Towards lab-based X-ray scattering tensor tomography with circular gratings

Reinforcement of the polymer matrix in injection-molded plastics with ﬁbers of various kinds can improve their mechanical properties. The resulting polymer composite material provides exceptional load-bearing capabilities with ﬂexibility in the molding of complex-shaped components. Mechanical properties of short-ﬁber-reinforced polymers are highly dependent on the ﬁber orientation in the matrix. Lack of maturity of injection molding simulation frameworks limits prediction of the complex ﬂow patterns during injection molding, while the requirement for resolving individual ﬁbers in conventional imaging methods sig-niﬁcantly restricts the imaged ﬁeld-of-view. Here we demonstrate 3D characterization of the ﬁber orientation in a complete FRP component enabled by a novel technique of X-ray tensor tomography. The obtained ﬁber orientation reconstruction can be efﬁciently used in reverse engineering applications, serving as a reference for simulations.


Introduction
Load-bearing capability is one of the main properties defining the performance of structural materials.Fiber-reinforced polymers (FRP) can deliver enhanced load-bearing performance using lightweight materials and are currently used in most structural engineering applications ranging from aerospace to medical applications [1].FRP components are usually made by adding fibers of different materials, such as glass and carbon, into a resin matrix and using molding processes in manufacturing complexshaped components.Unlike conventional structural materials such as steel and concrete, FRPs have anisotropic mechanical properties due to the high aspect ratio between the fiber length and diameter, with the highest mechanical properties of FRPs in the direction of the fiber alignment.Depending on the processing conditions, injection molding can result in complex flow patterns of the injected material, diverse fiber orientations, and unavoidable weld lines.Prediction of the flow and fiber orientation is a non-trivial problem that requires substantial computing resources and advanced numerical modeling [2].Characterization of the fiber orientation and validation of the simulation results has conventionally been performed using Xray computed tomography (XCT) [3].For examining fiber orientation, XCT reconstructions are required to resolve individual microfibers, limiting the imaged field-of-view by a few cubic millimeters.The recently developed technique of X-ray tensor tomography enables accessing information about anisotropic microstructure over a large (cubic centimeters) field-of-view [4].The contrast mechanism that enables this is the sample-induced small-angle X-ray scattering (SAXS).Grating interferometry can be utilized to measure the small-angle X-ray scattering with full-field X-rays, with which the acquisition is faster than rasterscanning SAXS methods [5].However, with conventional linear gratings, the sample has to be rotated while the rotation axis is tilted with respect to as well as around the beam axis (3 angular degrees of freedom) [6].This is because the probed SAXS signal depends on the relative orientation of the microstructure with respect to the grating lines and the incoming beam.A new design of the grating comprised of an array of circular unit cells allows to avoid rotation of the optics and reduce sample rotations by one degree of freedom, significantly relaxing the complexity of the acquisition geometry and total scanning time [7][8][9].In this paper, we first demonstrate the results of fiber orientation analysis of a complete FRP component using tensor tomography with a circular grating array performed at a synchrotron facility and an advanced robotic arm sample manipulation scheme.We then discuss ongoing developments of the lab-based tensor tomography for potential industrial applications.

Synchrotron-based experiment
A schematic overview of the XTT acquisition setup at the synchrotron is shown in Fig. 1a.A glass-fiber-reinforced polymer component (Fig. 1b) was scanned with a parallel monochromatic 25 keV X-ray beam.The probed autocorrelation length ξ of the setup was 0.53 µm.The sample was manipulated using Meca500-R3, a 6-axis robotic arm by Mecademic.The sample was mounted to the robot flange as a tool and was rotated around the tool center point (TCP).To ensure uniform sampling, the Fibonacci spiral method was used to generate 400 sample poses with orientations uniformly distributed over a unit sphere [10].Due to the small beam size relative to the sample dimensions, for each sample orientation, a stitch scan comprised of 2 × 9 tiles was acquired to cover a desired FOV of 32 mm × 32 mm.At each stitch scan sample pose, the image was acquired by an sCMOS detector with 6.5 µm pixel size and with an exposure time of 50 ms per image.A 1x microscope optics was coupled between the scintillator and the detector.The detector assembly was placed at a distance of L = 45 cm from the circular grating array.X-rays are diffracted by the circular grating array forming circular fringes downstream at the detector.The fringes are resolved with a 13 × 13 detector pixel grid (unit cell).

Lab-based experiment
Developed at the synchrotron facility, tensor tomography with circular gratings has been implemented with compact lab-based X-ray imaging setup for industry-friendly applications.A temporary lab-based X-ray scattering tensor tomography setup has been implemented at Xnovo Technology.A schematic overview of the setup is shown in Fig. 1c.A wooden toothpick assembly was chosen as the test sample (Fig. 1d) for the demonstration of the setup.Polychromatic X-ray cone beam was generated with 50 kVp and 4 W.An absorptive circular grating array has been used instead of the phase-shifting grating in order to maximize the visibility of the fringe with a polychromatic X-ray tube source with a low spatial and temporal coherence.The probed autocorrelation length ξ was approximately 30 nm.A 2-axis sample manipulator has been integrated into a commercial X-ray imaging setup with a micro-focal X-ray tube (ZEISS Xradia 520 Versa).The projection images were acquired at 270 sample poses with rotation angles at {0°, 4°, ..., 356°} while the rotation axis was tilted 0°, 22.5°, and 45°with respect to the beam direction.A detector coupled with 0.4× objective lens has been used and the effective detector pixel size was 84.5 µm.The circular fringes were resolved with 11 × 11 detector pixel grid window that constitutes a unit cell.The directional scattering signals have been extracted from each unit cell with eight sensitivity angles uniformly distributed between −180°and 180°.11th Conference on Industrial Computed Tomography, Wels, Austria (iCT 2022), www.ict-conference.com/2022

Tensor Tomographic Reconstructions
An iterative X-ray scattering tensor tomography reconstruction has been performed following the method described in [9].Local three-dimensional scattering distribution along multiple scattering sampling directions is tomographically reconstructed.For the robot-based sample manipulation scheme we used a tomographic consistency-based optimization algorithm that corrects for positioning uncertainties in projection data for all 6-degrees of freedom that define a sample pose.For the 2-axis sample manipulator acquired data, two shift parameters for each pose are sufficient for alignment and we expect a sufficiently accurate reconstruction for the fiber orientation with three tilt angles for the purpose of this study [11].The reconstruction code has been adapted to take the cone-beam geometry at the lab-based setup into account instead of the simple parallel-beam geometry at a synchrotron facility.The reconstruction iteration was stopped when a convergence was observed at the residual curve (eg. at iteration 65).Principal component analysis was performed in order to extract the fiber orientation vectors from the reconstructed scattering distribution [9].

Injection-molded glass-fiber-reinforced polymer acquired at a synchrotron facility
An overview tensor tomography reconstruction for the glass-FRP sample is shown in Fig. 2. The color represents the orientation according to the color sphere.As a qualitative validation, we could observe that the fibers are aligned towards the outlet points marked with white circles.The complex microfiber orientation has been mapped over the entire sample in centimeter-scale and this shows the great potential of tensor tomography for FRP industry.For example, tensor tomography can be utilized to predict mechanical properties of final products, to optimize production parameters, as well as to improve fiber injection molding simulation frameworks.More extensive analysis of synchrotron-based tensor tomography for such an FRP sample is currently in preparation for publication.

Toothpick assembly acquired at a compact lab-based setup
The toothpick assembly sample (Fig. 1d) has anisotropic micro-and nano-structures elongated along the longitudinal direction of each toothpick, along which we expect to see preferential fiber orientation.The wood contains micro-architecture with size of 100 − 300 nm [12] with which we expect sufficient scattering signal with the probed autocorrelation length of our setup.The reconstruction of the toothpick assembly acquired at the lab-based setup is shown in Fig. 3. Numerical convergence in the second norm of residual was observed (Fig. 3a) as well as in the projection images (Fig. 3b). Figure 3c clearly shows that the reconstructed fiber orientation is along the longitudinal direction of each toothpick.The micro-and nano-structure of the wooden sample was mapped over centimeter-scale volumetric field of view.The results shown here demonstrate the proof-of-concept of the lab-based X-ray scattering tensor tomography with circular gratings, implemented at Xnovo Technology.

Conclusions and outlook
To demonstrate the capability of X-ray scattering tensor tomography with circular gratings for a glass-fiber-reinforced polymer (FRP) sample, a synchrotron-based X-ray scattering tensor tomography has been performed and an overview of the reconstruction results has been shown.As the next step, a lab-based tensor tomography with circular gratings has been shown for a toothpick assembly sample.We are currently implementing a more flexible setup with higher autocorrelation lengths to measure a large variety of samples.In order to show the capability of the lab-based setup for FRP samples, the same glass-FRP will be measured and compared to the synchrotron result.

Figure 1 :
Figure 1: A schematic overview of the synchrotron-based acquisition setup is shown in (a).A photography of the measured glassfiber reinforced polymer sample is shown in (b).A schematic overview of the lab-based X-ray scattering tensor tomography setup is shown in (c).A toothpick assembly (d) was measured as the test sample.

Figure 2 :
Figure 2: Synchrotron-based X-ray scattering tensor tomography results.The local microfiber orientation has been reconstructed over the entire sample as shown in the rendered 3D views.The color represents the local preferential orientation (3D) of the fibers contained within each voxel.The orientation is color-coded by the color sphere.Scale bar: 1 cm

11thFigure 3 :
Figure 3: Lab-based X-ray scattering tensor tomography results.An iterative X-ray scattering tensor tomography reconstruction was performed until convergence is observed in the residual curve (a).The forward projection images of the reconstructed volumes at different iteration numbers are shown in (b), where the color represents the preferential fiber orientation (2D) according to the color wheel shown in the middle.As the residual curve converges in (a), more agreement was found between the forward projection and the measured projection.The tomographic reconstruction result (at iteration 50) is shown with a 3D rendered view in (c), where the color represents the fiber orientation (3D) according to the color-sphere.Scale bar: 1 cm