1
Synchrotron Imaging and Diffraction for In Situ 3D Characterization of Polycrystalline Materials
1.1. Introduction
The last few years have seen material science progressing rapidly into three-dimensional (3D) characterization at different scales (e.g. atom probe tomography [PHI 09], transmission electron microscopy tomography [WEY 04], automated serial sectioning tomography [UCH 12, ECH 12] and X-ray tomography [MAI 14]). A wealth of 3D data sets can now be obtained with different modalities, allowing the 3D characterization of phases, crystallography, chemistry, defects or damage and in some cases strain fields.
In the last 10 years, one particular focus of the 3D imaging community (like 2D in its time with the advent of EBSD characterization) has been on obtaining reliable three-dimensional grain maps. As most structural materials are polycrystalline and the mechanical properties are determined by their internal microstructure, this is a critical issue. There has been considerable effort to develop characterization techniques at the mesoscale, which can image typically 1 mm3 of material with a spatial resolution in the order of micrometers.
Among 3D characterization, an important distinction exists between destructive and non-destructive techniques. Serial sectioning relies on repeated 2D imaging (which may include several modalities) of individual slices, where a thin layer of material is removed between each observation (see Figure 1.1(a,b)). The material removal can be achieved via mechanical polishing [ROW 10], ion [DUN 99, JIR 12] or femtosecond laser ablation [ECH 15] in a dedicated scanning electron microscope (SEM). Considerable progress has been made in this line in the last decade, bringing not only high-quality measurements in 3D of grain sizes and orientations but also detailed grain shapes and grain boundary characters. The most serious threat of serial sectioning is, however, the destruction of the sample.
In parallel, the advent of third-generation synchrotrons worldwide, with ESRF at the forefront, brought hard X-rays, with their high penetrating power, to the structural material science community. X-ray computed tomography (CT) rapidly developed as a key observation tool, allowing the non-destructive bulk evaluations of all types of materials [MAI 14]. This made the in situ study of damage possible using specifically designed stress rigs [BUF 10]. Unfortunately, CT imaging relies on absorption and phase contrasts and remains blind to crystal orientation. Accessing crystallographic information in the bulk of polycrystalline specimens (average orientation per grain) was subsequently achieved using the high penetrating power of hard X-rays and leveraging diffraction contrast. The pioneering work of Poulsen took advantage of high-brilliance synchrotron sources to study millimeter-sized specimens by tracking the diffraction of each individual crystal within the material volume while rotating the specimen over 360°. This led to the development of 3DXRD [POU 04] and its several grain mapping variants (DCT [LUD 08], HEDM [LIE 11], DAGT [TOD 13]). Among them, the near-field variant called diffraction contrast tomography (DCT, see Figure 1.1(c)) will be detailed in section 1.2.7.
Figure 1.1. Various examples of grain mapping techniques (spatial resolution indicated in brackets): (a) serial sectioning through mechanical polishing (0.45 µm in the XY plane, 1.48 µm in the Z direction) [ROW 10], (b) tribeam (laser ablation, 0.75 µm) [ECH 15], (c) DCT near-field imaging (1.4 µm) [PRO 16a], (d) labDCT reconstruction with grain shape and orientations (5 µm, courtesy of Xnovotech). For a color version of the figure, see www.iste.co.uk/brancherie/microstructure.zip
This chapter is organized as follows: the fundamentals of 3D X-ray characterization of structural materials are reviewed in section 1.2 from pure absorption contrast to diffraction contrast tomography. Then, a recently developed stress rig adapted for DCT is presented in section 1.3. Finally, section 1.4 details how to use experimental 3D grain maps in finite-element crystal plasticity calculations.
1.2. 3D X-ray characterization of structural materials
1.2.1. Early days of X-ray computed tomography
As most materials are opaque to optic light, people have long relied on the observation of object surfaces and/or destructive cut to assess various evolutions of physical phenomenon or diagnostics. With the advent of X-rays in the 20th Century, both easy to produce and capable of penetrating significantly through matter, a large number of works have been devoted to develop a technique that would allow to see the invisible.
The theoretical bases of tomography were laid down by J. Radon in 1917 (see section 1.2.4), and the first part of the 20th Century saw the development of what we call today conventional tomography. This allowed to record sectional image through a body by moving X-ray source and the film in opposite directions during the exposure. However, the images remained difficult to interpret.
The development of computer power in the second part of the 20th Century let G. Hounsfield build the first computed tomography (CT) scanner in 1971. At that time, the spatial resolution achieved was about 3 mm and the resolution of the images produced after reconstruction by a Data General Nova 16 bits minicomputer was 80 × 80 pixels. The scanner was installed in an English hospital, where the first brain scan was performed on a patient.
In the United States, A. Cormack independently developed a similar system, and both of them received the Nobel Prize in Medicine in 1979 for their invention. Since then, CT scanning has been used in medical practice as a powerful tool to help diagnostic.
1.2.2. X-ray absorption and Beer Lambert's law
In the 1-100 keV1 energy range typically used for material science investigations, the interactions of an X-ray photon with matter are largely dominated by photoelectric absorption and elastic Thomson scattering. Unless specific conditions for diffraction are fulfilled (see section 1.2.7), the main contrast in a radiograph is due to absorption.
This phenomenon of primary importance non-restricted to X-rays was first observed by A. Becquerel in 1839 and later explained by A. Einstein in his famous 1905 paper using quanta of light (that we now call photon) conceptualized by M. Planck in 1900. This essentially allowed to explain that the electromagnetic field is quantified and can interact with electrons of the outer shell of an atom with the net result of being absorbed and that a photoelectron is emitted isotropically (fluorescence, Figure 1.2(a)). The absorption is thus related to the probability of the X-ray photon penetrating some material for being absorbed and will therefore strongly depend on the incident photon energy E and the number of electrons or equivalently the atomic number Z of the material.
Figure 1.2. (a) Absorption of an X-ray photon by an atom, and emission of a photoelectron (fluorescence); (b) attenuation of an X-ray beam of intensity I0 by a piece of material characterized by its absorption coefficient µ(x)
To quantify this effect, the linear absorption coefficient (µ) is introduced. By definition, µdx is the attenuation of the beam through a very small section of material of thickness dx lying at the distance x from the surface (Figure 1.2(b)). Thus, we have:
[1.1] Using the conditions I(x = 0) = I0 and I(x = L) = I leads to the Beer-Lambert law:
[1.2] If the attenuation of the object is not uniform, the Beer-Lambert law can be derived similarly with an integral:
[1.3] It is experimentally observed that the µ scale is approximately like Z4/E3 and displays strong jumps in energy (called absorption edges) when crossing the corresponding atom-electron binding energies (for instance, K edge is 1.56 keV for Al and 88.0 keV for Pb). In practice, the mass attenuation coefficient µ? = µ/? has been precisely measured and tabulated by crystallographers for all types of materials [SEL 93]. Reliable data sets are publicly available, such as the ones published by the NIST2. An example is plotted for both Al and Pb in Figure 1.3, which clearly shows why lead is used as shielding for X-rays: at 20 keV and for a thickness of 1 mm, the radiation is totally absorbed by the Pb sheet, whereas 42% of the photons have crossed in the case of Al.
Figure 1.3. X-ray attenuation through 1 mm of metal: Al vs Pb, note the absorption edges visible on the evolution of µ?. For a color version of the figure, see www.iste.co.uk/brancherie/microstructure.zip
1.2.3. X-ray detection
Once the radiation has been attenuated by the sample, it must be recorded in some way to form an image. Silver-coated photosensitive films have long been used for X-ray experiments (and are still in used in many cases for radiology). All detectors now produce digital data of the recorded signal. Three main X-ray detection technologies producing 2D images are usually distinguished:
