Tunable X-ray dark-field imaging for sub-resolution feature size quantification in porous media

X-ray computed micro-tomography typically involves a trade-off between sample size and resolution, complicating the study at a micrometer scale of representative volumes of materials with broad feature size distributions (e.g. natural stones). X-ray dark-field tomography exploits scattering to probe sub-resolution features, promising to overcome this trade-off. In this work, we present a quantification method for sub-resolution feature sizes using dark-field tomograms obtained by tuning the autocorrelation length of a Talbot grating interferometer. Alumina particles with different nominal pore sizes (50 nm and 150 nm) were mixed and imaged at the TOMCAT beamline of the SLS synchrotron (PSI) at eighteen correlation lengths, covering the pore size range. The different particles cannot be distinguished by traditional absorption µCT due to their very similar density and the pores being unresolved at typical image resolutions. Nevertheless, by exploiting the scattering behavior of the samples, the proposed analysis method allowed to quantify the nominal pore sizes of individual particles. The robustness of this quantification was proven by reproducing the experiment with solid samples of alumina, and alumina particles that were kept separated. Our findings demonstrate the possibility to calibrate dark-field image analysis to quantify sub-resolution feature sizes, allowing multi-scale analyses of heterogeneous materials without subsampling.

the artifact is aggravated by increasing correlation length . It is expected that the effect may be reduced by minimizing the difference in refractive index between the material and the medium, e.g. by using a non-gas material like in [Yang, F. et al. Advancing the visualization of pure water transport in porous materials by fast, talbot interferometry-based multi-contrast x-ray micro-tomography. Dev.
X-Ray Tomogr. X 9967, 99670L (2016)]. However, it should be noted that this may limit the sample size because of increased attenuation reducing the SNR. Further research would be required to explore the practical best practices in this respect. The impact of this edge artifact, as well as noise in general, is visible in Figure S2: in this graph, the AFS value is extracted per voxel, rather than per object, and displayed as box plots. Per-voxel values are evidently of very low precision, making it nearly impossible of reliably distinguishing a single 50nm disk voxel from a 150nm disk voxel. This limited low noise robustness is the motivation for the median aggregation of segmented structure voxels in the proposed procedure. Additionally, as object edges are disproportionally affected by artifacts, our procedure includes a method of mitigating edge artifacts specifically. This is accomplished by use of mathematical morphology (erosion): specifically, the segmentation masks are iteratively eroded, so that increasingly wider layers of the edge area are ignored before the AFS value is calculated. Out of the series of AFS values thus generated, one for each erosion iteration, the lowestvariance AFS value is retained to represent the object. This is found to improve precision: Figure S3 presents boxplots of the relative increase/decrease in variance of the AFS values for the objects in the mixed sample (Fig. 13) as the amount of edge voxels that were eroded prior to the signal analysis changes, compared to the situation when no edge voxels are removed. Notice how removing 30-40% of the edge typically results in a lower-variance estimation.

Supplementary Information S2: Pore space model
We model sub-resolution pores as idealized pulses of identical contrast H, cfr. Figure S4. The width W of these pulses, characteristic for the material's porosity, is our feature of interest. , which simplifies as:

~ ξ ²
On the other hand, for correlation lengths that exceed the largest pore size, i.e. ξ ≥ , we get: Which simplifies to:

~2
In other words, the values form, by approximation, a piecewise linear function of ξ that starts with a sloped segment and transitions into a constant segment, with the segment transition located around ξ = .
A real-world sample consists of a variety of pores sizes with non-spherical shapes, which result in the autocorrelation function being a weighted average of apparent cross-sectional pore sizes. If the correlation length ξ is below the smallest pore size , each contribution to the will be in the sloped regime of the function, and thus the total slope will be largest in magnitude. With increasing correlation length values ξ between and the largest pore size the slope would decrease until reaching 0 for ξ = . Due to the quantized nature of such measurements in finitesized voxels, the measured slope would therefore characterize a weighted average of typically encountered pore sizes and therefore be a characteristic for the pore size distribution. Furthermore, the slope may be robustly extracted from many sample points, i.e. dark field acquisitions at different correlation lengths ξ. This consideration is the motivation for extracting the slope as the characteristic feature.