. The tomographic reconstruction are compared to electron backscattering diffraction mappings recorded on the sample surface. Morphologic segmentation of the tomographic dataset opens the way to analyse and visualize the three-dimensional microstructure of the material.

11.1. Introduction The phenomenon of liquid metal embrittlement (LME) can be defined as the brittle fracture (or loss of ductility) of normally ductile metals and alloys when stressed while in contact with a liquid metal [PER 97]. LME leads to a significant deterioration of the mechanical properties of the solid metal: liquid Ga for example (melting point Tm= 29.8 °C) is known to induce brittle fracture in polycrystalline Al at stress levels far below the fracture strength of the material in air. Typically this deterioration is observed only over a limited range of temperatures near the melting point of the embrittling agent ('temperature trough' [NIC 79]). Strictly speaking, LME concerns metals and alloys under stress. However, it is known, that for some particular systems, intergranular penetration occurs even at zero applied stress. The observed kinetics are far beyond the rates expected for grain boundary diffusion. There is experimental evidence that the penetration may be considered as a wetting process which is governed by the ratio of the grain boundary energy ygb and the

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energy y sl of the liquid metal - solid metal interface. The equilibrium dihedral angle 6 at the solid metal is given by: 9 = 2cos"1(ygb/2ysl). Once the wetting condition 2ysl
Figure 11.1. a) grain boundary groove with equilibrium dihedral angle d >0 (TTJ c) Temperature dependance of surface and grain boundary energies d) resulting temperature dependance of equilibrium dihedral angle for different grain boundaries [Str 94]

However the formation of micrometer thick liquid layers cannot be explained within this simple framework and different models have been proposed to account for [Rab 97], [Gli 99]. As penetration and wetting are involved in the mechanism of LME, a more detailed characterisation of these processes might be considered as key steps towards a future understanding of LME. Up to now the occurrence of grain boundary wetting has mainly been detected by standard metallographic observation methods like Auger electron spectroscopy, TEM and SEM. Recently some in-situ transmission electron microscopical observations of the penetration process itself have been reported [Hug 98]. These observation methods provide excellent spatial resolution but are intrinsically limited to the sample surface or thin foils. Quite recently synchrotron radiation X-ray microtomography was applied for bulk observation of penetrated grain boundaries at the micrometer level [Lud 99]. The present paper gives an overview of results obtained with this new technique.

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11.2. Experimental set-up The experiments were carried out at the ID 19 beamline of the European Synchrotron Radiation Facility (ESRF). ID 19 is devoted to high resolution imaging and features an energy tunable (7-100 keV) X-ray beam of low divergence (~ 1 urad). The dedicated micro-tomographic set-up consists of a precision mechanics sample stage (rotation and translation) combined with a fast, high resolution detector system (Fig. 11.2). The detector system itself consists of a fluorescent screen (YAG: Ce) which transforms the X-rays into visible light and microscope optics to project the image on the cooled 10242 CCD camera, with a dynamic range of 14 bits, fast readout (0.22 s/frame) and low noise electronics (3 e-/s) [Lab 96]. The spatial resolution, limited mainly by scattering in the fluorescent screen, was determined with the knife-edge method to be 1.7 jam for the optical magnification used in our experiments (effective pixel size: 0.98 urn). The need for ultimate spatial resolution in order to resolve the Ga decorated grain boundaries of micron and submicron size restricts the maximum sample diameter to approximately 1 mm. In order to avoid beam-hardening artefacts, the incoming 'white' synchrotron radiation was monochromatized to 15 keV using a Rb-B4C multilayer with large energy bandwidth (AAA=10"2). Typical scan times (1000 projections over 180 degrees, 13 bit dynamic range) are of the order of 25 min. The samples were prepared from a piece of polycrystalline Al 5083 alloy (composition in at.%: Si (0.15), Fe (0.19), Cu (0.04), Mn (0.56), Mg (4.1), Cr (0.12), Ti (0.02)). The extruded material was annealed at 700 K for 24 h, then at 800 K for another 24 h. Optical inspection after surface etching (10 % HF) clearly

Figure 11.2. Schematic view of the experimental set-up showing multilayer monoehromator, sample rotation and translation stage and the detector system. The sample is rotated stepwise from 9 — 0° to 180° and 2D projections are recorded for each angular position 9. The spatial distribution of the linear attenuation coefficient in the sample can be reconstructed using a standard tomographic reconstruction algorithm

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revealed the columnar structure of the material. In the plane perpendicular to the deformation direction, grain size ranges from tens to hundreds of microns. Cylindrical samples with a diameter of 0.8 mm were machined from this material, with the cylinder axis parallel to the major axis of the elongated grains. In order to promote surface wetting by liquid Ga, the sample was dipped in a 10% NaOH solution for about 30 seconds in order to remove the Alumina layer covering the surface and a droplet of Ga was deposited on the surface immediately afterwards.

11.3. Results Figure 11.3 shows a reconstructed slice of a sample which has been penetrated by liquid gallium. Lighter gray-levels in the reconstructed images correspond to stronger absorption. The bright lines, dividing the image into smaller, closed cells can be clearly attributed to the presence of highly absorbing Ga (Z=31, attenuation length =19 |jm at 15 keV) layers inside the Al matrix (Z=13, attenuation length = 506 |tim at 15 keV). The spatial resolution of the used optical set-up is restricted to about 1.7 (am. However it is possible to detect features below this limit ('partial volume effect'): the locally enhanced absorption in a subregion of a volume element (voxel) will change the mean attenuation for the entire voxel. The fact that some of the wetted grain boundaries show only weak contrast might be attributed either to different composition or density of the absorbing material, or to a different (subpixel) size of the observed features. As Al shows a very limited solubility in Ga at low temperature [Mon 76] the differences in attenuation can not be explained by differences in composition of the liquid itself. As a consequence, the amount of liquid Ga present per unit volume must be different, or in other words, the width of some wetted grain boundaries is smaller than the pixel size. 11.3.1. Comparison with EBSD surface mapping As mentioned above, the thickness of the liquid layer varies considerably from grain boundary to grain boundary. A first attempt has therefore been undertaken to correlate the information from electron backscattering diffraction (EBSD) mapping (Fig 11.4c) before application of liquid Ga and microtomographic reconstruction of the same zone on the sample surface after penetration by liquid Ga (Fig. 11.4a,b). The sample was penetrated with liquid Ga and a first tomographic scan was performed after an initial anneal for 30 min at 50°C (Fig 11.4a). The second tomographic scan (Fig 11.4b) followed after an additional anneal of 60 min at 150 °C.

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Figure 11.3. a) Reconstructed tomographic slice (pixel size: 1 \jan) through a cylindrical Al sample which has been exposed to liquid Ga for 4h at 320K: the wetting of the grain boundaries by liquid Ga can be clearly detected; the magnified image shows the varying thickness of the Ga layer

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A series of circumstances contributes to the fact, that the contours observed with both techniques do not compare perfectly: a) corrosion and dissolution processes of excess liquid Ga on the sample surface may lead to changes in the size and shape of the surface grains, b) the tomographic cut defines a geometrically perfect plane whereas the (mechanically and electrochemically polished) sample surface analysed by EBSD does not, c) the spatial sampling rate of 5um in the EBSD mapping different from the pixel size of Ijam in the tomographic reconstruction d) spherical aberations of the electron microscope at low magnification leed to distorsions of the EBSD mapping. It is interesting to note, that grain boundaries which cannot be observed after the first anneal at 50°C, e.g. boundaries B'-B2 and C'-C2, are preferentially those of small relative misorientation (close colour codes in the EBSD mapping). These boundaries become visible after the second anneal at 150 °C - in accordance with the qualitaive model presented in Fig. 1 [Str94]. 11.3.2.3D visualization of a polycrystal The comparison of the tomographic reconstruction after the second anneal at 150°C with the EBSD surface mapping (Fig. 7.4b&c) shows, that even with the above mentioned limitations a good agreement is achieved. The tomographic reconstruction of a penetrated polycrystal might therefore be used to analyse and visualize the microstructure of the material in three dimensions. For this purpose the experimental raw-data need to be refined using morphological image processing methods. The applied three-dimensional segmentation algorithm, an implementation of the "watershed" method [Bou 99], transforms the experimetnal raw data (Fig. 7.5a) into binarised volume-data containing closed cells (Fig 7.5b). In a final step, a "label" (e.g. a number or a specific colour) is attributed to each voxel belonging to the same cell. The size and the shape of the individual grains in the polycrystalline material can now easily be analysed and visualized using a dedicated volume rendering software (Fig 7.5c). 11.4. Conclusions The present study is a new example showing the interest of synchrotron radiation X-ray microtomography as a non destructive bulk characterization method of materials at the micrometer scale. The presence of liquid Ga at the grain boundaries of polycrystalline Al leads to locally enhanced absorption and standard tomographic reconstruction methods allow to observe the wetting state of grain boundaries inside the material. The complementary information provided by micotomography and EBSD allows to correlate the wetting state of grain boundaries with the relative misorientation of the adjacent grains. We do not yet have enough data to draw quantitative conclusions, however we may confirm from our

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experimental results that, as expected, low angle grain boundaries are less penetrated than general grain boundaries.

Figure 11.4. Comparison of tomographic reconstructions (a,b) of the sample surface with electron backscattering diffraction mapping (c). The EBSD mapping was obtained before application of liquid gallium. Two tomographic scans were taken after the penetration: a) after an anneal of SOmin at 50 °C and b) after additional anneal of 60 min at 150 °C. Note that the low angle grain boundaries B'-B2 and C'-C2 are only detected after the second anneal

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Figure 11.5. a) tomographic reconstruction of same sample as in Figl after additional anneal of 2h at 300 °C. b) result of 3D morphological filtering and segmentation c) volume rendering of a small part of the sample Apart from the scientific interest in studying the penetration of liquid metals, one might also think of some practical use of the microtomographic observation of Ga decorated grain boundaries. Under the assumption that all boundaries are detected in the tomographic reconstruction, one can use the segmented volume-data in order to analyse and visualize the microstructure of a polycristal in three dimensions. This technique might for example be applied to study the influence of grain boundaries on the propagation of cracks in prefatigued aluminium alloys.

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Acknowledgements We would like to thank E. Boiler and P. Cloetens for their help during the microtomography experiments. We gratefully acknowledge useful discussions with J. Baruchel and Y. Brechet.

11.5. References [PER 97] FERNANDES P.J.L., JONES D.R.H., "Mechanisms of liquid metal induced embrittlement",International Material Reviews 42, p. 251, 1997. [NIC 79] NICHOLAS M.G., OLD C.F., " Review Liquid Metal embrittlement", Journal of Materials Science 14, p. 1-18, 1979. [Str 94] STRAUMAL B., GUST W., MOLODOV D., "Tie lines of the grain boundary wetting phase transition in the Al-Sn system", Journal of Phase Equilibria Vol.15, 4, p. 386-391, 1994. [Rab 97] RABKIN E., "Coherency strain energy as a driving force for liquid grooving at grain boundaries", Scripta materialia, vol.39, No.6, pp 685-690, 1998. [Gli 99] GLICKMAN E., NATHAN M., "On the kinetic mechanism of grain boundary wetting in metals", J.Appl.Phys.,85, p. 3185-3191, 1999. [Hug 98] HUGO R.C., HOAGLAND R.G., "In-situ observation of Aluminium embrittlement by liquid gallium", Scripta Materialia, 38, p., 523-529, 1998. [Lud 99]LUDWIG W., BELLET D., "Penetration of liquid gallium into the grain boundaries of aluminium: A synchrotron radiation microtomographic investigation", J. Mater. Science Eng. A, in press. [Lab 96] LABICHE J.C., SEGURA-PUCHADES J., VAN BRUSSEL D., MOY J.P., ESRF Newsletter 25 p. 41-43, 1996. [Mon 76] MONDOLFO L.F., Aluminium Alloys: Structure and Properties, Butterworths, London, 1976. [Bou 99] BOUCHET S., Segmentation et quantification d'images tridimensionnelles, rapport de stage 36me annee, Ecole des Mines de Paris, 1999.

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Chapitre 12

Determination of local mass density distribution

Closed cell aluminium based foams are chosen for performing digital transmission radiography and X-ray computed tomography (CT). Transmission radiography yields information on the heterogeneity of mass distribution, if two dimensional local mean mass density maps are computed. CT provides data of the three-dimensional mass distribution. Three-dimensional local mean density maps are proposed to identify hard and soft regions in a cellular structure and they are used for modeling the mechanical behavior of aluminium foams. High resolution CT enables a detailed structural analysis of the foam.

12.1. Introduction Industrial production of foamed metals [ASH 99, BAU 99, HOP 99, KRE 99, MIY 99, SEE 99] and metal matrix composites [DEC] is in progress for some specific applications. Being in a developmental stage there are no generally acknowledged specifications. Quality definitions for heterogenous materials have to be defined. One of the main structural features correlated to properties is the local distribution of the second constituent (either reinforcement or pores) within the matrix metal. The aim of this work is to develop a quantitative description of "heterogeneity" of such composites which can be applied for quality control. Foamed aluminium is chosen to demonstrate the determination of local mass distribution by transmission radiography or computed tomography. The recorded intensity distributions form the basis for the calculation of local mass density distributions. Since the density is the most important parameter which correlates the mechanical properties of a foam to those of the dense bulk material [GIB 97], density distributions can be used for the simulation of the mechanical behaviour and can be correlated to experimentally determined properties.

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12.2. Material Samples of foamed aluminium with closed cell structure were chosen for testing. The ALULIGHT material [MEP 96] of AlSiMg (AA 6xxx) wrought and of AlSil2Mg cast alloys were produced by LK-Ranshofen [KRE 99] and by SAS-Bratislava [SIM 99] and are originally covered by a skin of the matrix alloy, respectively its oxide.

12.3. X-ray radiography In transmission radiography the recorded intensities provide an integral information about the attenuation of the X-ray beam along its path, whereas in CT the recording of intensities from a large number of different views allows the mathematical reconstruction of the local attenuation at any point of the object [COP 94]. For both methods the attenuation values can be correlated to the mass density, if the material composition is known. The following X-ray sytems have been used: a Philips Tomoscan SR7000 medical tomograph to provide digital transmission radiography and computed tomography as well and a GAMMASCAN micro-CT system [BAM 97]. Medical CT can be applied to cellular aluminium yielding a resolution of about 0.7 mm (comparison with a metallographic light microscopic picture was given in [DEG 99]). The main features of the cellular structure can be revealed even in X-rayed cross sections of more than 250 mm in diameter. The GAMMASCAN 3d-tomograph with a 200 kV microfocus tube and a 12 inch area detector provide a resolution of about 1/1000 of the tested object's size (but not better than 10 um). High resolution technical CT is a powerful research tool to characterise structural features in 3D to correlate them to materials properties and their scatter [EVA 99, FOR 98] and to stimulate modelling [DAX 99]. Two dimensional (2D) local mean mass density maps of flat parts were derived from transmission images, where the pixel value p(/v) correlates to the mass density of the corresponding column traversed by the beam (Fig. 12.la). Averageing over a surrounding area of n x m pixels gives the local density value at the location of the central pixel. Calculating this average pa (rtj) for every point /v yields a local mean mass density distribution of the component:

where g(r ij-r ij) is the weight of each surrounding pixel. In Fig. 12.1bp(;c.) is a schematic onedimensional representation of the pixel values of the transmission image. A mean local density function pa(xi)is calculated by averaging over an intervall of [;t - 34, xi + 35] with a constant weigth function g = 1 /70.

nD efot eloca r sml idnemas ant ysi oi t d i s t r i bn u t i o7

16

Figure 12.1. (a) principle of calculating a local density value from a transmission image; (b) the density values p(x scheme) derived from the transmitted intensities t} (onedimensional are the result of a physical averaging over the column taversed by the beam. pa (xt ) illustrates the effect of mathematical averaging over an interval of Ax=70 pixel

CT experiments reveal a three dimensional (3D) data set of volume elements (voxels) containing the 3D mass distribution of the tested object. This data set is appropriate to investigate the cellular structure [GRO 99, COR 99] and to calculate a mean local density distribution [KRI 99a] by averaging over a certain surrounding volume of each voxel (Fig. 12. 2a). Considering a foam (or an MMC) the material is described by voxel values 0 in pores p(r ( p(rijk ) = pR for reinforcement) and ijk ) = p(r. jk ) = ps in cell walls, edges and nodes ( p(rijk ) = pM for the matrix). The mean local of the voxel functionlocated p at point rijk a (rjjk )density

is then defined

analogous to Eq. 1 as: p

Figure 12.2. (a) principle of calculating a local density value from a volumetric CT data set; (b) onedimensional representation p(xt) of a cellular structure and local density p(l(xj). Limits of the mean local mass density around the overall average density pm are indicated to discriminate "hard" (pa > pm + Ap) or "soft" (pa < pm - Ap) regions

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The averageing process neglects fine structural details and allows to determine local differences in the material distribution. Figure 12.2b shows a onedimensional model p(xi)of a cellular structure with massive walls of the solid density psand empty pores in between. Averaging the mass densities over intervals [*, - 34, ;c, + 35] with g = I/70yields the mean local density function pa(xi).

12.4. Results 12.4.1. Mass Distributions calculated from Transmission Radiographs

Figure 12.3. X-ray transmission of ALL/LIGHT plate: a) transmitted intensity of each pixel (0.6 x 0.6 mm2); b) contour plot of the 2D mean local mass density

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Figure 12.3a demonstrates a digital recording of the X-ray intensity transmitted through a part of a cellular wrought alloy slab (200 x 130 x 20 mm3), average density pm - 0.57 g/cm3) with skin. The intensity in each pixel oscillates strongly, which complicates the interpretation of local density variations. The contour plot of 2D local mean density is shown in Figure 12.3b varying from less than 0.2 g/cm3 to more than 2 g/cm3. It was computed by averaging the mass densities over areas of 11x11 pixels (6.4x6.4 mm2) as schematically shown in Fig. la. Although the original information is smeared over a given area, the averaging process reveals much clearer mesoscopic inhomogeneities in local mass distribution. The example reveals the drainage effect the sample suffered by foaming in an upright position. Some big pores (> 4mm) are visible too, where the mean mass density of a transmitted column of the averaging cross section amounts below one tenth of the average density of the whole sample.

12.4.2. Mass Distributions calculated from 3D-Tomograms 3D-CT recordings of cellular metals serve as basis for 3D density mappings. Fig. 12.4 depicts the mass distribution recorded by CT along 3 perpendicular cross sections of the same sample as shown in Fig. 12.3. Fig. 12.4a represents the mass distribution in each voxel. Fig. 4b shows the corresponding 3D mean local mass density maps for the voxels along these cross sections. The cells bigger than 4 mm can be clearly identified even in the 3D mean local density map. Mass density limits can be defined in 3D mean local density maps to distinguish regions with high or low mass density. Mean local density values of volumes of 2 x 2 x 2 m m 3 of an AlSil2-ALULIGHT compression test sample measuring 22x22x30 mm3 had been computed. A lower limit of 1.67-times the average density of the sample pm was chosen to discriminate "hard" regions of higher mass density. The iso-surface in Fig. 12.5a encloses the voxels with values above that limit. The density limit is reduced to 1.33-/?m and the averaging volume is increased 27-times yielding the iso-surface given in Fig. 12.5b. Figure 12.5a and b demonstrate the influence of the chosen density limit in combination with the averaging volume of the mean local density mapping. The influence of the selection of the averaging volume alone on the localisation of "soft" regions is compared in Fig. 12.5c and d, where the iso-surface at 0.67-pm encloses the regions of lower mean local mass density. The averaging volume is changed from a cube of 6 x 6 x 6 mm3 in Fig. 12.5c to a more than twice as big parallelepiped with one third of the edge lengths of the sample, i.e. 7x7x10mm 3 in Fig. 12.5d. The smaller the averaging volume the bigger the scatter in mean local density will be.

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Fig Ure l 2 4

, , ' - MaSS dmsity distribution on the surfaces of a parallelepiped recorded by technical CT: a) CT cross section images from the 3D data set (voxelsize 0.3x0.3x0 3 mm3) b) contour plots of 3D mean local density maps for an averaging volume of 4.3x4.3x4 3 mm> (15x15x15 voxels) for the voxels along the planes shown in (a)

The iso-surface presentations of the limits of mean local mass density reveal as well the 3D arrangement of the corresponding hard or soft regions across the sample Fig I2.5b shows a V-like interconnection of hard regions across the sample which will increase the compression resistance in the long direction. The soft regions of that sample - see Fig. 12.5c, d - are oriented rather parallel to the direction of compression suggesting, that they will not be identical with deformation bands [KRI yybj. Consequently the spatial distribution of hard and soft regions has to be considered too as an additional quality criterion. High resolution CT reveals details in the cellular structure as shown in Fig 126 The 3D iso-surface representation of the structure shows cell walls down to about 100 Mm thickness as well as tiny shrinkage pores in cell wall nodes. Such CT data can be used to study deformation mechanisms by computing 3D-displacements of structural features [FOR 98].

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Figure 12.5. Iso-surfaces of mean local mass densities in one AISU2-ALULIGHT test sample (a = b = 22 mm, c = 30 mm) of average density pm= 0.5 g/cm3: a) hard regions of mean local densities p > 1.67-pm in averaging volumes of 2x2x2 mm3, b) hard region of p > 1.33-pn, within 6x6x6 mm3, c) soft regions of p < 0.67- pm within 6x6x6 mm3, d) same density limit as c) but within 7 x 7 x 10mm3, i.e. 1/3 of the edge lengths of the sample

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Figure 12.6. Iso-surface at a level of 50% of pAI computed from high resolution 3D technical CT data (voxelsize: 40x40x 40 fjm3). The volume of 8x8x8 mm3 within a casting alloy ALL/LIGHT sample (same as in Fig. 5) shows a few cells, where the roughness of the cell walls, defects in cell walls, shrinkage pores in nodes and an ensemble of neighbouring cells can be seen

12.5. Application of the mean local density distribution X-ray transmission images of cellular samples are ambiguous in detecting big pores. Any conventional X-ray transmission system including medical CT is suitable for identification of density variations of cellular metallic parts of regular shape, based on 2D mean local density maps. 3D X-ray computed tomograms of even moderate resolution provide the basis for the calculation of 3D mean local mass densities to identify soft and hard regions and their spatial arrangement within the component. However the experimental results show, that the arrangement of these regions has to be considered when describing the mechanical behavior of foams (macroscopic anisotropic behavior). Fig. 12.7 shows the measured compressive stress-strain curves and the density distribution of two samples viewed from perpendicular directions. The density distribution is derived from medical CT measurements. The 2D local mean densities were calculated by averaging the CT data over columns having the thickness of the sample (20 mm) and a cross-section of 5 x 5 mm. The largest extension of the columns is oriented in y-direction (z-direction) for the xz-mapping Cry-mapping). The yield strength shows a variation of about 20%, although the average density of samples was fixed at 480 kg/m3. This high variation of yield stress can be explained by the density distribution. The sample A is characterized by the lower yield stress and shows a soft zone of low density, oriented nearly perpendicular to the loading direction; sample B has a higher strength and has a hard zone of high density

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parallel to the loading direction. Density variation limits can be chosen as quality criteria, but have to be combined with a reasonable choice of averaging volume.

Figure 12.7. Compressive stress-strain curves of Alulight (cast alloy) samples, both having an average density of 0.48 g/cm3 but different local density distribution as shown by mappings of these samples in xz and xy planes

Figure 12.8. The sample A: a) The cellular structure along the observed surface recorded by a digital camera at 2% overall strain. Deformation zones are marked by the white lines; b) distribution of the corresponding calculated equivalent plastic strain (PEEQ) on this surface

The assessment of the spatial arrangement of such hard and soft regions in mechanically loaded components can also be used for the meso-mechanical simulation of heterogeneous materials. The results of this simulation for foamed aluminium are presented, by demonstrating the calculated elastic-plastic behavior of a sample. A 3D density mapping which was calculated with an averaging volume of 5 x 5 x 5 mm3 was used in this finite element simulation. The detailed modeling has

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been described in [KRI]. Fig. 12.8b shows the distribution of the equivalent plastic strain [HIB 98] occurring on the sample's surface, which was observed in the experiment. The calculated maximum strain is in the same position as observed in the experiment by optical recording (Fig. 12.8a). The capability of the 3D model enables to follow the forming of deformation bands in the interior of the sample too. The calculated 3D plastic strain field for this sample is given in Fig. 12.9. Four stages, showing the growth of deformation bands, are depicted in this figure by indicating the regions having more than 1% equivalent plastic strain.

Figure 12.9. Simulated 3D propagation of plastic regions in the interior of the sample A at a) 0.7%; b) 1.0%; c) 1.23% and d) 1.5% overall strain. Direction x is the compression direction

Acknowledgements The authors gratefully acknowledge: the provision of ALULIGHT samples by Leichtmetall-Kompetenzzentrum Ranshofen (A) and Slovak Academy of Science, Bratislava (SK); the admission to use CT at the Department of Radiology, Division of Osteoradiology, University of Vienna (A) and the Ferderal Institute for Materials Research and Testing, Berlin (D). The work was funded by the Austrian Ministry of Science and Transport.

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12.6. References [ASH 99] ASHOLT P., "Aluminium Foam Produced by the Melt Foaming Route Process Properties and Applications", MetFoam99 a>, p. 133-140, 1999. [BAM 97] "Computertomographie", Leaflet, Federal Institute for Materials Research and Testing, Berlin (Germany), 1997. [BAU 99] BAUMGARTNER F., GERS H., "Industrialisation of P/M foaming process", MetFoam99 a>, p. 73-78, 1999. [COR 99] CORNELIS E., KOTTAR A., SASOV A., VAN DYCK D., "Desktop X-ray micro-tomography for studies of metal foams", MetFoam99 a), p. 233-240, 1999. [COP 94] COPLEY D., EBERHARD A., MOHR A., "Computed Tomography Part I : Introduction and Industrial Applications", Journal of Materials, vol. 46, no 1, p. 14-26, 1994. [DAX 99] DAXNER T., BOHM H.J., RAMMERSTORFER F.G., "Influence of microand meso-topological properties on the crash-worthiness of aluminium foams", MetFoam99 a>, p. 283-288, 1999. [DEG] DEGISCHER H.P., DOKTOR M., PRADER P., "Assessment of metal matrix composites for innovations - a Thematic Network within the 4th EUframework", Euromat 99 (to be published). [DEG 99] DEGISCHER H.P., KOTTAR A., "On the Non-Destructive Testing of Metal Foams", MetFoam99 a), p. 213-220, 1999. [EVA 99] EVANS A.G., HUTCHINSON J.W., "Mutifunctionality of Cellular Metal Systems", MetFoam99 a>, p. 45-56, 1999. [FOR 98] FOROUGHI B., "Study of cellular deformation of Al-Foam under Compressive Loading", Junior Euromat, 1998. [GIB 97] GIBSON L.J., ASHBY M.F., Cellular Solids : Structure and Properties, 2nd Ed., Cambridge University Press, 1997. [GRO 99] GROTE F., SCHIEVENBUSCH A., "Characterization of cast and compressed foam structures by combined 2D-3D analysis", MetFoam99 a), p. 227-232, 1999. [HIB 98] HIBBIT, KARLSSON and SORENSON INC., HKS ABAQUS/Standard user manual, Version 5.8, 1998. [HOP 99] HOPLER E., SCHORGHUBER F., SIMANCIK F., "Foamed aluminium cores for aluminium castings", MetFoam99 a>, p. 79-82, 1999. [KRE 99] KRETZ R., HOMBERGSMEIER E., EIPPER K., "Manufacturing and testing of aluminium foam structural parts for passenger cars demonstrated by example of a rear intermediate panel", MetFoam99 a>, p. 23-28, 1999. [KRI] KRISZT B., FOROUGHI B., KOTTAR A., DEGISCHER H.P., "Mechanical Behavior of Aluminium Foam Under Uniaxial Compression ", Euromat 99 (to be published). [KRI 99a] KRISZT B., KOTTAR A., DEGISCHER H.P., "Strukturanalyse von geschaumten Aluminium mittels Computertomographie", Metalle/Werkstoffwoche 98, Symposium 8, Ed.: R. Kopp, Bd. 6, p. 687-692, Wiley-VCH, 1999.

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[KRI 99b] KRISZT B., FOROUGHI B., FAURE K., DEGISCHER H.P., "Deformation behavior of aluminium foam under uniaxial compression (a case study)", MetFoam99 a), p. 241-246, 1999. [MEP 96] "Alulight", Leaflet, Mepura Ges.m.b.H., Ranshofen/Austria, 1996. [MIY 99] MIYOSHI T., ITOH M., AKIYAMA S., KITAHARA A., "Aluminuim Foam, 'ALPORAS': The Production Process, Properties and Applications", MetFoam99 a>, p. 125-132, 1999. [SEE 99] SEELIGER H.-W., "Application Strategies for Aluminum-Foam-Sandwich Parts (AFS)", MetFoam99 a>, p. 29-36, 1999. [SIM 99] SIMANCIK F., MINARIKOVA N., CULAK S., KOVACIK J., "Effect of foaming paramters on the pore size", MetFoam99 a\ p. 105-108, 1999. a)

Metal foams and porous metal structures, International conference, 14th-16th June 1999, Bremen (Germany), Ed.: J. Banhart, M.F. Ashby, N.A. Fleck

Chapitre 13

Modelling of porous materials evolution

By providing 3D images of the micro-geometry, synchrotron micro-tomography is offering huge possibilities to porous materials evolution modelling. Through three examples, reservoir rock diagenesis, carbon-carbon composite densification and ceramics sintering, this text illustrates those possibilities and puts into evidence the need of a strong theoretical framework. The volume averaging method is succinctly presented in the case of pressure solution in sandstone reservoirs. The fundamental concept of Representative Elementary Volume is introduced. Various problems specific of the considered materials are described and solutions to be used during the acquisition or in post-processing are outlined.

13.1. Introduction Understanding the evolution with time of natural or artificial porous materials is essential for many applications. In this paper, three examples that are presently studied within the research group CM3D* from Bordeaux, will be considered: - mineral diagenesis of reservoir rocks for oil exploration and production, - carbon/carbon (C/C) composites elaboration by vapour-phase densification of carbon-fibre preforms for thermostructural materials production, - sintering of advanced ceramics with controlled porosity distribution for mechanical properties enhancement (strength and toughness). In porous media physics the concept of "change of scale" is fundamental. The common objective of all the methods used to perform the change of scale (volume averaging, homogenisation, etc...) is to move from the local scale (the pore scale for the cases considered here) to a larger one where the porous material behaves as an

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equivalent continuous material characterised by effective properties. In this paper we will only make use of the volume averaging method. At the local scale everything is known; the equations and the associated boundary conditions governing the physical phenomena, the values of the physical parameters appearing in those equations and the geometry. Applying the change of scale operator to the problem defined at the local scale will make the equations of the large scale physics emerge. The effective properties appearing in those equations can be calculated from solutions of differential problems (eventually integrodifferential) stated at the local scale: the closure problems. Furthermore some methods, like volume averaging, provide conditions that must be verified to insure that the change of scale is possible. Depending on the treated problem, those conditions might be more or less difficult to verify, as illustrated by the three examples examined here. Mineral diagenesis of reservoir rocks is an evolution process for which the above-cited conditions are easily verified. Indeed, it is a very slow process that takes place within very large domains that are rather homogeneous. In this case the change of scale puts into evidence an evolution governed by physical phenomena at the pore scale but having a kinetics directed by fluxes at the large scale (fluid and mater fluxes). For C/C composite elaboration, the change of scale might be difficult if the size of the sample is too small with respect to the characteristic dimensions of the textile architecture or if the characteristic time of the chemical reactions responsible for the densification is too short. Practically, this last case is avoided because it produces reaction fronts and heterogeneities that drastically reduce the mechanical properties. For large enough samples, the change of scale is possible and all the remarks made for mineral diagenesis are valid. For small samples, direct simulation is still possible but one cannot define effective properties intrinsically related to the porous material. In particular, the equivalent properties would be affected by the boundary conditions applied at the "large" scale. In the case of sintering, particles are very small and the conditions for the change of scale are generally easily verified. Evolution, i.e. sintering, is caused by diffusion within the solid phase. This diffusion is governed by the local curvature of the interface and the kinetics is controlled by large-scale parameters, like temperature for instance. The corresponding coupling is weaker than for mineral diagenesis because those parameters are constant at the local scale (it is not the case for the local fluid flow for instance). Because of the differences pointed out above, the modelling strategy will slightly differ for each of these three examples. Nevertheless, a good characterisation of the 3D micro-geometry is always essential. For this purpose, computed microtomography (CMT) is a very well adapted tool, as it will be shown in the following paragraphs. In the first one, the volume averaging method is succinctly introduced

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through its application to an important mechanism of mineral diagenesis: pressure solution. The usefulness of CMT will appear clearly in this example and will be reinforced by the two ensuing paragraphs where preliminary but encouraging results will be given for C/C composites and ceramics sintering. 13.2. Evolution of sandstone reservoir rocks by pressure solution Rock deformation by pressure-solution is due to the variation of the chemical potential with stress; in some conditions, the stress concentration at the grain-tograin contacts (Figure 13.1) induces an accelerated dissolution of minerals in these zones and a precipitation of those minerals at interfaces under low stress. This microscopic mechanism is the origin of one of the three most important deformation modes of earth crust. More practically, research of a better understanding of this mechanism is justified by the fact that in deep hydrocarbon exploration (more than 3 km), compaction and cementation of the quartz matrix is the main cause of error in sandstone hydrocarbons reservoir quality evaluation. In the literature there are two main theoretical models for pressure solution: - the free-face pressure solution model adopted in this work, where dissolution occurs at the periphery of the contacts. This process leads to a reduction of the contact area, plastic deformation and finally collapse of this zone. - the water-film diffusion model where dissolution is supposed to occur within the grain-to-grain contacts. In this model, the existence of a thin water film able to support a high normal effective stress is necessary. Dissolved species are transported from the film to pore space by diffusion. 13.2.1. Local equations and volume averaging The pore scale configuration is pictured on Figure 13.2. The two phases, the fluid P and the solid a, are in contact at Apa their interface. A dissolved compound a is moving within the fluid phase by diffusion (to simplify the demonstration, convection is not considered here). The compound a can react with the solid phase at the interface. Equations [1] and [2] govern the transport at the local scale. —- = V.(DVCfl)

in the p phase

- npo . (DVCa) = k (Cfl - C*)

on the interface A po

[1] [2]

where Ca is the concentration of the compound a in the p phase, D its molecular diffusivity, t the time, k the reaction rate coefficient and C* the equilibrium concentration. C* is a function of pressure, temperature and stress.

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Figure 13.1. Tangential stress concentration in a simple spheres packing: all the zones having a stress value greater than 20% of the maximum are near the contacts

Figure 13.2. Local scale geometry and notations used in the text

The intrinsic phase average of the concentration is defined by: [3]

where V is the Representative Elementary Volume. Applying this operator to equations [1] and [2] and using the spatial averaging theorems [WHI 99], the following equation is obtained:

where GRAY's decomposition (equation [5]) of the local concentration has been used to distinguish the smooth part, the averaged concentration linked to transport at the large scale, from the spatial deviation produced by the reaction at the interface and by the interface itself.

In a similar way the equilibrium concentration is decomposed into two terms:

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Assuming that k is constant and introducing the specific surface av defined by:

equation [4] can now be written:

[8]

Equation [8] is still containing microscopic terms that have to be eliminated. Following the same way as [WHI 99], it can be inferred that the spatial deviation of the concentration can be represented by the following expression:

where f is a vector and s a scalar. Both are solutions of partial-differential problems, called closure problems, which have to be solved at the local scale:

f mainly takes into account the effects of the micro-geometry on diffusion and s takes into account the previous effects plus the effects of the chemical reactions on the interface. The problems [10] are similar to the closure problems classically obtained for diffusion-reaction in porous media, the specific effects of local stress concentration being carried by £*• Indeed, this term is not an unknown for transport equations. It is a local property of the interface that has to be calculated by solving an almost decoupled local mechanical problem. The transport equation at the large scale can now be written:

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where the effective diffusivity is given by [WHI 99]:

[12]

The local flux of matter can be expressed by equation [13]. A new version of the model, taking into account the existence of water films at the grain-to-grain contacts, is under development. When the new expression of the local flux will be available, it will be possible to compare the evolutions predicted by both models for the same starting geometry. In parallel, both models will be confronted to real data (§ 13.2.3) looking for arguments supporting a choice between them. 13.2.2. REV size of a Fontainebleau sandstone sample The concept of Representative Elementary Volume (REV) is central in the change of scale theory for porous media. However there is no explicit formula for the REV size ro, the main indication being that it has to be large enough compared to the local scale characteristic lengths (lp, la) and small enough compared to the largescale characteristic length (L): [14]

Using tomography data acquired at the ESRF (ID 19) for ELF-EP, the 3D microgeometry shown on Figures 13.3 and 13.4 has been reconstructed. It is the central zone of cylindrical sample (diameter of 6 mm). The voxels are cubic (edges of 10 jam) and the complete data set comprises 256 x 256 x 256 voxels. To estimate the REV size of this Fontainebleau sandstone sample for different physical properties, the following numerical experiments have been performed: - cubic volumes of different sizes have been extracted from the central region of the complete data set; - those volumes being considered as REV, three physical properties have been computed: - porosity values from the number of voxels belonging to the fluid phase, - effective diffusivity values using equation [12] after resolution of problem [10],

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-permeability values after resolution of a closure problem obtained by volume averaging of Stokes equations [BER 95]; - the results have been plotted as functions of the volume size (Figure 13.5).

Figure 13.3. Visualisation of the solid phase of a Fontainebleau sandstone sample

Figure 13.4. Visualisation of the fluid phase of the same sample

Figure 13.5. Evolutions of porosity, effective diffusivity and permeability with the size of the computational cell

Figure 13.5 is clearly demonstrating that it is possible to obtain, using CMT, 3D images of porous samples large enough to attain the REV size for various physical properties. 10 times the average grain diameter can be considered as a good

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approximation of the size of a cubic REV for simple granular porous media. The subsequent and complementary question is now: what is the required size for the voxel in order to characterise correctly the fluid-solid interface? The importance of this aspect is obvious when dealing with more complex porous media and with coupled phenomena like in the following example.

13.2.3. An example of pressure solution evolution from the ELLON field Among the numerous difficulties encountered when studying mineral diagenesis of reservoir rocks, two are rather specific: - phenomena occur at very different scales (from the pore scale -100 urn- to the basin scale -500 km) and are strongly coupled, - the initial state (-100 My) is not directly accessible. Having that in mind, the Ellon field (Alwyn area, North Sea) can be considered as exceptional. Indeed, in an early stage of its evolution this sandstone reservoir has been drastically and quickly modified by two diagenetic events [POT 97]: first a general and almost complete calcite cementation (porosity initially around 40% and about 3% after cementing) and secondly a localised dissolution of the calcite cement leading to an heterogeneous formation composed of two domains separated by sharp fronts (Figure 13.6). During the subsequent evolution, the cemented zone remained unchanged (Figure 13.7) and the uncemented zone has been modified by pressuresolution and quartz over-growths (Figures 13.8-9) giving an actual porosity of about 20%. Those processes mainly occurred at the pore level with small coupling with

Figure 13.6. Sandstone sample from the ELLON field showing the interface between the cemented (C) and the non-cemented (NC) zones

Figure 13.7. Reconstructed section (512x512 pixels of 6.5 um) of a sample at the C-NC interface. Black zones are pores, dark grey zones are quartz grains and light grey zones correspond to calcite

Modelling of porous materials evolution

Figure 13.8. Reconstructed section (512x512 pixels of 6.5 urn) of a NC sample. Black zones are pores and dark grey zones are quartz grains

Figure 13.9. Reconstructed (700x700 pixels of 1.8 urn) of sample. Quartz overgrows are visible. Pores are partly filled with

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larger scales (no water flow) and the initial state has been frozen by calcite cementation; it is why the Ellon field is an exceptional case to study. Examining Figures 13.7 and 13.8, it is evident that this sandstone is more complex than Fontainebleau sandstone. Porosity is larger but spatial distribution of this porosity is completely different. Both were rather similar quartz grains packing at the deposit (Figure 13.7) but the diagenetic mechanisms that took place have been different. The combination of classical CMT (Figures 13.7, 13.8) and local CMT (Figure 13.9) seems to be a promising solution to problems where more than one scale are relevant.

13.3. C-C Carbon/carbon composites are well known high-performance materials for thermostructural applications, such as rocket nozzles or aeroplane brakes, and their market is in appreciable extension. They are usually produced either by impregnation of a preform made of carbon fibres by pitches or mesophases or by vapour-phase densification of the same preform [NAS 99]. Chemical Vapour Infiltration (CVI) is a variant of the Chemical Vapour Deposition (CVD) process involving the cracking of gaseous species (precursors) which lead to the deposition of a solid phase on a hot substrate by heterogeneous reaction. For instance, a mixture of hydrocarbons and hydrogen is used to obtain a pyrocarbon deposit. The gaseous species are transported inside the preform by viscous flow or by diffusion.

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In isobaric-CVI (I-CVI) low pressures are required in order to avoid diffusional limitations and premature pore plugging. As high temperatures are required for the heterogeneous reaction, one has to deal with transport by Knudsen diffusion (also called Klinkenberg effect or slip flow), in addition to ordinary diffusion. The role of porous medium transport is fundamental for the monitoring and optimisation of the CVI process, and it may be easily understood that it depends in a rather complex way on the preform geometry. One of the possible ways to investigate precisely the properties of interest either for direct use (mechanical, thermal) or for processing (permeability, diffusivity, and specific surface...) is to obtain accurate data concerning the 3D architecture of real fibrous preforms at various stages of densification. For this, CMT is a powerful tool having a resolution compatible with the size of an isolated single carbon fibre (7-8 jam diameter). Furthermore, the 3D character of the method yields essential information about the connectivity of the porous and solid phases. It is possible, on the basis of the reconstructed 3D images where the interface between porous and solid phases is visible, to compute the essential geometrical properties [LEE 98] and effective transport properties (permeability [BER 95], diffusivity and Knudsen diffusivity [VIG 95] as well as mercury penetration curves for the porous phase [HAZ 95], thermal and electrical conductivity [QUI 93], stiffness tensor... [POU 96]). It is also possible to perform on such images simulations of the evolution of the microstructure under some constraints, such as matrix deposition or infiltration, and chemical or mechanical degradation. Images were collected at the ID 19 beam line of ESRF (European Synchrotron Radiation Facility). The outstanding quality of the secondary beam allows to image samples at resolutions as good as 0.8 jam. Here, a resolution of 1.8 urn has been chosen because of the wide size of the representative elementary volume (REV). Accordingly, the studied samples measured less than 2 mm in diameter. Even though, it is not claimed that 2-mm width images do contain a C/C composite REV. The problem that arises specifically when imaging C/C composites is linked to the low absorption coefficient of C at the frequencies of use, together with the important coherence of the beam, and the order of magnitude of the resolution. In such conditions, the X-ray beam reveals itself much more sensitive to phase shifts than to intensity absorption [CLO 96]. When the detector plane is placed roughly at one centimetre behind the sample, the image obtained after reconstruction displays an enhancement of the void-solid interface under the form of a bright-dark double band, while the interior of both phases are of approximately the same mean grey level (Figure 13.10). The double band is the result of an interference between the coherent rays passing close to the interface, through both phases, one of which (carbon) induces a phase shift.

Modelling of porous materials evolution

Figure 13.10. Image extracted from one slice of a reconstructed 3D image

of a C/C partially densified preform obtained by synchrotron X-ray CMT

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Figure 13.11. 3-D rendering of the treated sample. The circles focus on rounded fibre

tips

If phase contrast allows the immediate production of human-understandable images, it is not suited for subsequent computations. Nevertheless, the important information is contained inside the phase-contrast image, since for most of the interface, the bright layer is always on the same side of the interface (e.g. the fluid side) and the dark layer lies always on the other. A variety of methods may be designed to answer to the following question: for any point (voxel) of the discretized image, what is the phase it lies in? An image treatment sequence has been designed to extract pertinent information from C/C tomographs displaying a phase contrast structure enhancing the void/solid interface. After having applied the algorithm, the image is fully binary. However, some closed porosity remains, which is removed using a classical percolation algorithm. It is also possible to use a mask and recover the exact grey-level values of the pixels that touch the interface, in order to use subsequently accurate surface tessellation procedures. Phase contrast images, after this treatment, are now suited for subsequent physico-chemical computations at pore-scale. As an example, the image treatment suite has been applied to a 200x200x50 voxels sub-sample of a tomography taken at 2 urn spatial resolution (Figure 13.10). The result is shown on Figure 13.11.

13.4. Ceramics sintering Sintering is known as the process allowing the transformation of a powder into a compact material presenting at least some mechanical properties. The widest used model of solid state sintering considers the ceramic as ideal packing of facetted

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grains with its associated porosity situated on the grain edges [COB 61]. In this representation the process is completely defined by two parameters: the grain size and the density. In real systems, grain size distribution and initial packing inhomogeneities lead to internal stress gradients originating differential densification phenomena (constrained densification). Pore evolution should be considered and analysed independently of density and grain size changes. Sintering of a powder deposited on a dense and rigid substrate (Figure 13.12) of the same chemical nature, is an interesting model to investigate constrained sintering at the macroscopic scale.

Figure 13.12. Schematic of (a) a powder sintered without constraints (free sintering) (b) a powder layer sintered on a rigid substrate (constrained sintering)

A new phenomenological model of solid state sintering has been developed. In this model, pore size is explicitly considered [LET 94]. This parameter can be determined by image analysis of polished sections permitting a correct description of free and constrained sintering. If macroscopic shrinkage behaviour is correctly obtained, development of localised microstructural phenomena cannot be taken into account in this framework. For instance, large pore defects present in the compact can evolve and lead to de-sintering phenomena [LAN 89, HEI 94]. So, improvement of sintering modelling needs more detailed analyses of densifying phenomena, especially at a lower scale. It requires a more accurate description of porosity and CMT associated to 3D reconstruction techniques looked as the up to date technique to investigate pore structures in real ceramic samples at the appropriate scale (grain scale). Experiments were carried out on a well-known ceramic material, i.e. A12O3. The powder was a pure alumina powder (Baikowski DF 1200) and the average grain size was estimated to be 4 urn. A reference sample (initial state) was obtained by cold pressing this powder and then by sintering it at low temperature (1300°C, 1 min.) in order to get a minimum of cohesion without any significant change of the microstructure. The role of a constraint on the sintering and on the development of the porosity was investigated by comparing free sintered samples (powder naturally sintered) to constrained samples (powder layer deposited on a sapphire, i.e. dense alumina, substrate and sintered). The sintering temperature was fixed to 1600°C and

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the sintering times were 15 min, 30 min and 60 min. Due to the large initial grain size, relative density of the free samples remained low and was 62.5 %, 67 % and 68 % respectively. The constrained samples were assumed to exhibit even lower relative densities [LET 94].

Figure 13.13. 2D sections of alumina ceramics samples sintered during 30 min. at 1600°C (image size: 512x512 pixels, pixel size: 0.9 /jm2). Free sintering on left and constrained sintering on right (on left is the sapphire substrate)

Figure 13.14. 60 min. free sintered ceramic. Formation of a denser shell around a large pore is observed while no change in density can be noticed along the crack. The first effect may be attributed to differential sintering (localised constrained sintering phenomenon) as it has been predicted and observed in 2D within ceramic and composites [LAN 96]. Image processing analysis tools will be used to obtain 3D information on the micro structure around the pore to relate them to previous studies

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X-ray computerised micro-tomography of these alumina ceramics were performed on ID 19 beamline. All specimens were scanned at the two lowest resolutions (0.8 jam and 1.8 jam), using a monochromatic beam (17.5 keV) obtained using a bent multilayer device (strong decrease of the acquisition time). A control section was numerically reconstructed for each sample. Unfortunately, as it can be seen on Figure 13.13, phase contrast artefacts related to the low grain size (large number of solid-gas interfaces) superimposed themselves on the grey level image, preventing an easy separation of the two phases (pore/matter) at the micrometer scale. Nevertheless, numerical analysis of the 3D signal is planned to extract average density fields close or far from the substrate. Furthermore, some samples reveal interesting features justifying an deeper exploration: for example, a free sintered specimen (60 min.) shows two types of internal defects producing different microstructural modifications (Figure 13.14). 13.5. Conclusions and forthcoming works The three examples presented in this paper illustrate the huge possibilities offered by CMT to porous materials evolution modelling. Recent developments (phase contrast imaging, local tomography) increased notably the quality of the data that can be obtained and extended the domain of application to new fields. CMT is opening very promising perspectives but one must keep in mind that, because of the amount of data that is generated, an efficient use of this tool requires consequent intellectual and material investments. CM3D* is orienting its activities towards those goals: volume averaging is providing the theoretical framework for data exploitation, new codes are developed for data treatment and physical properties computation, new computing facilities will be available in near future (next camera will generate images of 2048 x 2048 pixels!!!). If data acquisition has been rather simple for rocks (§ 13.2), it has not been the case for C/C composites and ceramics (§ 13.3 and 13.4). More elaborated acquisition procedures are required there. Development and testing of those procedures will be possible only if the collaboration between users and beamline staff continues.

Acknowledgement All the micro-tomography data has been acquired at the ESRF (European Synchrotron Radiation Facility, Grenoble, France) on the ID 19 beamline. 3D * CM3D is a thematic research group from Bordeaux (Caracterisation et Modelisation 3D de 1'evolution des milieux poreux reels. Contact: [email protected]).

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reconstructions have been performed using VOLUMIC, a code developed by CREATIS (UMR5515, CNRS-INSA Lyon).

13.6. References [BER 95] BERNARD D., "Using the Volume Averaging Technique to Perform the First Change of Scale for Natural Random Porous Media", Adv. Methods for Groundwater Pollution Control, Gambolati & Verri Eds., p. 9-24, Springer Verlag, New York, 1995. [CLO 96] CLOETENS P., BARRETT R., BARUCHEL J., GUIGAY J., SCHLENKER M., "Phase Objects in Synchrotron Radiation Hard-X-ray Imaging", J. Phys. D : Appl. Phys., vol 29, p. 133-146, 1996. [COB 61] COBLE R.L., "Sintering Crystalline Solids. I. Intermediate and Final State Diffusion Models", J. Appl. Phys., vol. 32, p. 787-792, 1961. [HEI 94] HEINTZ J.M., SUDRE O., LANGE F.F., "Instability of Polycrystalline Bridges than Span Cracks in Powder Films Densified on a Substrate", J. Am. Ceram. Soc., vol. 77 [3], p. 787-91, 1994. [LAN 89] LANGE F.F., "Powder Processing Science and Technology for Increased reliability", J. Am. Ceram. Soc., vol. 72 [1], p. 3-16, 1989. [LEE 98] LEE S-B., STOCK S.R., BUTTS M.D., STARR T.L., BREUNIG T.M., KINNEY J.H., "Pore Ggeometry in Woven Fiber Structures: 0°/90° Plain-Weave Cloth Lay-up Preform", J. Mater. Res., vol 13(5), p. 1209-1217, 1998. [LET 94] LETULLIER P., Ph.D. thesis, n°1247, University Bordeaux I, 1994. [NAS 99] NASLAIN R., "Key Engineering Materials", CSJ Series - Publications of the Ceramic Society of Japan vol. 164-165, Switzerland, Trans Tech Pub., p. 38, 1999. [HAZ 95], HAZLETT R.D., "Simulation of Capillary-Dominated Displacements in Microtomographic Images of Reservoir Rocks", Transport in Porous Media, vol 20(1-2), p. 21, 1995. [QUI 93] QUINTARD M., WHITAKER S., "Transport in Ordered and Disordered Porous Media: Volume-Averaged Equations, Closure Problems, and Comparison with Experiment", Chem. Eng. Sci., vol 48, p. 2537, 1993. [POT 97] POTDEVIN J-L., HASSOUTA L., "Bilan de matiere des processus d'illitisation et de surcroissance de quartz dans un reservoir petrolier du champ d'Ellon (zone Alwyn, Mer du Nord)", Bull. Soc. Geol. France, vol 168 (2), p. 219-229, 1997. [POU 96] POUTET J., MANZONI D., HAGE-CHEHADE F., JACQUIN C.G., BOUTECA M.J., THOVERT J-F., ADLER P.M., "The Effective Mechanical Properties of Reconstructed Porous Media", Int. J. Rock. Mech. Min. Sci. & Geomech. Abstr., vol 33(4), p. 409-415, 1996. [VIG 95] VIGNOLES G.L., "Modelling Binary, Knudsen, and Transition Regime Diffusion inside Complex Porous Media", J. de Physique IV, vol 65(1), p. 159166, 1995.

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[WHI 99] WHITAKER S., The Method of Volume Averaging : Theory and Applications of Transport in Porous Media, Theories and Applications of Transport in Porous Media, V. 13, Kluwer Acad. Pub., Dordrecht, 1999.

Chapitre 14

Study of damage during superplastic deformation

Superplastic deformation of some industrial alloys can induce damage, leading to premature fracture. This damage by cavitation is generally divided in three main steps: nucleation of the cavities, their growth and finally coalescence between cavities. Up to now, this latter point has been poorly documented due to a difficulty to get reliable data with conventional techniques of 2D characterisation of damage. In this work, high resolution X-ray micro-tomography is used as a technique of quantification of the population of cavities due to the ability to obtain 3D information and a particular attention is given to the coalescence process. In the case of a superplastically deformed Al-Mg alloy, it is shown that coalescence occurs in a large strain interval and that just before fracture, most cavities are connected together. A parameter is proposed to quantify the coalescence process.

14.1. Introduction to damage in superplasticity Superplasticity is frequently defined as the ability for a polycrystalline material deformed at high temperature, to reach elongations to fracture at least one order larger than those obtained in conventional plasticity. Elongations larger than 1000 % may be obtained in the case of metallic alloys. A sample of aluminium alloy deformed in superplastic tensile conditions is presented in figure 14.1. Superplasticity requires both specific experimental (temperature and strain rate) and microstructural conditions. It is associated to the predominance of grain boundary sliding (GBS) as the main mechanism of deformation. The movement of grains during superplastic deformation is illustrated by figure 14.2, which displays a

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SEM micrograph of a Pb-Sn alloy of which a marker line was drawn on the polished surface before testing [DUP 97]. After deformation, the marker line inside the grains is not significantly modified whereas gaps are detected through the grain boundaries, which indicates that movements of grains are predominant.

Figure 14.1. Example of an aluminium alloy deformed in superplastic conditions

In consequence, superplastic properties are promoted by a reduction of the mean grain size of the material. For metallic alloys, grain sizes of about 10 microns are generally needed to exhibit superplasticity at strain rates compatible with industrial processes. From a rheological point of view, superplastic deformation occurs under very moderate flow stresses (typically less than 10 MPa) and is associated with a large plastic stability resulting from the high value of the strain rate sensitivity parameter m, deduced from the conventional viscoplastic law between the flow stress a and the strain rate e :

m a = Ke

[1]

Superplastic forming (SPF) is to day an industrial forming process to produce components with complex shapes. It concerns particularly titanium and aluminium and major applications have been developed in the aeronautical industry. However, in the case of single-phase materials, like aluminium alloys, superplastic deformation induces damage through the microstructure, leading to premature fracture but also to a reduction of service properties of the alloy after SPF. Such damage can be observed on figure 14.3, which shows a SEM micrograph of a superplastically deformed aluminium-magnesium alloy [LAR 98]. To day, industrial SPF overcomes this difficulty by forming components under superimposed pressure, which inhibits the damage process but increases the cost and limits the maximum size of the components to be shaped. In consequence, a way for the promotion of SPF in the future, is the ability to superplastically form aluminium alloys under atmospheric pressure. Previous works have demonstrated that some

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benefits in terms on damage sensitivity can already be obtained by appropriate heat treatments before testing [BLA 96]. Nevertheless, a better understanding of the damage process in superplastic conditions appears as a key parameter for the future development of SPF.

Figure 14.2. SEM micrograph showing the predominance ofGBS in superplasticity

Figure 14.3. Example of strain-induced cavitation when an aluminium alloy is superplastically deformed

The damage process induced by superplastic deformation is usually divided in three main steps: nucleation of the cavities, growth and coalescence leading to fracture. Cavity nucleation is attributed to microcracking or vacancy agglomeration and is mainly located at triple junctions or near intergranular particles through the alloy [RAJ 77], as a result of stress concentrations generated by GBS. Cavity growth is generally interpreted thanks to models initially developed for materials deforming in creep conditions [RIC 69, HAN 76, CHO 86]. For small cavities, diffusion is expected to contribute predominantly to cavity growth whereas for larger cavities, it is considered that growth is controlled by plastic deformation of the matrix surrounding the cavity. In this case, the variation with strain 8 of the volume V of a cavity is given by [HAN 76]: V = V0 exp(r|ge)

[2]

with V0 a constant and r|g the cavity growth parameter. Some expressions of the parameter r)g have been proposed in the past, in particular as a function of the rheology of the matrix [PIL 85]. However, it must be underlined that, since these models of cavity growth were initially developed for alloys deforming in creep conditions, they do not take into account GBS, although it is the predominant mechanism of deformation in superplasticity.

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Despite its crucial link with fracture, the coalescence process between growing cavities has been poorly documented [STO 83, STO 84]. Very limited experimental data have been reported and few models were proposed. Moreover, these models are generally based on very strong hypotheses about the spatial distribution of the cavities through the microstructure and the criteria of coalescence. In particular, they assume that the cavities are randomly distributed through the microstructure and that all the cavities have a spherical shape and the same diameter. Lastly, coalescence is supposed to occur only when cavities impinge. It is expected that such assumptions are not experimentally satisfied. As already mentionned, many cavities are preferentially nucleated near grain boundary particles, which are not randomly distributed through the microstrucutre. Moreover, the shapes of large cavities after superplastic deformation can be very irregular, as it is illustrated by figure 14.4, which shows an optical micrograph of the fracture zone for a superplastically deformed aluminium alloy. This irregularity of the shape of the cavities is attributed to GBS, which is the main mechanism of deformation in superplasticity. Experimental data about strain-induced damage in superplastic conditions are frequently quantified from the variation with strain of the cavity volume fraction Cv. Figure 14.5 displays such a variation of Cv with s in the case of superplastic deformation of Al-Mg alloy. The cavity volume fraction continuously increases with strain and after a period of apparent incubation in which the level of cavitation remains limited (less than 1 %), a sharp increase is obtained and the cavity volume fraction, when fracture occurs, is generally high in the case of superplastic alloys, typically more than 10 %.

Figure 4. Optical micrograph of the fracture zone after superplastic deformation of an Al-Li alloy

Figure 14.5. Variation with strain of cavity volume fraction Cv during superplastic deformation of an Al-Mg alloy

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The variation with strain of Cy is frequently rationalised according to: Cv = Cvo exp(Tiapps)

[3]

where CVo is a constant and r\app the apparent parameter of cavitation sensitivity. The logarithm of experimental values of Cv is plotted as a function of strain and in most cases, a straight line can be roughly obtained in a relatively large strain interval, which allows the measurement of a slope r|app. The corresponding value of r)app is then compared to those deduced from the cavity growth models. Indeed, these models predict also an exponential variation of the volume of the cavity with strain under two main assumptions : firstly, growth is controlled by plastic deformation of the matrix which surrounds the cavity; secondly, the number of cavities is roughly constant in the corresponding strain interval. The differences between such predictions and r)app are frequently discussed in detail and sometimes interpreted in terms of continuous nucleation of cavities during deformation. Despite the fact that these assumptions (in particular the constancy of the number of cavities per unit volume in a large strain interval) appear very questionable, it is difficult to draw conclusions with data deduced from conventional 2D techniques of characterisation.

14.2. Usual techniques of characterisation Two techniques of characterisation are generally used to quantify strain-induced cavitation in superplasticity: variation of relative density of the alloy and quantitative metallography from polished sections. Both techniques have some notable disadvantages. Density variation measurements are relatively easy to perform and allow detection of low cavity volume fraction, typically less than 0.01 %, but the measurements appear doubtful for large cavity volume fraction. Indeed, for such levels of cavitation, some cavities may connect with the outer surfaces of the specimen and artificially modify the results, leading to an apparent slackening of the cavitation increase with strain. Moreover, some precautions must be taken to systematically check the possible variation of density of the alloy during an heat treatment similar to that undergone by the sample during high temperature deformation [VAR 89]. Finally, this technique is a global one since only Cv can be measured whereas no data about the population of the cavities (number, size...) are available.

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By quantitative metallography, such data about the population of cavities can be obtained but it requires to polish the surfaces to observe, which can modify the apparent size and shape of the cavities, particularly in the case of aluminium alloys. Moreover, as already mentioned, irregular shapes of cavities can be detected in superplastically deformed alloys, as shown in figure 14.4. In such conditions, the study of cavity coalescence from two-dimensional data appears very hazardous. From these remarks, it appears very interesting to use a technique of threedimensional (3D) characterisation of the population of cavities in a superplastically deformed alloy. In consequence, high resolution X-ray micro-tomography seems to be a very promising technique since it can provide three-dimensional images of the bulk of materials [HIR 95, BUF 99].

14.3. Experimental procedure Tomography experiments were carried out at the ID 19 beamline of the European Synchrotron Radiation Facility (ESRF). ID 19 is devoted to highresolution imaging. The beam energy was 17.5 keV. The samples were set on a goniometer allowing a precise positioning of the sample. A scan of the samples consisting of the recording of 800 two-dimensional radiographs was performed during a 180° rotation around the vertical axis. Those radiographs were recorded on a 1024 x 1024 CCD camera developed at ESRF [LAB 96]. The average exposure time for a radiograph was 0.3 s and the whole scan lasted about 10 minutes. The pixel size of the camera was 2x2 (irn2. The detector was set 3 mm behind the sample. For each sample, the investigated volume was approximately 0.6 x 0.6 x 0.6 mm3, knowing that for large strains, the final thickness of the sheet after SPF in industrial conditions, may be less than 1 mm. In a first step of characterisation, only cavities with a volume larger than 10 voxels were taken into account. It is the reason why the interpretation of the results deduced from X-ray micro-tomography were focused on the coalescence process, in order to deal with relatively large cavities. The studied material was an aluminium-magnesium alloy (Al - 4.2Mg - 0.7Mn 0.2Fe - 0.1 Cr, wt %). The alloy was provided in the form of sheets of 2.5-mm thickness. The mean grain size was about 10 urn. The superplastic properties of this alloy have been investigated in detail [MAR 99] and the cavitation behaviour in superplastic conditions was studied by deforming the alloy at 525°C and 10"4 s"1 at different strains. In these conditions of deformation, the elongation to fracture was about 400 %. Density variation measurements were performed on a micro-weighing machine allowing detection of relative density variation close to 0.005 % and limited quantitative metallography on polished sections were also carried out on SEM micrographs.

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14.4. X-ray micro-tomography results Figure 14.6 shows a 2D section of the superplastically deformed alloy, deduced from X-ray micro-tomography data. Well-contrasted sections are obtained, resulting from the difference of X-ray attenuation between the cavities (in dark) and the aluminium alloy.

Figure 6. 2D section showing good contrast between cavities and the aluminium alloy.

Figure 7. Comparison of the variations with strain of Cv between X-ray microtomography and density variation measurements

Figure 14.7 compares the variations with strain of the cavity volume fractions deduced from density variation measurements and from X-ray micro-tomography. A good correlation between the results obtained by these two independent techniques is found. It confirms the validity of X-ray micro-tomography as a fruitful technique of quantification of strain-induced cavitation in superplastic alloys. Moreover, this technique allows to get information about the population of cavities as illustrated by figure 14.8, which displays the reconstructed image of the spatial distribution of the cavities after an elongation of about 170 %. This condition corresponds to a mean cavity volume fraction of about 1 %. For this elongation, most cavities are isolated through the microstructure, although some connections between cavities can be detected.

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Figure 14.8. Reconstructed image of the population of cavities after an elongation of about 170% (Cv »1 %)

14.5. Quantification of the coalescence process As already mentioned, the increase with strain of the cavity volume fraction is frequently described according relation [3]. A value of the parameter r|app is thus estimated and compared to those predicted by the cavity growth models. In the case of the investigated alloy, a value of r|app close to 5 was obtained which is agreement with previous works (IWA 91, FRI 96). Nevertheless, this value is significantly larger than the values predicted by cavity growth models (typically close to 2 in this case). This difference is generally attributed to continuous nucleation of cavities during strain, resulting in an apparent increase of the value of r\. To confirm this interpretation, the variation with strain of the number of cavities nA per unit area is frequently determined from SEM micrographs. This work was carried out in the case of the studied alloy [LAR 98] and an increase of nA with strain was obtained in the investigated strain interval. From the 3D data obtained in tomography, it is also possible to estimate the variation with strain of the apparent number of cavities par unit area for a given family of planes (as it is shown in figure 14.6). This estimation has been performed in the case of the investigated alloy and an increase of this apparent number of cavities per unit area is obtained whatever the planes of observation. Consequently, these results have confirmed those deduced from SEM micrographs, even if, for a given strain, the values of nA may depend on the technique of characterisation: nA is lower in the case of the tomography results, which can be attributed to the fact that small cavities (i.e. equivalent diameter smaller than 5 urn) have not be taken into account in the treatment of the X-ray tomography data.

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However, as already mentioned, when strain is increased, the cavities become very irregular and consequently, the interpretation of the associated variation of nA may be delicate. It appears more reliable to study the variation with strain of the number of cavities nv per unit volume. Figure 14.9 shows the variation with elongation of the number nv of detected cavities per unit volume. Between elongations from 200 % to more than 400 %, a continuous decrease of nv is obtained, which points out the extent of the coalescence process. The data obtained for an elongation close to 150 % has to be considered with caution since it must be kept in mind that only cavities with a volume larger than 10 voxels were taken into account in the procedure of counting. It means that an apparent increase of ny with strain can be partially attributed to cavity growth.

Figure 14.9. Variation with elongation of the number of cavities per mm

From the results presented in figure 14.9, it can be concluded that the effects of strain on the variation of nv and nA (deduced from SEM observations or from X-ray micro-tomography data) are contrary. These differences between the dependencies on strain of nv and nA confirm the fact that a the usual interpretation of the cavitation processes from 2D characterisations is very hazardous. The variation with strain of the number of cavities per unit volume is a first approach to quantify the coalescence process. However, this parameter gives only limited indications about the mechanism of coalescence. A way to get additional data is to follow the variation with testing conditions of the largest cavity in the investigated volume, since it may give indication about the extent of connection between cavities through the microstructure. In this view, figure 14.10 displays a 3D observation of the largest cavity obtained after an elongation of about 400 % for which Cv is about 14.5 %. It can be seen in figure 10 that the largest cavity admits a very irregular shape and extends beyond the studied volume. Moreover, this cavity

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corresponds to about 75 % of the total cavity volume fraction, which indicates that most cavities have connected together. From this conclusion, a coalescence parameter CP can be defined, according to: CP =

volume of the largest cavity — -x 100 total volume of cavities

Figure 14.10. Reconstructed image of the largest cavity through the microstructure after an elongation of about 400 %

Figure 14.11 shows the variation with strain of the coalescence parameter CP. The value of this parameter remains limited up to a strain of about 1.2 and then sharply increases. This strain corresponds, for the associated conditions of deformation (525°C and 10"4 s"1) to a value of Cv close to 5 %. These results confirm the importance of the coalescence process during superplastic deformation since they point out that coalescence occurs in a large domain of strain. It must be kept in mind that between 8 « 1.2 (i.e. an elongation « 230 %) and fracture, which is obtained after an elongation « 400 %), the mean cavity volume fraction Cv increases roughly from 5 % to 15 %. Consequently, it indicates that coalescence takes place not only in a large strain interval but also in a large domain of Cv.

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Moreover, from figures 14.9 and 14.11, it seems that two domains of coalescence may be identified. In a first step (for strains between 1.0 and 1.3), a significant reduction of nv is obtained whereas the value of the coalescence parameter remains low. It means that a large number of cavities coalesce but that the volume of the largest cavity through the volume remains comparable to the mean cavity volume of the cavities. It is expected that this step of coalescence is associated to a nonuniform spatial distribution of cavities. Indeed, as already mentioned, cavity nucleation takes place preferentially near intergranular particles which are not uniformly dispersed in the alloy, as a result of the thermomechanical process undergone by the material. Indeed, previous work has shown that, as a result of the thermomechanical process undergone by the material, some stringers of second phase particles are present in the studied alloy [LAR 98]. In a second step (when 8 > 1.3), the value of CP increases sharply. It indicates that a large fraction of cavities connect together, leading to one cavity which concentrates the coalescence process.

Figure 14.11. Variation with strain of the coalescence parameter .CP

Complementary data are nevertheless required before any definite conclusion about the relevance of these two steps of the coalescence process.

14.6. Conclusions High resolution X-ray micro-tomography appears as a very promising technique to characterise cavitation induced by superplastic deformation of industrial alloys. It confirms the difficulty to interpret experimental information deduced from conventional 2D quantitative metallography. In the case of an Al-Mg alloy, the coalescence process has been preferentially investigated since it is concerned with large cavities.

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Despite a quite limited number of experiments, significant results have been obtained. Coalescence occurs in a large strain interval, which means that the frequently used assumption that the number of cavities remains roughly constant during deformation is not valid. Just before fracture, most cavities are connected together. A parameter CP was proposed to quantify the degree of connection of the cavities in the superplastically deformed alloy and the variation with strain of CP suggests a coalescence process in two steps. Additional experiments are however needed to establish a reliable quantitative correlation between CP and the experimental conditions of testing of superplastic alloys.

14.7. References [BLA 96] BLANDIN J.J., HONG B., VARLOTEAUX A., SUERY M., L'ESPERANCE G., Acta Mater., vol. 44, p. 2317-2326, 1996. [BUF99JBUFFIERE J.Y., MAIRE E., CLOETENS P., LORMAND G., FOUGERES R., Acta Mater., vol. 47, p. 1613, 1999. [CHO 86] CHOKSHI A.H., J. Mater. Sc., vol. 21, p. 2073, 1986. [DUP 97]DUPUY L., DEA INP Grenoble, 1997. [FRI 96] FRIEDMAN P.A., GHOSH A.K., Metall. Mater. Trans., vol. 27A, p. 3827, 1996. [HAN 76] HANCOCK J.W., Metal Sc., vol. 10, p. 319, 1976. [HIR 95] HlRANO T., USAMI K., TANAKA Y., MASUDA C., J. Mater. Res., vol. 10, p.

381, 1995. [rwA91]IWASAKi H., HIGASHI K., TANIMURA S., KOMATUBARA T., HAYAMI S., Proc. of Int. Conf. on Superplasticity of Advanced Materials (ICSAM), p. 447, 1991. [LAB 96]LABICHE C., SEGURA-PUCHADES J., VAN BRUSSEL D., MOY J.P., ESRF

Newsletter, vol. 25, p. 42, 1996. [LAR 98]LARIVIERE D., DEA INP Grenoble, 1998. [MAR 99] MARTIN C.F., Thesis INP Grenoble, 1999. [PIL 85] PILLING J., RIDLEY N., Res. Mechanica, vol. 23, p. 31, 1985. [RAJ 77] RAJ R., Acta Metall., vol. 25, p. 995, 1977. [RIC 69] RICE J.R., TRACEY D.M., J. Mech. Phys. Solids, vol. 17, p. 201, 1969. [STO 83] STOWELLMJ., Metal. Sc., vol. 17, p. 1, 1983. [STO 84] STOWELL M.J., LIVESEY D.W., RIDLEY N., Acta Metall., vol. 32, p. 35, 1984. [VAR 89] VARLOTEAUX A., BLANDIN J.J., SUERY M., Mater. Sc. Tech., vol. 5, p. 1109, 1989.

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