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Phillipp Schneeberger
X-ray diffraction (XRD) is a popular non-destructive qualitative and quantitative technique aimed at characterizing crystal lattice parameters (Drickamer et al., 1967), local strain (Gailhanou et al., 2007), microstructure evolution (Oliveira et al., 2022) or phase constituent proportions (Peng et al., 2005) from analysed specimens (e.g. metals, polymers and ceramics). Although XRD has been primarily emphasized as an efficient tool for qualitative analyses, it is often used to perform quantitative measurements of the phase concentrations within a material. The Rietveld refinement method (McCusker et al., 1999) is generally applied to conduct quantitative analysis of XRD patterns, but it requires the diffraction profiles for all possible phase constituents to be collected appropriately during the preparation stage, so that the individual components can be adequately identified afterwards. From a practical point of view, this preparation is rather demanding for (complex) heterogeneous specimens. Moreover, the preferred orientation effects of XRD measurement (Dickson, 1969; Campbell Roberts et al., 2002) are extremely difficult to deal with experimentally.
In this work, a proper orthogonal decomposition (POD) algorithm, suitably extended to incorporate inequality constraints such as positivity (and referred to as positive-POD or p-POD), is proposed to circumvent the aforementioned challenges: For all phase constituents whose diffraction profiles are experimentally available (denoted `known constituents'), the POD technique (Chatterjee, 2000) is first applied to construct the experimental diffraction spectrum while taking the preferred orientation effect into account. Then, by enforcing positivity constraints, the phase concentrations for the known constituents can be estimated through a quadratic minimization with convex positive constraints using the sub-gradient projection algorithm (Boyd et al., 2003). Finally, the phase concentration and experimental diffraction data for the unknown constituents can be obtained.
At a given load, the XRD profiles are recorded, scanning through the specimen along the tensile direction, i.e. across the strain localization bands, so as to elucidate the on-going phase transformation(s) from the progressive changes in the diffraction spectra. For NiTi shape-memory alloys, depending on the forming process and chemical composition, when subjected to mechanical loads the R phase sometimes appears as an intermediate phase in a two-step phase transformation. It usually co-exists with austenite at the macroscopic scale (whatever the stress or thermal load), which impedes the measurement of its individual diffraction spectrum. In contrast, it is possible to find specific conditions for which pure austenite or martensite phases exist in the specimen.
The missing R-phase diffraction pattern and the undetermined preferred orientation of martensite variants prevents the Rietveld refinement method from achieving any comprehensive results. To overcome this limitation, the proposed p-POD algorithm permits the estimation of the concentration of different phases along the sample and provides an estimated R-phase diffraction spectrum. The proposed method is extremely versatile, since it requires neither complete knowledge of the diffraction data for all constituents nor challenging experimental processing to remove the signature of preferred orientations in the specimen.
The paper is organized as follows. Section 2 presents the combined in situ XRD and digital image correlation (DIC) measurement setups and the associated strain fields and raw diffraction spectra acquired during 1D tensile loading. The Rietveld processing method is recalled briefly in Section 3. Section 4 introduces the positive POD algorithm to conduct phase field reconstruction. Section 5 applies the proposed algorithm to the spectra of NiTinol recorded in scans along the tensile axis at different stages of loading. Section 6 draws some conclusions.
The raw spectra collected at the four interruption stages are plotted in Fig. 5. For the sake of readability, here and in the following an offset proportional to the y coordinate of the studied spot is added to the spectra so that they do not overlap.
The crystallography of NiTinol alloys has been extensively investigated. With the published crystallographic information [A and M (Bhattacharya, 2003), and R (Zhang & Sehitoglu, 2004)], it is possible to use powder diffraction theory (Cullity, 1956) to construct theoretical diffractograms (as illustrated in Fig. 6) and provide qualitative characterization of the phase constituents present at the four different stages.
With careful sequential Rietveld refinement, thermally induced martensite detwinning diffraction profiles and associated concentrations can be accurately estimated, as reported by Oliveira et al. (2021). However, in the present case, the combination of missing knowledge of the R diffraction profiles and possible preferred orientation effects due to martensite detwinning makes it very difficult to use the Rietveld refinement method to conduct any reliable quantitative analysis. To highlight these difficulties better, and how we propose to circumvent them with the positive POD algorithm, the Rietveld method and its limitations are briefly recalled in the following section.
Several assumptions are made here: diffraction profiles between different phase constituents follow a natural mixture law, and other XRD parameters are known (e.g. background, FWHM, asymmetry parameters, unit-cell dimensions, preferred orientation etc.). Rietveld refinement aims to determine the phase concentration through a nonlinear least-squares fitting,
(i) Noise. The quadratic cost function implemented in Rietveld refinement rests upon the assumption that the noise in the XRD measurement follows a Gaussian distribution with a uniform variance (in 2θ) for to be optimal. However, for X-ray detectors, it is commonly reported that noise follows a Poisson distribution. In such a case, the chosen quadratic cost function in Rietveld refinement is not `wrong' but neither is it optimal. Hence the nature of the measured noise needs to be characterized first and the cost function needs to be adapted accordingly, with a weight proportional to the inverse intensity.
(ii) Correlated parameter fitting. To the best of the authors' knowledge, most commercial program codes using Rietveld refinement require a sequence of parameter refinement (background, unit-cell crystal parameters, asymmetry parameters, preferred texture factor etc.) to reach the sought phase concentration. Therefore, when dealing with data from a mixture of different phases with a pronounced texture preference, the nonlinear least-squares fitting is prone to secondary minimum trapping, and the set of correlated parameters (concentration, preferred orientation) are ill-estimated. Thus, simplifying the Rietveld refinement protocol and determining the phase concentration with fewer model parameters is appealing.
If the crystal lattice orientation of the phase is unique and represented using a rotation quaternion denoted n1, the diffraction profile is denoted , which can be seen as the set of intensities of the different hkl Bragg diffraction peaks.
(i) Equation (9) can be considered as the generalized formulation of the experimental diffraction profiles of a pure phase (either a single crystal or a polycrystal with a pronounced texture preference are particular cases).
(ii) Hereinafter, is used to represent the XRD profile for a single phase with a particular orientation ei (after convolution accounting for instrumental acquisition). In this respect, other model parameters are no longer to be identified independently for phase concentration determination. Thus, the number of unknowns in the refinement is drastically reduced.
(iii) When the preferred orientation effect is present in the tested sample, it requires at least two (and at most three) sets of preferred orientation factors; hence two or three diffraction profiles are needed for the same phase to guarantee a trustworthy phase concentration estimation. The preferred orientation correction implemented in the Rietveld refinement method is a first-order correction considering the principal crystal lattice orientation
In the following, a similar methodology is applied to determine the unknown R-phase spectrum: assuming that at this stage the austenite and martensite spectra are already known, by selecting the appropriate ratio between the pressure and penalty terms, the austenite/martensite XRD contributions can be extracted from the experimental diffraction spectrum while the residual is expected to be `positive'. Thus the residual can be further interpreted as the missing R-phase spectrum weighted by its concentrations.
In the following, the discussed test case is such that three phases and only three are expected in this material. With the additional assumption that their orientation is transversely isotropic, it would be expected that no more than three modes are needed to account for the entire set of data. This obviously does not mean that the angular modes Un(2θ) for n = 1, 2 or 3 should coincide with the pure-phase spectra, but rather that the linear combinations of these three modes should be sufficient to match any composition of the three phases.
This simple presentation rests on the assumption of a unique type of orientation distribution per phase. Here transverse isotropy is the most likely, but a single orientation or an isotropic distribution would also lead to the same result that no more than three modes are needed to account for all acquired experimental spectra (with the exception of noise which can be considered as an additional mode). In cases where the orientation or orientation distribution is evolving along the loading direction, then more modes should be added with a maximum of three modes per phase, as discussed above (neglecting symmetries that can reduce this number).
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