Characterization of oriented microstructures through anisotropic small-angle scattering by 2D neutron dark-field imaging

Within neutron imaging, different methods have been developed with the aim to go beyond the conventional contrast modalities, such as grating interferometry. Existing grating interferometers are sensitive to scattering in a single direction only, and thus investigations of anisotropic scattering structures imply the need for a circular scan of either the sample or the gratings. Here we propose an approach that allows assessment of anisotropic scattering in a single acquisition mode and to broaden the range of the investigation with respect to the probed correlation lengths. This is achieved by a far-field grating interferometer with a tailored 2D-design. The combination of a directional neutron dark-field imaging approach with a scan of the sample to detector distance yields to the characterization of the local 2D real-space correlation functions of a strongly oriented sample analogous to conventional small-angle scattering. Our results usher in quantitative and spatially resolved investigations of anisotropic and strongly oriented systems beyond current capabilities. Acquiring orientation-resolved neutron images currently requires the sample or system to be rotated, precluding single-shot measurement. Here, the authors achieve small-angle scattering information with spatial resolution in a single shot through the inclusion of a circular diffraction grating.

N eutron imaging is a nondestructive technique, which is able to probe bulk objects providing information about their internal structure with macroscopic resolution. In contrast to X-ray radiation, neutrons are sensitive to some light elements such as hydrogen and lithium, while they can penetrate many heavy elements, e.g. lead, according to the specific neutron cross sections 1,2 . Due to their intrinsic properties, imaging techniques based on neutron beam became versatile tools in many research fields, such as observation of water distribution in fuel cells, soil-root systems, geomaterials, mapping the elemental composition of fuel pellets and resolving magnetic properties in bulk samples [3][4][5][6][7][8] . A groundbreaking contribution to the imaging research field came with the development of the grating interferometry technique (GI) around two decades ago, with both Xray (xGI) and neutron (nGI) radiations [9][10][11][12][13][14][15][16] . The GI approach yields simultaneously three different and complementary contrast signals: the transmission image (TI), the differential phase contrast image (DPCI) and the ultra-small-angle scattering referred as the dark-field image (DFI) 17 . Since its establishment, the nGI technique created significant impact in neutron imaging research, opening the way to probe and characterize different features of distinct materials in a wide variety of applications that were otherwise inaccessible. The first nGI experiment yielded to the visualization and quantification of the phase shift induced by nuclear interaction probed by DPCI 14 . A decade later the phase shift induced by the magnetic potential was measured by means of a grating interferometer in combination with polarized neutrons (pnGI) 18 . Even a concept to address fundamental physics problems such as the measurement of the neutron electric charge neutrality employing a Talbot-Lau interferometer in a highintensity pulsed neutron beam has been proposed 19 . However, most of the investigations carried out with nGI have mainly focused on the small-angle scattering information provided by the DFI 20 . This contrast modality provides information on the microstructures of the sample on (sub)micrometer length scales, beyond the resolution capability of conventional neutron imaging [21][22][23][24][25] . A major breakthrough for the development of this approach could be established with the pursuit of quantitative evaluation of the neutron dark-field signal providing information about the macroscopic variations of the microstructure [26][27][28][29] . Furthermore, the DFI contrast proved to be a powerful tool for the investigation of magnetic domains in particular in bulk ferromagnetic grain-oriented electrical steels and of magnetic phenomena in superconductors as well [30][31][32][33][34][35][36][37][38][39][40] . Another important application field of DFI is related to the porosity and early crack formation detection in engineering materials, becoming particularly important in additive manufacturing studies 23,41 .
Remarkable efforts have been put in place to develop the nGI technique, improving the instrumentation quality and pushing the boundaries of the accessible length scales toward the hundreds of nanometers range [42][43][44][45][46][47][48] . Besides providing information about the characteristic microscopic structure sizes in the probed object, the dark-field signal due to the underlying small-angle scattering (SAS) is sensitive to structural anisotropy and orientation. However, standard nGI measurements are based on line gratings which imply SAS sensitivity only perpendicular to the grating lines 49 .
Within the X-ray community, the earlier implementations to achieve directional sensitivity were based either on rotating the interferometer setup or the sample 50,51 . Furthermore, different approaches including new grating designs, such as checkerboard and speckle patterns paved the way for further developments [52][53][54][55][56][57][58][59] . Among all the disparate grating geometries the circular design has been implemented to characterize the transverse coherence of Xray beams, demonstrating the advantages of this solution, and opened up the possibility for additional applications 60 .
In order to overcome this limitation in directional sensitivity, here we demonstrate a broadband, single-shot imaging approach with 2D dark-field SAS resolution based on a neutron far-field interferometer characterized by a grating design consisting of small pitch circular gratings arranged into a square array.
To achieve single-shot imaging the interference pattern generated by the diffractive phase grating is directly resolved on the high-resolution neutron detector, without requiring an absorbing analyzer grating. An angle dependent local fringe analysis is then performed to retrieve the local directional dark-field signal of the sample for each independent circular array of the modulation pattern. The grating design can be understood as a 2D periodic repetition of a circular unit cell. This is in analogy to a circular grating interferometry approach introduced at highly coherent synchrotron radiation and eventually generalized to polychromatic and incoherent X-ray sources by redesigning the unit cell 61,62 . Here, we adapted the circular grating interferometer idea for application with a cold neutron beam. Furthermore, we extend the approach towards spatially resolved quantitative 2D-SAS analyses and structural characterization. In order to achieve this we varied the sample to detector distance (L s ), varying accordingly the autocorrelation length (ξ) probed by each specific measurement, which is thus enabling a systematic quantitative scattering study of the locally strongly oriented system, in analogy to the case of isotropic microstructure characterization established in refs. 27,28 .

Results
Working principle of the 2D-Diffractive far-field neutron grating interferometer. The neutron interferometer is composed by an absorbing source grating (G 0 ) and a phase shifting grating (G 1 ). A schematic of the circular unit cell nGI setup is depicted in Fig. 1a. Despite the presented method does not intrinsically rely on coherent illumination, conventional neutron beams usually do not provide sufficient collimation in order to observe the interference pattern. Therefore, an absorbing source grating is required in order to create an interference pattern at the detector distance which leads to the condition 62 : where g 0 is the beam aperture, which corresponds to the holes diameter of the 2D array as described in the Method section and shown in Fig. 1a. The parameter l corresponds to the distance between the two gratings, d is G 1 to detector distance and p 1 the annular periodicity of G 1 as depicted in Fig. 1. The 2D pinholes array geometry of G 0 is designed such that a superimposition of the individual generated patterns corresponding to each aperture is achieved (Lau condition) 63 . Each unit cell of the phase grating G 1 is composed by one or more concentric annuli containing a diffractive substructure. The design of the phase grating G 1 is characterized by three periodicities: a global one W 1 and two local ones p 1 and g 1 . W 1 corresponds to the repetition period of a unit cell as shown in Fig. 1a, b. The coarse local period p 1 corresponds to the annular period of the diffractive annuli, as mentioned above. Finally, the fine local period g 1 corresponds to the periodicity of the diffractive circular structure of the phase shifting annuli shown in detail in Fig. 1c.
The distance (d), depicted in Fig. 1a, between G 1 and the detector can be calculated, in the non-parallel beam case, as: where z equals to p 1 g 1 2λ , and λ the neutron wavelength.
The global periodicity W 1 determines the projected experimental interference fringe matrix on the detector, which defines the spatial resolution of the retrieved dark-field signal with an effective pixel size corresponding to a single unit cell. The recorded pattern for each unit cell, assuming uniform diffraction properties of the circular grating in the range of one unit, can be approximated as a radial cosine function expressed in polar coordinates as: where A corresponds to the average intensity of the modulation pattern and V(ξ, ω) the radial visibility as a function of the autocorrelation length ξ. Assuming an oriented fiber structure the angle dependent visibility reduction can be modeled at each angle ω as following 62,64 : where b 0 corresponds to the average scattering intensity, b 1 the oriented component of the scattering function, and ϕ d the direction of the underlying fibers structure. The ratio b 1 /b 0 can be regarded the degree of anisotropy of such a system. A standard experiment requires the acquisition of two images: an open beam image without the sample and one with it. Once the set of images has been recorded each unit cell is identified with a segmentation algorithm from the open beam image. This segmentation parameters are afterward used as a mask for the sample image. The segmented units are subsequently analyzed individually in the context of the corresponding open beam segments. The reconstruction algorithm performs as a first step a 2D-Fourier Transform to each unit cell and then converts it from Cartesian coordinates to polar coordinates in order to analyze the angular variation of the visibility, which is encoded in the intensity of the second harmonic 62 . Hence, the data extraction is performed in a similar manner to conventional grating interferometry at each angle ω, i.e., the visibility reduction is given then by: where V s and V f are the radial sample and open beam visibility, respectively. TheÎðqÞ is the power spectrum, the subscripts s and f refer to the sample and to the open beam, respectively, with q m,n being the scattering vector in Cartesian coordinates and q N ω utilizing the polar representation, where N is the ration of the unit cell size W 1 and the coarse period p 1 , which corresponds to two. The range of sensitivity for the characterization of microstructures changes by varying the sample to detector distance as reported in ref. 26,28 . The relation between the probed autocorrelation length ξ and L s can be expressed as: where p 1 also corresponds to the period of the concentric intensity pattern recorded on the detector. Therefore, by varying L s directly implies a scan of the autocorrelation length for a constant wavelength at which the image is recorded. The angular and autocorrelation lengths dependencies of the dark-field signal yields to quantitative information about the locally strongly oriented system.
Scattering models of anisotropic and strongly oriented systems. Assuming that the scattering from a single point results into a 2D-Gaussian scattering intensity profile, in the specific case of a quasi-1D structure like such as long oriented fibers, the visibility reduction can be expressed as 64 : where α corresponds to the proportion of the beam that has been scattered, σ 1 and σ 2 the widths of the 2D-Gaussian distribution in the two axial directions.
In the studies presented in ref. 64 , the directional DFI variations of fiber structures have been studied for weakly and strongly oriented orders with conventional xGI, inferring that in the former case a sinusoidal behavior is expected as a function of the angle, while the latter display a nonsinusoidal trend, peaked at the preferred scattering direction. In an analogous way we studied the directional DFI variations of strongly oriented system, however, additionally as a functions of the probed autocorrelation lengths.
The change in DFI signal as a function of ξ yields additional quantitative information about the microstructures underlying the system and can be expressed as 26 : where Σ is the macroscopic scattering cross-section, t the sample thickness and G(ξ) is the one-dimensional projection of the autocorrelation function of the scattering structure calculated via the Abel transform of the autocorrelation function. The macroscopic scattering cross-section is defined as: where Δρ corresponds to the difference in scattering length density of the scatterer and the medium. ζ represents the volume fraction of the scatterer and χ its characteristic size. Different models have been derived in order to describe G(ξ), in particular for the case of oriented anisotropic system such as the infinitely long cylinder, having its side parallel to the x axis and its diameter in the zy-plane, the correlation γ(r) and autocorrelation functions are given by 65 : where D is the cylinder diameter.
Dark-field 2D real-space analysis of carbon fibers. A carbon fiber plate scaffolded into a polymer resin was selected as a model sample to study the dark-field signal from a locally strongly oriented system, as shown in Fig. 2a, b. The specimen thickness is 1 mm. The fibers of the sample are uniformly oriented along one direction parallel to the face of the sheet, a scanning electron microscopy (SEM) image of the sample can be seen in Fig. 2b. The plate has been cut in three pieces and oriented along exemplary directions, as depicted in Fig. 2a: the red one at 0°, the green one at 120°and the blue one at 240°with respect to the vertical direction. The corresponding dark-field signal arising from the fibers depends on their orientation. Therefore, the Yshaped pie chart arrangement of the cut pieces provides a 120°a ngular displacement of the fibers orientations in the imaging plane can be regarded serving as validation for our purposes. Figures 2c, d show the results of the measurement, performed at 4.1 Å and ξ equals to 0.3 μm probed correlation length.
The two images are the attenuation contrast image in Fig. 2c and the color coded directional DFI image (DFI directional ) in Fig. 2d.
The conventional attenuation based transmission image in Fig. 2c allows to identify the edges of three distinct regions within the field of view (FoV), but without providing any useful information about the fiber structure and its orientation. On the other hand, the color coded directional DFI image in Fig. 2d clearly shows the triple-fold orientation variation corresponding to the different sample regions.
The directional DFI has been retrieved from all the individual scattering images at different angles and in order to represent the fibers orientations from the most prominent scattering direction. The directional information, including the scattering strength and the angle, encoded into the directional DFI uses the hue saturation value (HSV) color space in an analogous manner as adopted in refs. 61,62,64 . The hue corresponds to the characteristic direction of the oriented system; the saturation is set to 1 and the value is defined as the normalized amplitude of the scattering directionality, where the dark areas correspond to the ones with no directional scattering. In addition, our correlation length scans for quantitative SAS analyses were spanning a range from 20 nm to 670 nm as shown in Fig. 3. Figure 3a shows the 2D real-space correlation functions retrieved from our DFI data. The pattern in Fig. 3a is averaged over the whole FoV which is corresponding to the real space correlation function a conventional small-angle neutron scattering (SANS) measurement would yield, integrated over the whole sample. In this representation of the results, Fig. 3a, we can draw some conclusions regarding the presence of three main orientations of the microstructures within the sample, nevertheless, without being able to properly identify if there is an orientation variation due to different regions or due to mixed fiber orientations throughout the sample. This is a common issue of conventional scattering methods when probing a nonhomogeneous sample and the most widely adopted solution to tackle this problem is to perform a scanning pencil-beam SANS measurement in order to retrieve the spatially resolved scattering information, which is time consuming and cannot be performed in a single-shot acquisition. The results presented in the darkfield 2D real-space correlation functions of Fig. 3b-d show the directional dark-field signals averaged over the three segmented COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0308-4 ARTICLE COMMUNICATIONS PHYSICS | (2020) 3:42 | https://doi.org/10.1038/s42005-020-0308-4 | www.nature.com/commsphys areas in red for the 0°fibers orientation, green for the 120°and blue for the 240°, respectively, as depicted in the insets. The three dark-field 2D real-space correlation functions, Fig. 3b-d, clearly show highly eccentric scattering behaviors, for each region, parallel to the fibers orientation in the respective segmented areas, which is a distinct characteristic of strongly ordered systems.
Furthermore, the dark-field signal has been analyzed as a function of the angle and the autocorrelation lengths ξ, as shown in Fig. 4a, b.
The experimental results of the fibers, which exemplify a strongly oriented system, exhibit an angular dependency in terms of probed autocorrelation lengths, as shown in Fig. 4b, showing a sinusoidal behavior for shorter ξ and a nonsinusoidal one for longer ξ. Conventionally, this sinusoidal or nonsinusoidal behavior has been attributed to the orientation degree of the scatterer, for weakly and strongly oriented system, respectively, measured at a single specific sample to detector distance, as reported in ref. 64 . However, our case demonstrates how an autocorrelation length scan combined with an analysis of the angular dependency is essential to make an assessment of the orientation degree of the scatterer. The experimental results have been fitted according to Eq. (7) showing good agreement with the model at four different autocorrelation lengths: 0.04 μm, 0.09 μm, 0.26 μm and 0.51 μm. The retrieved parameters σ 2 1 , σ 2 2 , ψ and α for all points are presented in Table 1. The parameters σ 2 1 , σ 2 2 show an increment in the eccentricity of the 2D-Gaussian scattering profile as a function of the autocorrelation length.
Moreover, the consistency of ψ allows us to properly determine the angle of the fibers orientations at (116.6 ± 0.7)°, beyond a correction factor of ±180°due to phase wrapping.
As a last step of our investigation the experimental data have been described and fitted with the oriented infinitely long cylinder model according to Eqs. (8) and (10), as shown in Fig. 4c. Since the probed autocorrelation lengths ξ span from 20 nm to 670 nm and the diameter of the carbon fibers corresponds to 5 μm, as retrieved by means of the SEM image  in Fig. 2b, this characteristic size falls out of the probed range and hence cannot be fully assessed. Nevertheless, we can assume the diameter D, the volume fraction ζ values from SEM and use as a priori knowledge in order to calculate the macroscopic scattering cross-section Σ of Eq. (9) subsequently estimate the scattering length density contrast Δρ.
Due to the saturation of the dark-field signal at longer autocorrelation lengths, which can be observed in the logarithmic representation of the dark-field value in Fig. 4c, a rejection threshold has been set in order to evaluate only the points with ξ < 0.25 μm. The composition of the sample can be approximated as carbon for the fibers (the scatter) and Hydrogen for the polymer resin (the medium), resulting into a Δρ ¼ 49:537 10 À6 Å À2 for a wavelength of 4:1 Å 66 . The resulting scattering length density value from the fitting of the experimental points corresponds to 61 ± 7:8 10 À6 Å À2 which is in reasonable agreement with the crude estimate and underlines the potential for quantitative analysis of our measurements.

Discussion
The authors have presented a neutron grating interferometer able to perform in a single-shot mode 2D directional dark-field image acquisition in addition to the conventional attenuation contrast. The single-shot approach in contrast to the existing methods does not require any rotation of the sample nor the gratings during acquisition compared to existing methods. The feasibility of the method was demonstrated by imaging a batch of carbon fibers oriented along three exemplary directions. The retrieved directional dark-field image clearly reveals the fibers orientations in the three distinct regions. Thus, the method achieves in a single step what was has been used to require several acquisitions previously. Furthermore we showed how the measurements at different correlation lengths, through the sample to detector scan, allow quantitative SAS analyses and the creation of a dark-field 2D real-space correlation function for every region of interest in the image. We demonstrated that the experimental results pave the way for further investigations leading to characterization and quantification of anisotropic microstructures, providing a model to fit the real-space correlation function and describing the fiber structure addressing the issue of signal saturation and how to circumvent the problem. The proposed method approaches the investigative strength of conventional scattering techniques, however, spatial resolution capabilities, and tremendously simplifies the access to multidirectional dark-field information enabling the study of non-homogeneous and anisotropic scattering from systems with preferred microstructural orientations such as in polymer science, soft condensed matter physics, rheological studies and nondestructive testing of additively manufacturing specimens.

Methods
2D-Diffractive far-field neutron grating interferometer setup. The validity of the proposed technique is demonstrated with experimental measurements conducted at the cold beamline ICON, at the Swiss Spallation Neutron Source (SINQ), at the Paul Scherrer Institut (PSI) 67 . A turbine based energy selector was used to get a monochromatic beam at a wavelength of 4.1 Åwith a energy resolution of Δλ λ ¼ 12%. The source grating G 0 is a gadolinium sputtered absorbing mask positioned at the neutron beam aperture, it consists of 2D array of g 0 = 1.3 mm holes in an upright square lattice, with lattice constant p 0 = 5.5 mm, as shown in Fig. 1. A phase shifting grating (G 1 ) with a radial period p 1 of 300 μm and unit cell period g 1 of 2 μm was fabricated in house by e-beam lithography and etching of Si. The grating was etched to a depth of 37 μm, which at 4.1 Å produces a phase shift of π in the neutron wave packet. A scanning electron microscopy (SEM) image of the fabricated phase grating can be seen in Fig. 1b, c. In order to directly resolve the interference pattern on the detector a Fiber Optic Taper (FOT) has been adopted in combination with a 10 μm Gadox-based scintillator screen and a 100 mm objective, resulting into a spatial resolution of about 20 μm measured with a Siemens star test object 68,69 . The described nGI setup reaches a performance of 42% in visibility when combined with the velocity selector and around 20% with white beam. A total number of 55 directional dark-field images covering the range [0, π] were retrieved from the singleshot measurements. The G 1 was placed at 7 m from the source grating G 0 , whereas the detector at 0.4 m from G 1 according to Eq. (2). The sample was always placed in between the phase grating G 1 and the detector, as illustrated in Fig. 1a.
Data acquisition and reduction. A stack of 60 images, each with an exposure time of 320 s, has been reduced by median projection before applying the single-shot algorithm. The total acquisition time was mainly dictated by the flux reduction due to the velocity selector, the absorption grating G 0 , and by the low light yield of the Gadox-based scintillator.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.