René Tannert
email: rene.tannert@dlr.de

Marina Schwan

Ameya Rege

Mario Eggeler

Julio Cesar Da Silva

Marion Bartsch

Barbara Milow

Mikhail Itskov

Lorenz Ratke

The three-dimensional structure of flexible resorcinol-formaldehyde aerogels investigated by means of holotomography

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Aerogels are nanostructured open-porous materials with a very low density that renders them excellent materials for thermal insulation purposes [START_REF]Aerogels Handbook[END_REF]. The gel structure can be suitably tailored by adjustment of chemical parameters during synthesis (concentration of reagents, pH value). Among all aerogels, organic gels based on resorcinol-formaldehyde (RF gels) play a prominent role as they show the lowest thermal conductivity determined so far [START_REF] Lu | Thermal Conductivity of Monolithic Organic Aerogels[END_REF]. In order to improve the thermal, mechanical, and other properties an in-depth knowledge of the three-dimensional (3D) structure of aerogels is a prerequisite as it allows modeling and simulation studies [START_REF] Pekala | Organic aerogels: microstructural dependence of mechanical properties in compression[END_REF][START_REF] Lu | Correlation between structure and thermal conductivity of organic aerogels[END_REF]. To this end, the porous nature of aerogels is most commonly assessed quantitatively using methods that rely on (nitrogen) sorption [START_REF] Reichenauer | Structural Characterization of Aerogels[END_REF]. However, methods used to determine the pore-size distribution are limited by underlying assumptions -the popular Barrett-Joyner-Halenda (BJH) model, for instance, involves a cylindrical pore shape and does not give information about the full range of pore-width and volume [START_REF] Barrett | The Determination of Pore Volume and Area Distributions in Porous Substances. I. Computations from Nitrogen Isotherms[END_REF]. Although data on porosity obtained via the BJH model have yielded multi-scale mechanical models of aerogels based on silica [START_REF] Fu | Simulation of the microstructural evolution of a polymer crosslinked templated silica aerogel under high-strain-rate compression[END_REF] and cellulose [START_REF] Rege | Micro-mechanical modelling of cellulose aerogels from molten salt hydrates[END_REF], the exploitation of modeling tools is still limited by such simplified pore models. Alternative computational methods that combine density functional theory calculations and sorption experiments, although established for several classes of rigid, ordered porous solids, cannot be confidently applied to aerogels as their porous network undergoes deformation upon adsorption [START_REF] Reichenauer | Nitrogen adsorption in compliant materials[END_REF][START_REF] Landers | Density functional theory methods for characterization of porous materials[END_REF].

Another common method that can provide pore-size distributions over a large range (from a few nm to micrometer) is mercury porosimetry [START_REF] Giesche | Mercury Porosimetry: A General (Practical) Overview[END_REF]. It is, however, only reliable for very stiff aerogels as more flexible samples can be significantly compressed during the intrusion experiment [START_REF] Johnston | Compression effects in mercury porosimetry[END_REF][START_REF] Scherer | Deformation of aerogels during characterization[END_REF].

Electron microscopy is well-established as a tool to provide two-dimensional (2D) images of aerogels on the nanometer scale [START_REF] Reichenauer | Structural Characterization of Aerogels[END_REF]. A combination of focused ion beam milling and scanning electron microscopy has proven to represent a powerful technique to obtain highly resolved 3D structures of porous materials [START_REF] Holzer | Three-dimensional analysis of porous BaTiO3 ceramics using FIB nanotomography[END_REF][START_REF] Wilson | Three-dimensional reconstruction of a solid-oxide fuel-cell anode[END_REF]. Although it has been successfully applied to mesoporous carbon [START_REF] Ziegler | Direct three-dimensional reconstruction of a nanoporous catalyst layer for a polymer electrolyte fuel cell[END_REF][START_REF] Balach | A direct and quantitative image of the internal nanostructure of nonordered porous monolithic carbon using FIB nanotomography[END_REF][START_REF] Balach | A Direct and Quantitative Three-Dimensional Reconstruction of the Internal Structure of Disordered Mesoporous Carbon with Tailored Pore Size[END_REF], it has not yet been used to provide 3D images of RF gels, since they are more difficult to investigate due to a rapid electric charging during measurements hampering suitable image detection.

Tomographic methods that use X-ray irradiation have been used to visualize porous materials such as metal foams [START_REF]X-Ray Tomography Applied to the Characterization of Highly Porous Materials[END_REF], and also some low resolution 3D-structural images of composite materials containing RF gels have been described [START_REF] Haghgoo | Use of X-ray microtomography to study the homogeneity of carbon nanotube aqueous suspensions and carbon nanotube/polymer composites[END_REF][START_REF] Laskowski | Synthese und Eigenschaften von Aerogel-Aerogel-Verbundwerkstoffen[END_REF]. However, RF gels are rapidly destroyed in the classical experimental setup of X-ray absorption when higher energy levels are used. Thus, no 3D-nanostructure has yet been extracted from X-ray absorption experiments. In that context, nanoscale zoom holotomography provides an alternative X-ray imaging method that does not rely on absorption, but instead on the phase-contrast [START_REF] Mokso | Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics[END_REF]. Holotomography is capable of imaging materials with a high sensitivity for electron density, high isotropic spatial resolution (down to ~30 nm) and small voxel size (down to ~5nm). As very light materials (such as aerogels) do not absorb much of the Xrays, the phase-contrast image plays a key role to efficiently visualize and quantify pore geometries. This technique has already been successfully applied to the structural elucidation of porous systems including Metal Matrix Composites [START_REF] Kenesei | Structure-property relationship in particle reinforced metalmatrix composites based on holotomography[END_REF][START_REF] Girardin | Characterization of Porosity in a Laser Sintered MMCp Using X-Ray Synchrotron Phase Contrast Microtomography[END_REF], Solid Oxide Fuel Cell electrodes [START_REF] Sar | Three dimensional analysis of Ce0.9Gd0.1O1.95-La0.6Sr0.4Co0.2Fe0.8O3-δ oxygen electrode for solid oxide cells[END_REF][START_REF] Hubert | Role of microstructure on electrode operating mechanisms for mixed ionic electronic conductors: From synchrotron-based 3D reconstruction to electrochemical modeling[END_REF], and Arabidopsis seeds [START_REF] Cloetens | Quantitative phase tomography of Arabidopsis seeds reveals intercellular void network[END_REF].

Aerogels have not been studied using holotomography, and our article describes efforts to this end. For first tests, superflexible resorcinol-formaldehyde (sf-RF) gels were selected [START_REF] Schwan | New soft and spongy resorcinol-formaldehyde aerogels[END_REF], as they possess rather large particles and pores in the sub-micrometer range. They should be easier to visualize than classical Pekala-type RF aerogels which have both particles and pores in the low nanometer range [START_REF] Pekala | A Synthetic Route To Organic Aerogels -Mechanism, Structure, and Properties[END_REF].

2 Experimental setup

Preparation and characterization of RF aerogel samples

Monoliths of sf-RF aerogels were synthesized and dried using supercritical carbon dioxide as reported previously [START_REF] Schwan | New soft and spongy resorcinol-formaldehyde aerogels[END_REF]. Aerogel monoliths were ground with a planetary micro mill (Pulverisette 7, Fritsch, Germany) filled with 17 zirconium oxide balls (of approximately 10 mm diameter). For grinding the mill was sealed and rotated for 7 minutes at a velocity of 700 rpm. The milling process was repeated three times with 3 minutes cooling interval in-between milling cycles. The envelope density of monolithic sf-RF was calculated from measured volume and weight of an aerogel piece, whereas the envelope density of sf-RF powder was measured with a GeoPyc 1360 analyzer (Micromeritics, USA). The skeletal density was measured by helium pycnometry using an AccuPyc machine (Micromeritics, USA). The internal surface area of the aerogel monolith and powder was calculated from the nitrogen adsorption-desorption isotherms measured on a TriStarII machine (Micromeritics, USA) using the method from Brunauer, Emmett, and Teller (BET, [START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF]) implemented in the Micromeritics software. Before analysis, the samples were outgassed for 6 hours at 110°C and 10 -2 mbar. The microstructure of the aerogels was investigated using a Merlin scanning electron microscope (SEM; Zeiss, Germany). RF aerogels, which are electrically non-conductive, were coated with gold prior to SEM. The average particle diameter was determined by applying the linear intercept method on SEM images [START_REF] Heyn | Short reports from the metallurgical laboratory of the royal mechanical and testing institute of Charlottenburg[END_REF]. This method, which was originally developed for metals and alloys, can be adapted to aerogels as long as they possess a particulate structure. Light microscopy was performed using a SteREO Discovery.V12 stereomicroscope (Zeiss, Germany). Mercury porosimetry (conducted by Porotec, Germany) failed to produce reliable intrusion data due to substantial deformation of the sample during the experiment.

Preparation of the sample for nanoholotomography

Thin-walled quartz glass capillaries of 100 µm outer diameter and 10 μm wall thickness (Hilgenberg, Germany) were densely filled with aerogel powder dried in an oven at 120 °C and ambient pressure over night. Subsequently, the capillary was cut to 10-15 mm length, hermetically sealed with paraffin wax, and fixed with Loctite Super Glue 3® onto sample holders for nanotomography (see Figure 1).

[Figure 1: Aerogel-filled glass capillary glued onto the sample holder (a) and magnified view on the glass capillary (b)]

Nanoscale zoom holotomographic analysis on sf-RF aerogels

The holotomography experiments were performed at the ID16A Nano-Imaging beamline recently built at ESRF, Grenoble [START_REF] Zabler | Optimization of phase contrast imaging using hard x rays[END_REF][START_REF] Hubert | Role of microstructure on electrode operating mechanisms for mixed ionic electronic conductors: From synchrotron-based 3D reconstruction to electrochemical modeling[END_REF]. The 17.05 keV X-ray beam was nano-focused using Kirkpatrick-Baez mirrors to a spot size of about 30 nm, and the magnified projections of the samples were recorded by a FReLoN detector [START_REF] Labiche | Invited article: The fast readout low noise camera as a versatile x-ray detector for time resolved dispersive extended x-ray absorption fine structure and diffraction studies of dynamic problems in materials science, chemistry, and catalysis[END_REF]. The scans were carried out using the zoom tomography approach where the samples of sf-RF aerogel were imaged at four different focus-to-sample distances [START_REF] Mokso | Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics[END_REF][START_REF] Hubert | Role of microstructure on electrode operating mechanisms for mixed ionic electronic conductors: From synchrotron-based 3D reconstruction to electrochemical modeling[END_REF]: 24.160 mm, 25.197 mm, 29.343 mm and 37.951 mm. Such distances were chosen to achieve an effective voxel size of 60 nm. 2000 projections were acquired over 180 degrees for each distance. The focus-to-detector distance was 1.2 m. A total of 8000 projections were therefore acquired with exposure time of 1s which took about 4 h to complete. The experiments were performed at room temperature.

Three-dimensional reconstruction

The obtained holotomographic projections are phase contrast images taken at the holographic regime. Using the Fresnel fringes on the images at the four distances with different magnifications, it is possible to recover the phase-shifts due to the specimen [START_REF] Mokso | Nanoscale zoom tomography with hard x rays using Kirkpatrick-Baez optics[END_REF]. Such phase maps are proportional to the projection of the electron density of the material. First, the projections are rescaled down to the smallest pixel dimension recorded during the scan, and then, the projections recorded at four distances are aligned. The phase retrieval at each tomographic angle, as described in a previous work by Cloetens and coworkers [START_REF] Cloetens | Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays[END_REF], was carried out with in-house software using the GNU Octave programming environment (http://www.octave.org) and the public domain image analysis program ImageJ (http://rsbweb.nih.gov/ij/). The 3D volumes were reconstructed from the phase maps using the filtered back-projection (FBP) [START_REF] Bracewell | Inversion of fan-beam scans in radio astronomy[END_REF] with the ESRF software PyHST. The resulting volume contained 2048 x 2048 x 2048 pixels.

Image analysis

The raw volume was visualized and analyzed using the "Avizo" software package (version 9.2; FEI Company, Hillsboro, United States). The extracted subvolume was filtered with an edge-preserving smoothing image filter (Bilateral filter) in order to reduce noise and to facilitate generating a binary image for separating macropores and denser material. A 2D-and 3D-threshold preview was used to set the lower and upper threshold interval of the "Interactive Thresholding" module in Avizo. Different threshold values were tried resulting in different computed material porosity and size and numbers of macropores. A threshold interval ranging from -1200 to 110 was chosen for further analysis assuring that all macropores are captured and pores are not connected to each other. Even though some pores might be connected, this selected threshold data should provide a reasonable assumption for macropore-size distribution. The "Volume Fraction" and "Label Analysis" tools in Avizo were then used for the determination of the material porosity and labelling of each pore, respectively. Furthermore, different colors were assigned to independent pores.

Results and Discussion

A first issue was to produce aerogel powder grains small enough to enter the microcapillary envisioned for the tomography experiment, yet large enough to retain the microstructure including pores. To this end, we employed a planetary ball mill and compared several properties of monolith and powder [START_REF] Piedboeuf | Influence of the textural parameters of resorcinolformaldehyde dry polymers and carbon xerogels on particle sizes upon mechanical milling[END_REF]. As indicated in table 1, the inner surface area (BET area), the envelope (ρ e ) and the skeletal (ρ s ) densities, as well as the resulting porosity of the monolith and the powder matched well which indicates that there is no alteration of the microstructure. Also, scanning electron micrographs show no significant change in the microstructure (Fig. 2). The microstructural analysis of both aerogel monolith and powder revealed an average particle size of about 0.11 µm, while large pores in the range of 2-7 µm were observed. Since the monolith and the powder display similar structure and properties, the powder appears to be a viable surrogate for the monolithic sf-RF and was thus employed in tomography. a the porosity was calculated from envelope and skeletal densities [START_REF] Tannert | Reduction of shrinkage and brittleness for resorcinol-formaldehyde aerogels by means of a pH-controlled sol-gel process[END_REF] [Figure 2: Microstructure of monolithic (a) and powdered (b) sf-RF samples]

Figure 3 shows the reconstructed 3D structure obtained by X-ray holotomography. Three orthogonal slices are displayed, and a small cubic region is highlighted to give an idea of scale. In a first qualitative interpretation, such a 3D representation suggests a higher density of material between the powder grains which was puzzling since those regions should have been filled only with air and thus should appear darker than the solid material. A possible explanation of this phenomenon is that upon milling loose sf-RF particles may have been accumulated between the highly porous grains leading to a higher material density outside the grains than inside.

[Figure 3: tomogram of sf-RF aerogel; a representative region of interest was selected from a RF grain (red bounding box with cube edge length of 1.5 µm)]

Within a bigger grain of sf-RF a cubic region of interest with edge length of 1.5 µm was selected for further analysis (red bounding box in figure 3, figure 4a). The extracted subvolume was binarized in order to distinguish between material and macropores. As this binarization step was in particular prone to give biased results, care was taken to identify pores unequivocally by only selecting areas with sharp decrease in gray-values as pores (see Figure 4b). Pores identified in this way were then analyzed with respect to the pore diameter and volume.

[Figure 4: Image of extracted subvolume after applying bilateral filter (a); pores identified are highlighted in blue (b)]

In total, 1056 macropores with volumes ranging from 2.16•10 -4 µm 3 (volume of one voxel) to 51.7 µm 3 were identified and labeled (Figure 4). A rather low material porosity of 14.8 % was computed using volume fraction analysis. The huge difference in porosity as computed from 3D x-ray data to porosity values that were determined using pycnometry (96 %, see table 1) may be mainly attributed to the presence of mesopores and very small macropores that were out of reach for our experimental setting. An analogous experiment using a voxel size of 30 nm in local tomography mode did not yield a useful 3D reconstruction for sf-RF. Since the fieldof-view is smaller in that setting, all material surrounding the field-of-view contribute as noise to the experiment thus hampering further analysis of the raw data. An alternative sample envisioned for future X-ray experiments could consist of a single aerogel grain that is glued to the sample holder and thus is only surrounded by air.

[Figure 5: Visualization of the pore space (for clarity, colored pores are shown without material)]

A visual inspection of macropores as identified in the region of interest indicated that a spherical geometry might be used to further analyze pores quantitatively (Figure 5). A corresponding analysis shows that the majority of identified pores, with diameters between 74.4 nm and 4.62 µm, are in the low macropore regime (diameters below 0.5 µm) whereas relatively few larger pores (diameter > 1.5 µm) were found (Figure 6a). Most of the macropore volume, however, seems to be located in pores of 1 to 4 µm diameter as suggested by the bimodal distribution of relative pore-volume illustrated in Figure 6b. As pore-sizes and corresponding volumes affect the kinetic profiles of transport processes taking place in the aerogel, these findings might have implications for the use of RF gels, for instance as matrix for metals in catalysis [START_REF] Moreno-Castilla | Carbon Aerogel-Supported Pt Catalysts for the Hydrogenolysis and Isomerization of n-Butane: Influence of the Carbonization Temperature of the Support and Pt Particle Size[END_REF] or as carrier of metal-organic frameworks for sorptive applications [START_REF] Wickenheisser | Hierarchical MOF-xerogel monolith composites from embedding MIL-100(Fe,Cr) and MIL-101(Cr) in resorcinol-formaldehyde xerogels for water adsorption applications[END_REF]. Furthermore, the structural data obtained so far might be useful for studies towards a modeling of the mechanical and thermal properties of RF-aerogels.

[Figure 6: representations of distributions of the absolute pore-diameter (a; histogram with a class width of 0.2 µm) and the relative pore volume (b)]

4 Conclusions and Outlook Superflexible resorcinol-formaldehyde aerogels were proven as suitable probes for zoom holotomography. Insights into the 3D structure of such gels could be obtained at the macropore level. For microstructural characterization, macro-porosity was treated as an entity of individual spherical pores. Distributions of both pore size and pore volume were determined. Accordingly, most pores are small macropores (diameter < 0.5 µm) whereas most of the pore volume is located in larger macropores of 1 -4 µm diameter.

Comparative studies on other aerogel and xerogel samples are ongoing. These include the investigation of more rigid gels in order to validate the pore-size distribution extracted from holotomography using mercury porosimetry. Current efforts are also devoted to the extraction of physical quantities for the description of pore shape (e.g. Feret diameters) and connectivity (Euler numbers) and will be published in due course. Further X-ray holotomographic experiments are planned using smaller samples that might allow for the use of smaller voxel sizes and, consequently, for structural data at higher spatial resolution that also allow to investigate the mesopore regime.

  Fig 1a

Table 1 :

 1 Properties of sf-RF aerogel determined for monolith and powder

	Sample	BET, m 2 /g	ρ e , g/cm 3	ρ s , g/cm 3	Porosity a , %
	sf-RF monolith	52	0.0650	1.53	95.8
	sf-RF powder	45	0.0623	1.47	95.8

Acknowledgments

The authors wish to thank Benjamin Ignatzi for the production and characterization of powders and Guillermo Requena and Galina Kasperovich for assistance and computer resources provided during preliminary image analysis. We thank the European Synchrotron Radiation Facility for access to synchrotron radiation facilities at ID16A beamline (proposal SC4154 granted to R.T., M.S., A.R., B.M., M. I., and L.R.). Financial support from the German Aerospace Center (program "Terrestrial vehicles", project "Next Generation Car" for R.T., M.S., B.M., and L.R.) is gratefully acknowledged.


Julio C Da Silva
email: jdasilva@esrf.fr

Jan Haubrich

Guillermo Requena

Maxime Hubert

Alexandra Pacureanu

Leonid Bloch

Yang Yang

Peter Cloetens

High energy near-and far-field ptychographic tomography at the ESRF

Keywords: X-ray ptychography, ptychographic X-ray computed tomography, PXCT, near-field ptychography, far-field ptychography, high-energy x-rays, tomographic reconstruction methods, synchrotron X-ray imaging

In high-resolution tomography, one needs high-resolved projections in order to reconstruct a high-quality 3D map of a sample. X-ray ptychography is a robust technique which can provide such high-resolution 2D projections taking advantage of coherent X-rays. This technique was used in the far-field regime for a fair amount of time, but it can now also be implemented in the near-field regime. In both regimes, the technique enables not only high-resolution imaging, but also high sensitivity to the electron density of the sample. The combination with tomography makes 3D imaging possible via ptychographic X-ray computed tomography (PXCT), which can provide a 3D map of the complex-valued refractive index of the sample. The extension of PXCT to X-ray energies above 15 keV is challenging, but it can allow the imaging of object opaque to lower energy. We present here the implementation and developments of high-energy near-and far-field PXCT at the ESRF.

INTRODUCTION

In computed X-ray tomography, 2D projections taken in multiple angular directions covering a 180 • -range are used to reconstruct a 3D image of an object by applying the inverse Radon transform to the projection data set. [START_REF] Natterer | The Mathematics of Computerized Tomography[END_REF] The quality of the 3D images strongly depends on the quality of the 2D projections. Thus, different X-ray imaging techniques are used nowadays to improve the contrast and resolution of such projections: in-line holography, [START_REF] Cloetens | Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays[END_REF] propagation-based imaging, [START_REF] Cloetens | Hard x-ray phase imaging using simple propagation of a coherent synchrotron radiation beam[END_REF][START_REF] Snigirev | On the possibilities of xray phase contrast microimaging by coherent highenergy synchrotron radiation[END_REF] and ptychography. [START_REF] Da Silva | Elementary signals in ptychography[END_REF][START_REF] Thibault | High-resolution scanning x-ray diffraction microscopy[END_REF] The common point in all those techniques is the use of coherent X-rays, like the one provided in the 3rd generation synchrotron sources.

One technique which has recently become very popular for its high resolving power and sensitivity is ptychography. [START_REF] Da Silva | Elementary signals in ptychography[END_REF][START_REF] Thibault | High-resolution scanning x-ray diffraction microscopy[END_REF][START_REF] Da Silva | Mass density and water content of saturated never-dried calcium silicate hydrates[END_REF][START_REF] Da Silva | Assessment of the 3d pore structure and individual components of preshaped catalyst bodies by x-ray imaging[END_REF][START_REF] Dierolf | Ptychographic x-ray computed tomography at the nanoscale[END_REF] Such a technique can provide simultaneously the sample's complex-valued refractive index and the illumination, which allows taking into account inhomogeneities in the beam. [START_REF] Da Silva | Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution[END_REF][START_REF] Thibault | Probe retrieval in ptychographic coherent diffractive imaging[END_REF] Actually, the frequency mixture of the structured beams would even improve the quality of the reconstructed images as previously discussed. [START_REF] Da Silva | Elementary signals in ptychography[END_REF][START_REF] Stockmar | Near-field ptychography: phase retrieval for inline holography using a structured illumination[END_REF][START_REF] Guizar-Sicairos | Role of the illumination spatial-frequency spectrum for ptychography[END_REF] For years, part of the mathematical fundamentals of the technique remained poorly understood, but a new theoretical framework was recently proposed. [START_REF] Da Silva | Elementary signals in ptychography[END_REF] It uses the notion of decomposition of a signal into elementary signals proposed by D. Gabor [START_REF] Gabor | Theory of Communication[END_REF] and the concept of frames, [START_REF] Daubechies | Painless nonorthogonal expansions[END_REF][START_REF] Duffin | A class of nonharmonic fourier series[END_REF] which is the core of the theory of wavelets. Such a framework allows us deriving the sampling requirements and other experiments parameters in order to optimize the ptychographic scans even in challenging conditions as the use of high energy X-rays.

Ptychography at ID16A nano-imaging beamline of ESRF is implemented at high-energies (17.05 keV and 33.6 keV ) with very high photon flux. This enables 2D and 3D high-resolution imaging of highly absorbing and bulky samples. With some changes in the experimental setup and very few changes in the phase-retrieval algorithms, [START_REF] Stockmar | Near-field ptychography: phase retrieval for inline holography using a structured illumination[END_REF][START_REF] Enders | A computational framework for ptychographic reconstructions[END_REF][START_REF] Thibault | Reconstructing state mixtures from diffraction measurements[END_REF] such a scanning microscopy technique can be implemented in either near-and far-field regimes. The redundancy of information offered by the overlapping of different scanning spots and oversampling in the detector plane is explored for the ptychographic phase-retrieval procedure. [START_REF] Da Silva | Elementary signals in ptychography[END_REF][START_REF] Thibault | High-resolution scanning x-ray diffraction microscopy[END_REF] Both variants are currently available at ID16A nanoimaging beamline.

Finally, the combination of ptychography with tomography gives rise to the ptychographic X-ray computed tomography (PXCT) technique. [START_REF] Dierolf | Ptychographic x-ray computed tomography at the nanoscale[END_REF][START_REF] Stockmar | X-ray nanotomography using near-field ptychography[END_REF][START_REF] Guizar-Sicairos | Phase tomography from x-ray coherent diffractive imaging projections[END_REF] PXCT is part of the most recent developments on the ID16A beamline of ESRF. It can provide a 3D map of the complex-valued refractive index (n( r) = 1 -δ( r) + iβ( r)) of the sample with the high spatial resolution and high sensitivity. We can obtain the electron density of a sample from the refractive index decrement, δ( r), and its linear attenuation coefficient from the absorption index, β( r). Therefore, PXCT is a fully quantitative technique. [START_REF] Da Silva | Mass density and water content of saturated never-dried calcium silicate hydrates[END_REF][START_REF] Da Silva | Assessment of the 3d pore structure and individual components of preshaped catalyst bodies by x-ray imaging[END_REF] At the ID16A beamline, high-energy PXCT has already been applied to many problems in life and material sciences. We present here the developments in high-energy PXCT at ID16A beamline as well as possibilities of applications in materials sciences. We outline the pipeline used to process the data and obtain the high-quality tomography.

HIGH-ENERGY PTYCHOGRAPHY

In the implementation of far-field ptychography, the sample is scanned with a spatially localized beam and overlapping adjacent scanning spots. An interference speckled-pattern, related to the Fourier transform of the probed part of the sample, is recorded at the detector plane for each scan position in the Fraunhofer regime. [START_REF] Da Silva | Elementary signals in ptychography[END_REF][START_REF] Thibault | High-resolution scanning x-ray diffraction microscopy[END_REF][START_REF] Rodenburg | Hard-x-ray lensless imaging of extended objects[END_REF] In this case, the resolution of the retrieved image is independent of the focus. In the near-field ptychography, transverse diversity and redundancy needed for the phase retrieval are induced by transversely translating the sample within a coherent and structured beam, which is typically larger than the sample. [START_REF] Stockmar | Near-field ptychography: phase retrieval for inline holography using a structured illumination[END_REF] An in-line hologram is recorded on the detector, in the holographic Fresnel regime, for each sample position. Due to the small focal spot provided at the ID16A beamline, we usually record magnified holograms in order to obtain high-resolution images. In both variants of ptychography, the images are reconstructed by iterative algorithms which try to reinforce the consistency between the different scanning positions and the acquired data for each of them, as well as the consistency among the different overlapping scanned regions. Software tools are already available to the community for such reconstructions, and at the ID16A beamline we use the Python package Ptypy. [START_REF] Enders | A computational framework for ptychographic reconstructions[END_REF] One of the main requirements for ptychography is the high degree of coherence of the X-ray beam impinging into the sample. The higher the energy, the harder to obtain such a coherence. In third generation synchrotron sources, the beam is not coherent when it goes out of the ring. Thus, coherence filtering is needed and is implemented by creating a secondary source, which can be a pair of slits at a distance L upstream the sample. The degree of transverse coherence ξ t can be estimated as ξ t = λL/(2s), where λ is the wavelength and s is the aperture of such a secondary source. We can notice that L needs to be relatively long for high energies (short wavelengths) in order to recover enough coherence. This is exactly what it is done at ID16A nano-imaging beamline of ESRF, where the sample is about 185 m from the source and about 145 m from the secondary source. Such a long beamline provides a relatively high degree of coherence of the X-ray beam at the sample position for the ptychography experiments at two discrete energies: 17.05 keV and 33.6 keV .

Secondly, the sampling of ptychographic data requires that there are at least two pixels in the detector for each speckle in the far-field regime or at least two pixels for each Fresnel fringe in the near-field regime. Since the size of a speckle or a Fresnel fringe is smaller as the X-ray energy gets higher, one needs an efficient detector with relatively small pixel. At ID16A beamline, we use indirect detection systems as the FReLoN CCD cameras, [START_REF] Labiche | Invited article: The fast readout low noise camera as a versatile x-ray detector for time resolved dispersive extended x-ray absorption fine structure and diffraction studies of dynamic problems in materials science, chemistry, and catalysis[END_REF] which enables experiments with high photon flux and high energies provided by the beamline. Furthermore, the criterion for optimizing the overlap, the illumination size, scan pattern, sample-to-detector distance, and the detector pixel size have been previously published. [START_REF] Da Silva | Elementary signals in ptychography[END_REF][START_REF] Stockmar | Near-field ptychography: phase retrieval for inline holography using a structured illumination[END_REF] Recently, ptychography at ID16A has successfully carried out using 33.6 keV X-ray photons, which is currently the world's highest energy in which ptychography has been performed. [START_REF] Da Silva | Efficient concentration of high-energy x-rays for diffraction-limited imaging resolution[END_REF] Such achievement opens new possibilities for the application of high-energy PXCT to important scientific problems where the sample is bulky or opaque to lower energies.

THE PRE-PROCESSING STEPS

Before the actual tomographic reconstruction, there are several pre-processing steps to follow: removal of the linear phase ramp and phase offset, phase unwrapping when necessary, vertical and horizontal registration of the projections. Those steps are described in the following subsections.

Linear phase ramp removal and phase offset

In the ptychographic reconstruction, an ambiguity on the linear phase term arises from the fact that both the object and illumination are simultaneous retrieved from data taken at different scanning positions. This ambiguity is intrinsic to ptychography. Since the wavefield past the sample is assumed to be the multiplication of the complex-values functions of the object and the of the illumination, a linear phase ramp multiplying the object can be compensated by a linear phase ramp of opposite sign multiplying the illumination. Additionally, any information about an overall phase shift provided by the sample is lost because we can only measure intensity patterns. Thus, the reconstructions should be corrected by an additive constant and linear phase terms. At ID16A beamline, we used a method proposed by Guizar-Sicairos et al. [START_REF] Guizar-Sicairos | Phase tomography from x-ray coherent diffractive imaging projections[END_REF] for those corrections. A priori information of a known part of the sample is required in order to estimate the additive phase constant and we typically use a region of air as reference.

Phase unwrapping

Ptychography is a technique which allows the phases to go over several π's. [START_REF] Da Silva | Mass density and water content of saturated never-dried calcium silicate hydrates[END_REF] However, the phases are retrieved using expression involving arctangent, i.e, extracted from a trigonometric function e iϕ , where ϕ is the phase. Therefore, the resulting phases are given in modulo 2π and discontinuities appear when the phases reach π or -π. The unwrapping procedure consists in resolving such discontinuities and restoring the desired continuous phase function. At ESRF, we typically used algorithms like the classical Goldstein's 23 and Flynn's 24 algorithms or others like the one implemented the Python package Scikit-image [START_REF] Van Der Walt | scikit-image: image processing in Python[END_REF] proposed by Herraez et al. [START_REF] Herráez | Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path[END_REF] The latter is more often used because it is faster than the first two ones. Nevertheless, for the more complicated cases we use the Goldstein's algorithm and, when the result is still not satisfactory, we use Flynn's algorithm, which is quite slow, but efficient.

Registration of the projections

In PXCT, the sample is transversely displaced by piezo motors for each projections over a rotation of 180 • . The long experiments in addition to imprecisions of the motors will cause misalignments of the projections. In order to reconstruct high-resolution 3D images using tomography, one has to register such a set projections with less then one pixel precision. At ID16A beamline, we align the projections independently in the vertical and horizontal directions with sub-pixel precision using algorithms adapted to the information contained in the ptychographic projections. [START_REF] Guizar-Sicairos | Phase tomography from x-ray coherent diffractive imaging projections[END_REF]

Vertical registration

For vertical alignment of the projections, we use the phase-contrast projections obtained by ptychography and the Helgason-Ludwig consistency condition. [START_REF] Ludwig | The radon transform on euclidean space[END_REF][START_REF] Helgason | The radon transform on euclidean spaces, compact two-point homogeneous spaces and grassmann manifolds[END_REF] Such a condition states that:

R(f )(θ, p)dp = f (x, z)dxdz, (1)
where R(f ) is the Radon transform (sinogram) of the 2D function f (x, z), with f (x, z) being the 2D sub-space of the 3D function f (x, y, z), p is the radial coordinate of the sinogram, θ is the projection angle, z is the direction of the X-ray beam, and the x, y-coordinates define the plan of the projection.

Equation ( 1) is basically the Plancherel's theorem for the Radon transform [START_REF] Durrani | The radon transform and its properties[END_REF] and states that the integral of any projection along p is independent of the angle θ. Thus, we can define the vertical mass distribution vmd θ (y) as the integration of the phase along horizontal direction x of the projections:

20 vmd θ (y) = ϕ θ (x, y)dx = - 2π λ δ(x cos θ -z sin θ, y, z cos θ + x sin θ)dxdz, (2)
where ϕ θ (x, y) represents the phase-contrast projections and δ(.) is the refractive index decrement of the sample. Therefore, based on this, Guizar-sicairos et al. [START_REF] Guizar-Sicairos | Phase tomography from x-ray coherent diffractive imaging projections[END_REF] derived an iterative alignment approach to maximize the correlation of the vertical mass distributions of the sample and align the projections vertically. This has the advantage that we only use a 1D function for the alignment rather than using the complete 2D images. At ESRF, we implemented such algorithm in Python and we use for the alignment of the near-and far-field ptychography projections.

Horizontal registration

We perform the horizontal alignment of the projections using the tomographic consistency condition, which states that there is an unique relationship between the sinogram and the reconstructed tomographic slice. [START_REF] Guizar-Sicairos | Quantitative interior x-ray nanotomography by a hybrid imaging technique[END_REF][START_REF] Mayo | Software image alignment for x-ray microtomography with submicrometre resolution using a sem-based x-ray microscope[END_REF][START_REF] Dengler | A multi-resolution approach to the 3d reconstruction from an electron microscope tilt series solving the alignment problem without gold particles[END_REF] Based on such a condition and on previously published algorithms, [START_REF] Mayo | Software image alignment for x-ray microtomography with submicrometre resolution using a sem-based x-ray microscope[END_REF][START_REF] Dengler | A multi-resolution approach to the 3d reconstruction from an electron microscope tilt series solving the alignment problem without gold particles[END_REF] Guizar-sicairos et al. [START_REF] Guizar-Sicairos | Quantitative interior x-ray nanotomography by a hybrid imaging technique[END_REF] proposed an improved iterative alignment in which, a tomographic reconstruction of the misaligned projections is obtained following by a forward Radon transform to obtain the sinogram from that reconstruction. The consistency condition states that such a sinogram will be the original sinogram only if both are consistent with a 3D representation. This will not be the case given that the projections are misaligned. Thus, we can use that consistency to robustly refine the misaligned sinogram to match the sinogram obtained from the reconstruction at each iteration to align the projections. For the inverse Radon transform, we use the modified filtered-back-projection algorithm described in the next section.

THE TOMOGRAPHIC RECONSTRUCTION

There are several tomographic reconstruction algorithms, and filtered-back-projection (FBP) [START_REF] Bracewell | Inversion of fan-beam scans in radio astronomy[END_REF] is the most used one. Since it is not iterative, it does not present problems of convergence and it is robustly fast. In addition, it allows to apply frequency cutoffs and filters to the data. At ESRF we use a modified version of FPB which calculates the tomographic slices from the derivatives of the projections. By modifying the filter, such modified FBP can even be insensitive to the phase wrapping [START_REF] Guizar-Sicairos | Phase tomography from x-ray coherent diffractive imaging projections[END_REF] and, therefore, it is of great interest to PXCT reconstructions.

Modified Filtered Back Projection (FBP) algorithm

The standard FBP algorithm implements the reconstruction by: 20

- 2πu λ δ( r) = π 0 F -1 f { | f | F x [ϕ θ (x, y)] }dθ, (3)
where F[.] is the forward 1D Fourier transform, F -1 {.} is the inverse Fourier transform, u is the voxel size, and | f | is the ram-lak filter applied to the projections. By using the derivative property of the Fourier transform, F x {∂ϕ/∂x} = i2πuF x {ϕ}, we can rewrite the equation (3) as: 20

- 2πu λ δ( r) = π 0 F -1 f 1 i2π sgn(f ) F x ∂ϕ θ (x, y) ∂x dθ, (4)
where ∂ϕ θ (x, y)/∂x is the derivative of the phase projections and sgn(.) is the signal function. The new filter is now 1/[i2π sgn(f )], which is sometimes referred as Hilbert filter. For the PXCT reconstructions at ESRF, we used such a modified version of FBP.

EXPERIMENTAL RESULTS

PXCT is routinely offered at ID16A beamline of ESRF in both near-and far-field regimes and has been applied to a wide range of scientific problems in life and material sciences. A recent example of applications of near-field PXCT to fuel cells was demonstrated in a previous work [START_REF] Stockmar | X-ray nanotomography using near-field ptychography[END_REF] and another example of far-field PXCT application to materials research is shown below.

Applications to materials science

Far-field PXCT was recently applied to study interfaces between metals and polymer for applications in aerospace industry. The sample was made of a titanium alloy bounded to a polymer and a deposition layer of platinum on the top coming from the focused-ion-beam (FIB) preparation. The experiments were performed at ID16A beamline using an energy of 17.05 keV and 1 % bandwidth. We used a binary acquisition strategy 34 with 4 interlaced sub-tomograms acquired over a range of 180 • . 360 projections were acquired with angular step of 0.5 • . Each projection was obtained from a far-field ptychography experiment with a beam size of 1.4 µm at the sample position and scanning step size of 600 nm. The coherent diffraction patterns were acquired with a FReLoN CCD detector [START_REF] Labiche | Invited article: The fast readout low noise camera as a versatile x-ray detector for time resolved dispersive extended x-ray absorption fine structure and diffraction studies of dynamic problems in materials science, chemistry, and catalysis[END_REF] with effective pixel size of 19.6 µm located at 1.2 m downstream the sample. The exposure time in each position was 0.5 s and the beam divergence past the KB mirrors was 0.6 mrad. Thence, we can estimate the sampling ratio, which was 1/αβ = 7.6, where α = 0.43 is the ratio step-size/beam-size and β = 0.31 is the inverse of the oversampling factor in the reciprocal space. [START_REF] Da Silva | Elementary signals in ptychography[END_REF] This indicates that the sampling requirements were respected. The ptychographic phase-retrieval was carried out using the package Ptypy 17 and tomographic processing steps were performed using inhouse developed Python scripts follow the steps described in the sections above.

Figure 1 displays the results of the PXCT data processing. Figure 1(a) shows an axial slice, without any post-processing, of the volume where we can distinguish the different materials. The voxel size is 13.06 nm and the diameter of the sample is 18.3 µm. Figure 1(b) presents the 3D rendering of the volume where the different materials are indicated: blue for a protecting platinum layer used during sample preparation, cyan for the polymer layer, gray for the titanium. We have also identified some porosity regions formed during processing of the titanium surface. This example is an important scientific case where high-energy X-rays PXCT represents a powerful imaging technique.

CONCLUSIONS

We described here the implementation of near-and far-field ptychography at ID16A nano-imaging beamline of ESRF. We tackled the challenge of performing such a technique with high-energy X-rays (17.05 keV and 33.6 keV ) and sufficiently high degree of coherence required by ptychography. This will therefore allow better resolved 3D imaging of highly absorbing and thick samples, commonly found in metallurgy, energy related materials or packaged semiconductor devices, for example. The PXCT benefits from the high flux to image the object with enough sensitivity, i.e., high sensitivity to weak variations of electron density of a sample. Several applications are already taking advantage of the high-energy PXCT at ESRF as shown in section 5.1 and figure 1. We also showed that the capability of ptychography to provided the complex-valued refractive index of the sample opens new opportunities for methodological developments in terms of tomographic data processing.

Figure 1 .

 1 Figure 1. Hybrid metal-polymer material. (a) An slice of the tomographic volume obtained by PXCT. (b) 3D rendering of the volume. The different material phases are indicated. The voxel size is 13.06 nm and the diameter of the cylinder is 18.3 µm.

ACKNOWLEDGMENTS

The authors acknowledge the technical support and instrumentation assistance of L. Andre, F. Villar, R. Baker, P. van der Linden, E. Gagliardini, Y. Dabin, A. Vivo, R. Barrett, C. Morawe, C. Jarnias, J.C. Labiche, J. Morse, and O. Hignette. We also thank S. Chalkidis, P. Thibault, and B. Enders for the assistance with Ptypy. Additionally, we are grateful to M. Guizar M. Guizar-Sicairos, A. Diaz and A. Menzel for the discussions on the registration and tomographic reconstruction algorithms and to T. Elberfeld for the implementation of Goldstein's and Flynn's algorithms.