Dominique Bernard
email: dominique.bernard@icmcb.cnrs.fr

F Léonard

E Plougonven

G Bruno

On the use of autocorrelation functions, permeability tensors, and computed tomography to characterise the anisotropy of diesel particulate filter materials

Keywords: Anisotropy, Autocorrelation function, Computed tomography, Permeability tensor, Diesel particulates filter, Ceramics

On the use of autocorrelation functions, permeability tensors, and computed tomography to characterise the anisotropy of Diesel particulate filter materials

Introduction

3D images such as the reconstruction of X-ray tomography data can provide a wealth of information on the internal structure of materials, but quantification of specific geometrical or topological characteristics linked to some bulk physical property is far from straightforward. It is well known that properties of the crystal lattice of such materials are anisotropic, but the degree of anisotropy of the macroscopic properties depends on the meso-structure (i.e. the arrangement of the material and the pore space).

This study focuses on methods to quantify the differences in physical properties as a function of direction, i.e. their anisotropy, and how it can be linked to measures of anisotropy of the internal structure of the material.

The methods to measure the anisotropy of meso-structures through image analysis have mostly originated from the study of 2D images to characterize deformed grains in the domains of geosciences [START_REF] Sander | An introduction to the study of fabrics of geological bodies[END_REF] or metallurgy [START_REF] Gokhale | Quantitative Characterization and Representation of Global Microstructural Geometry[END_REF]. More recently, in the medical domain research aiming at characterizing the structure of trabecular bones has produced various secant-based methods to compute standardized histomorphometry parameters [START_REF] Parfitt | Bone histomorphometry: Standardization of nomenclature, symbols, and units: Report of the asbmr histomorphometry nomenclature committee[END_REF]. Such methods are based on shooting parallel lines at different angles in the object, and counting the number of traversals through the porous or solid phase versus the lengths of the segments. The most widely used is called Mean Intercept Length (MIL) [START_REF] Whitehouse | The quantitative morphology of anisotropic trabecular bone[END_REF], others include Intercept Segmentation Deviation (ISD) [START_REF] Chiang | Quantifying the directional parameter of structural anisotropy in porous media[END_REF], or Line Fraction Deviation (LFD) [START_REF] Varga | Sampling sphere orientation distribution: An efficient method to quantify trabecular bone fabric on grayscale images[END_REF]. There are also point-based methods, that perform local measurements (again by shooting lines) from different points in the image, such as Volume Orientation (VO) [START_REF] Odgaard | Estimation of structural anisotropy based on volume orientation. A new concept[END_REF], Star Chord Length (SCL) [START_REF] Chiang | Local thickness and anisotropy approaches to characterize pore size distribution of three-dimensional porous networks[END_REF], Star Volume Distribution (SVD) [START_REF] Cruz-Orive | Characterizing anisotropy: A new concept[END_REF], or Star Length Distribution (SLD) [START_REF] Odgaard | Fabric and elastic principal directions of cancellous bone are closely related[END_REF]. The main drawback of these methods is that they require segmented images, i.e. images that have been processed to separate the background (the pore space) and the foreground (the bone in these works).

The segmentation step can be a strong source of error for images with high level of noise or affected by acquisition artefacts. Methods, circumventing the need to segment and thus working directly on the grey-level images, have been investigated. Examples are the Fast Fourier Transform (FFT) for 2D radiographs [START_REF] Millard | Power spectral analysis of vertebral trabecular bone structure from radiographs: Orientation dependence and correlation with bone mineral density and mechanical properties[END_REF], Gabor wavelets [START_REF] Xiang | Comparative assessment of bone mass and structure using texture-based and histomorphometric analyses[END_REF], or Tensor Scale Distribution (TSD) [START_REF] Saha | A robust method for measuring trabecular bone orientation anisotropy at in vivo resolution using tensor scale[END_REF]. The main problem of these methods is their expensive computational time that makes their extension to 3D almost prohibitive.

These practical limitations to 2D require a priori knowledge of a symmetry plane of the studied material to be unbiased. An attractive alternative, combining possibility to process grey-level images and computational efficiency, is the spatial Auto-Correlation Function (ACF) [START_REF] Rotter | Autocorrelation analysis of bone structure[END_REF][START_REF] Wald | Study of trabecular bone microstructure using spatial autocorrelation analysis[END_REF].

The ACF gives a similarity measure in the volume as a function of distance and direction. It consists in a cross-correlation of the image with itself. The ACF belongs to the family of the two-points correlation functions. These functions condense a lot of information about the morphology of the processed images, making then suitable for applications as different as advanced segmentation of 3D images of granular materials [START_REF] Faessel | Segmentation of 3D microtomographic images of granular materials with the stochastic watershed[END_REF] or reconstruction of different realisations of similar heterogeneous materials [START_REF] Jiao | Modeling heterogeneous materials via two-point correlation functions: Basic principles[END_REF].

In these examples, as in most of the applications, the studied materials are considered as isotropic, making the two-points correlation function depend on the distance only.

Regarding the anisotropy estimate, autocorrelation has been compared with MIL by Wehrli et al. in [START_REF] Wehrli | Quantitative MRI for the assessment of bone structure and function[END_REF][START_REF] Wald | Spatial autocorrelation and mean intercept length analysis of trabecular bone anisotropy applied to in vivo magnetic resonance imaging[END_REF]. For high resolution images of trabecular bones MIL is considered as the gold standard, but the authors favour autocorrelation because of its speed and lower sensitivity to noise. The ACF can be easily computed for 3D images after segmentation or not. As shown in this paper, when the full 3D ACF is analysed the different directions of anisotropy can be revealed without a priori. We focus on this method to quantify the anisotropy of Diesel Particulate Filter (DPF) materials from 3D reconstructions of X-ray tomography data and compare the results with the ones given by another approach based on their filtration properties.

Diesel Particulate Filter (DPF) materials are usually porous ceramics, often containing micro-cracks [START_REF] Adler | Ceramic diesel particulate filters[END_REF][START_REF] Bruno | Not All Microcracks are Born Equal: Thermal vs. Mechanical Microcracking in Porous Ceramics[END_REF]. The reasons for this choice are manifold: i-they can be used at very high temperatures (typically around 800 °C, but sometimes in excess of 1000 o C); ii-they have very good thermal shock resistance in their temperature range of application (especially if micro-cracked); iii-they are inert to many gases and combustion products; iv-they can be manufactured with tailored porosity fraction.

They are typically produced by extrusion of a slurry into the desired filter shape, with successive sintering at high temperature [START_REF] Kasai | Cordierite ceramic filter and method of producing the same[END_REF][START_REF] Beall | Cordierite porous ceramic honeycomb articles with delayed microcrack evolution[END_REF]. This extrusion process causes anisotropy at both microscopic (grains, pores) and macroscopic (properties) levels. In other words, the crystallographic (grain orientation), meso-structural (pore space orientation), and material properties (mechanical, thermal, and filtration) acquire a preferential orientation along the extrusion axis. This orientation dependence extends at the component level, and is exacerbated by the cellular geometry of the filter. Indeed, a dependence of material properties on direction has been reported at the component level (Young's modulus and thermal expansion, [START_REF] Bubeck | Direction dependent mechanical properties of extruded cordierite honeycombs[END_REF]) and at the material level (Coefficient of Thermal Expansion [START_REF] Bruno | On the stress-free lattice expansion of porous cordierite[END_REF], Fracture toughness [START_REF] Shyam | The Thermal Expansion, Elastic and Fracture Properties of Porous Cordierite at Elevated Temperatures[END_REF][START_REF] Gordon | The Relationship Between Microstructure and Fracture Toughness for Fibrous Materials for Diesel Particulate Filters[END_REF], stress-strain curves [START_REF] Lichtner | Effect of Macropore Anisotropy on the Mechanical Response of Hierarchically Porous Ceramics[END_REF]). Recently, an analysis of both X-ray refraction and computed tomography images [START_REF] Kupsch | Evaluating Porosity in Cordierite Diesel Particulate Filter Materials, Part 1-X-ray Refraction[END_REF][START_REF] Onel | Evaluating Porosity in Cordierite Diesel Particulate Filter Materials, Part 2 Statistical Analysis of Computed Tomography Data[END_REF][START_REF] Kupsch | Microstructure characterisation of ceramics via 2D and 3D X-ray refraction techniques[END_REF] has shown that the orientation of the pore space is directly correlated with the anisotropy of the mechanical and thermal properties.

The pore space in these materials is totally interconnected and has a very complex geometry making its decomposition in representative pore elements difficult.

Extracting relevant average properties allowing the use of models based on 'unit cells' containing voids of defined geometry [START_REF] Leigh | Modelling of elastic constants of plasma spray deposits with ellipsoid-shaped voids[END_REF][START_REF] Roberts | Elastic properties of model porous ceramics[END_REF] is consequently questionable. Instead, we need to consider, as done in this work, approaches that tackle the problem of determining global properties of the materials using the complete 3D images.

Besides mechanical and thermal properties, certainly important for the structural stability of such DPFs, the functional properties (and especially filtration) are extremely relevant for their performance. Not much work is available in the open literature dealing with the effect of the meso-structure of such filters on the filtration properties.

Relevantly for the present study, synthetic cordierite meso-structures have been investigated using X-ray tomography data in [START_REF] Schmidt | Generating 3D model of microporous ceramics[END_REF]; however, while global porosity fraction and permeability values have been obtained, no information about the permeability tensor orientation has been given. Otherwise, much work has been dedicated to numerical models of permeability and pressure drop, see e.g. [START_REF] De Carvalho | Pore-Scale Numerical Investigation of Pressure Drop Behaviour Across Open[END_REF], the latter being relevant for industrial applications.

While the meso-structure of these materials has been classically investigated by means of scanning electron microscopy (see e.g. [START_REF] Gordon | The Relationship Between Microstructure and Fracture Toughness for Fibrous Materials for Diesel Particulate Filters[END_REF][START_REF] Masoudi | Predicting Pressure Drop of Wall-Flow Diesel Particulate Filters -Theory and Experiment[END_REF]), since the resolution needed lies within the realm of SEM, a global view of the meso-structure is also needed, yet with the highest possible resolution. Optical microscopy (OM) can yield relevant information, but is 2D and lacks resolution. On the contrary, X-ray computed tomography (CT) has the potential to disclose features in the bulk (3D analysis), with a resolution in between SEM and OM, and a large field of view, comparable to OM. As with other imaging techniques, if one is to go beyond qualitative assessment of the features of a structure, image processing algorithms are needed to extract quantitative information.

The scope of this work is to apply specific methods (Autocorrelation function; Permeability calculations) to 3D CT reconstructions in order to characterize the structural and filtration anisotropies. We consider samples extracted from filters of different materials (cordierites and silicon carbide) having similar porosity fraction but different arrangement of the pore space. We determine their permeability tensors as well as the autocorrelation function of the computed tomography reconstructions and use these results to quantify the anisotropy of the materials by two different methods. These global approaches have the advantage of being directly comparable with analytical and numerical global models, designed to extract or predict properties.

Characterization methods

Computed Tomography

X-ray computed tomography (CT) is a non-destructive imaging technique based on differences in X-ray absorption from the multiple constituents of a specimen (different materials, densities, and/or features such as inclusions, pores, cracks, matrix). The specimen is placed between an X-ray source and a detector, then rotated so that a series of 2D radiographic images (usually thousands) can be acquired over a rotation of the specimen on 180° or 360°. These 2D radiographic images are then processed so that a 3D digital reconstruction of the specimen can be obtained. The reconstructed volume is a 3D image composed of 3D pixels with a grey value representing the X-ray attenuation in the corresponding volume in the specimen. Therefore, to be able to resolve a certain feature, two conditions are required: first, the feature must be significantly larger than the pixel size of the reconstructed volume, and second, there should be enough contrast in the grey values corresponding to the different features of interest.

Laboratory X-ray computed tomography

Laboratory XCT experiments were performed on a GE v|tome|x L 300/180 equipped with a 180 kV source, a tungsten transmission target (actual focal spot size below 2 µm as determined with JIMA test pattern RTC02), and a GE 2000 × 2000 pixel DXR-250 detector. The source was operated at a voltage of 60 kV and a current of 170 µA. The sample projections were taken at 1500 angular positions per 360° (increments of 0.24°), and an exposure time of 3 seconds. The samples were placed 8 mm downstream from the source, with a source-detector distance of 800 mm, and a binning of 2×2 pixels, so that the effective magnification obtained was 50. The resulting pixel size was 4.0 µm and the scan time 1h15min. In order to ensure the projection of the entire sample, pieces of 2 × 2 cells were cut out of the honeycomb filter.

Synchrotron radiation X-ray computed tomography

Synchrotron-CT measurements were performed at the beamline BAMline [START_REF] Görner | BAMline: the first hard X-ray beamline at BESSY II[END_REF][START_REF] Rack | High resolution synchrotron-based radiography and tomography using hard X-rays at the BAMline (BESSY II)[END_REF] at the synchrotron source BESSY II of the Helmholtz-Zentrum Berlin für Materialien und Energie (HZB), Berlin, Germany. The beam energy was set to 15 keV, in order to achieve optimal contrast for the investigated porous filter structures. A PCO4000 CCD camera (4008 × 2672 pixels) was used in combination with an Optique Peter microscope equipped with a CdWO4 scintillator and a 10-fold magnifying objective, resulting in a pixel size of 0.434 µm. A series of 3200 projections were acquired per 180° rotation, with a 4 s exposure. A small sample-detector distance of 15 mm was set to ensure a limited scattering range of refractive edge artefacts and the entrance slits were narrowed to the field of view (FoV = 1.7 × 1.1 mm 2 ) in order to avoid detector backlighting [START_REF] Lange | Numerical correction of X-ray detector backlighting[END_REF][START_REF] Kupsch | Einfaches Korrigieren des Hinterleuchtens von Röntgendetektoren[END_REF]. To avoid the sample size exceeding the FoV, small splinters of less than 1.5 mm size were prepared from each sample, containing just one crossing region and one protruding wall of the filter.

Autocorrelation Function

The autocorrelation function of a real-valued image I is defined as:

ACF 𝐯 〈I 𝐮 . I 𝐮 𝐯 〉 (1)
where I(u) is the value of the image at the position specified by the vector u, I(u+v) the value of the image at the position u plus the lag vector v, and < > is the averaging operator over the support of the image. With this expression ACFI(v) is the expectation of the product of the image I with a copy of I that is offset by v.

Most of the time what is called autocorrelation function does not correspond to

expression [START_REF] Sander | An introduction to the study of fabrics of geological bodies[END_REF], but to the NACFC, the normalised autocorrelation function of the centred image (I(u) - I ) [START_REF] Thissen | An autocorrelation method for threedimensional strain analysis[END_REF], noted CI(v):

C 𝐯 〈 〉 𝐯 〈 𝐮 〈 〉 . 𝐮 𝐯 〈 〉 〉 〈 𝐮 〈 〉 〉 (2)
where Var(I) is the variance of the image I.

Both expressions (1 and 2) quantify, as a function of v, the correspondence of I with its copy offset by v. A maximum always occurs for v = 0 and, in case of a periodic image, for any v = l p , l p being the lattice vectors defining the periodic cell. For all other values of v, ACFI and CI contain information about the effect of distance (‖𝐯‖) and orientation (𝐯 ‖𝐯‖ ⁄ ) on the correlation of the image I with itself. This encouraged some authors to attempt extracting dimensional information (grain size for instance) from ACFI or CI, but with a limited success for real or realistic 3D images of materials (see [START_REF] Thissen | An autocorrelation method for threedimensional strain analysis[END_REF] for a review on usage in geosciences, and Appendix A for a note concerning the use of CI for this purpose). On the contrary, anisotropy estimates seem to be more achievable even if some aspects remain unclear [START_REF] Bernard | Constrained sintering of glass films: Microstructure evolution assessed through synchrotron computed microtomography[END_REF][START_REF] Heilbronner | The autocorrelation function: an image processing tool for fabric analysis[END_REF].

To illustrate the possibilities offered by the use of the autocorrelation function, we consider the example of the dense packing of ellipsoidal particles represented by The starting point was a random close packing of 10 5 non-overlapping spheres produced using a generator based on [START_REF] Zinchenko | Algorithm for random close packing of spheres with periodic boundary conditions[END_REF] (see Appendix A for more details). All the spheres, having the same density (set equal to 1) and the same diameter (set equal to 2.2995 10 -2 in relative units to the cube edge), were placed within a unit cube (edges equal to one). Periodic boundary conditions were applied and the resulting packing had a theoretical porosity fraction equal to 0.3634. The positions of the sphere centres were modified by dividing the B coordinate (green axis Fig. 1) by 2, and the C coordinate (blue axis) by 10. An ellipsoidal particle (Fig. 1.b) was then placed at each modified centre position, and a rotation of 23° around the A axis (red axis) applied. Finally, the discrete binary image IEl (1000 3 pixels) presented in Fig. 1.a was generated from the central region of this ellipsoid packing.

The resulting image is simpler than 3D images of real materials because:

 all particles are identical (geometry, density, and orientation),

 there is no overlap between particles  all distances between near neighbours are different (because of randomness), but similar (because of compactness).

On the other hand, this image is much more relevant than examples regularly used in the literature (single 2D/3D objects or loose aggregates [START_REF] Thissen | An autocorrelation method for threedimensional strain analysis[END_REF][START_REF] Heilbronner | The autocorrelation function: an image processing tool for fabric analysis[END_REF][START_REF] Heilbronner | Image Analysis in Earth Sciences. Microstructures and Textures of Earth Materials[END_REF]) as it represents a very dense random packing (porosity fraction of the final discretised packing = 0.3279) with very anisotropic particles.

The corresponding NACFC, CIEl(v), was computed in about 3 minutes on a middle-range PC using a classic Fast Fourier Transform algorithm [START_REF] Frigo | The Design and Implementation of FFTW3[END_REF]. CIEl(v) is a 3D real-valued function with complicated visual representation. For isotropic images, it is sufficient to represent CIEl(r.w) as a function of r for any direction w (see examples in Appendix A).

For anisotropic images, 1D or 2D representations can be misleading if the anisotropy principal directions are not known [START_REF] Bernard | Constrained sintering of glass films: Microstructure evolution assessed through synchrotron computed microtomography[END_REF]. This is the reason why approaches to the determination of the pore space anisotropy such as the one presented in [START_REF] Onel | Evaluating Porosity in Cordierite Diesel Particulate Filter Materials, Part 2 Statistical Analysis of Computed Tomography Data[END_REF] do not fully describe the orientation of the meso-structure. In Fig. 2.a, the shape of CIEl(v) is deformed because the axis C, perpendicular to the section, is not a principal direction of anisotropy. In Fig. 2.b, the axis perpendicular to the section is A, a principal direction of anisotropy, and the 2D image is not deformed. Consequently, determining the principal directions of anisotropy appears mandatory before any further analysis. For that purpose, we developed the following procedure: first, VIEl(c), the volume made by the pixels with a correlation function (or its normalised version) value larger than a given threshold c (c = 0.5 for the example given in Fig. 2.c), is built. Second, the tensor of inertia of this volume is computed. Third, 1, 2, 3, the principal directions of this tensor are computed, giving the three principal directions of anisotropy.

For each principal direction of anisotropy, it is then easy to represent the evolution of NACFC as a function of the lag along this direction. The resulting curves are presented Fig. 3 for the image IEl. Interestingly, the fact that DA_Ellips > DA_Inertia implies that the shape of the ellipsoid VIEl(c) is closer to a disc than to a needle (see Appendix B).

A priori, both quantities are relevant, and it is only in the frame of a comparison between different methods for estimating the degree of anisotropy that a choice between them can be done. In this work, comparison is done with the values extracted from the permeability tensor computed from binary 3D images.

Computation of permeability tensor

In his seminal experimental work, Darcy [START_REF] Darcy | Les fontaines publiques de la ville de Dijon, Victor Dalmont[END_REF] showed that when water is flowing through sand filters there is a linear relation between the flux and the pressure drop divided by the thickness of the filter. The coefficient of proportionality is the hydraulic conductivity. It comprises two terms: the first, viscosity, depends on the fluid, and the second, permeability, depends on the porous medium. More recently, several theoretical derivations of Darcy's law have been proposed leading to the vector form that is now generally used:

𝐕 𝐊 . 𝛁p (3)
where V is Darcy's velocity (m s -1 ), K the permeability tensor (m 2 ), µ the fluid dynamic viscosity (kg m -1 s -1 ),  the gradient operator, and p the fluid pressure (kg m -1 s -2 ).

If, at the pore scale, the fluid flow is driven by Stokes' equations, Eq. 3 can be derived by change of scale using volume averaging [START_REF] Whitaker | The Method of Volume Averaging, Theory and Applications of Transport in Porous Media[END_REF]. With this method we not only derive Eq. 3, but also establish conditions for Darcy's equation to be valid. In so doing, we introduce the concept of REV (Representative Elementary Volume, i.e. the smallest volume over which the computed permeability tensor will be representative of the porous material), and construct a partial differential problem (the closure problem). The solution of this problem can be used to compute the permeability tensor directly from the local geometry [START_REF] Bernard | Permeability assessment by 3D interdendritic flow simulations on microtomography mappings of Al-Cu alloys[END_REF]. The closure problem can be written as:

𝛁𝐝 ∇ 𝐃 𝐈 0 ; 𝛁. 𝐃 0 ( 4
)
where I is the identity tensor, d an unknown vector and D an unknown tensor. This set of equations, very similar to Stokes equations, is completed by the following boundary conditions:

𝐃 0 at A ; 𝐝 r l 𝐝 r ; 𝐃 r l 𝐃 r , i 1,2,3 (5)
where Asl is the solid-liquid interface. The porous medium is supposed to be periodic, and (li, i=1,2,3) are the components of the lattice vector defining the periodic cell.

From D, the permeability tensor can be expressed as:

𝐊 ε 〈𝐃〉 ( 6
)
where  is the global porosity fraction, and - l is the notation for the averaging operator applied to the pore space only (see [START_REF] Whitaker | The Method of Volume Averaging, Theory and Applications of Transport in Porous Media[END_REF] for details).

This change of scale is possible when the sample is large enough to be representative of the porous medium. The 3D image is then an REV, and periodic boundary conditions (Eq. 5) can be used without noticeable perturbations of the closure problem solution [START_REF] Bernard | Using the volume averaging technique to perform the first change of scale for natural random porous media[END_REF]. Consequently, when the conditions for the change of scale are verified, it is possible to compute the permeability tensor of a porous sample directly from a segmented 3D image by solving the closure problem (Eq. 4 to 6).

From Eq. 4 to 6 it is obvious that, through the boundary condition at the solidliquid interface Asl, the permeability tensor is an implicit function of the geometry at the meso-scale only. Simplifying this implicit function, for instance to a reduced set of measureable characteristic parameters, is the object of a longstanding and continuous effort of research with rather limited advances [START_REF] Xiao | Geometry models of porous media based on Voronoi tessellations and their porosity-permeability relations[END_REF][START_REF] Hommel | Porosity-Permeability Relations for Evolving Pore Space: A Review with a Focus on (Bio-)geochemically Altered Porous Media[END_REF].

For many materials, direct measurement of permeability in a single direction is arduous, and estimating the full permeability tensor is generally impossible. Combining modern imaging techniques to acquire high quality 3D images describing the local meso-structure and efficient numerical code to solve the closure problem, a good approximation of the complete permeability tensor can be obtained. Although in practical applications solving the closure problem is always computationally demanding, this approach produces a relevant characteristic for anisotropic media that is experimentally unreachable.

In this work the closure problem is solved using an enhanced version of permea3D, a code developed at ICMCB (Bordeaux, France) and presented in [START_REF] Bernard | Permeability assessment by 3D interdendritic flow simulations on microtomography mappings of Al-Cu alloys[END_REF]. In the frame of this work, the parallelization for very large 3D images was implemented. Qualitatively, Cord1 to Cord3 are similar materials, as they have similar chemical composition, pore former content and sintering schedule. In fact, they show similar pore space features: "pore elements" (intended as the dark objects visible in the CT slices) are sharp-edged and have a wide size distribution (accepting the legacy of the classic approach of measuring the pore size distribution by means of mercury porosity).

Diesel particulate filter materials and their meso-structure

A
On the contrary Cord4 possesses much smoother pore/material interfaces and a larger porosity fraction (though finer pore space features). For SiC, traces of the powder particles composing the final material are clearly visible (analogous to IE1); correspondingly, the "pore elements" have a rounded shape. The total porosity fraction is lower than for the cordierite samples. Interestingly, the homogeneity of the "pore elements" visible in the CT slices is higher for SiC and Cord4 than for Cord1 to Cord3. This is due to the fact that the SiC sample was probably manufactured using a sieved SiC powder (i.e., with narrow

a) b) c) d) e) f)
particle size distribution), and Cord4 was manufactured with smaller pore former particles than in the case of Cord1 to Cord3.

No clear global orientation of the pore space is visible from the CT slices.

Results

Analysis of Computed Tomography Data

The slices presented in Fig. 4 are extracted from the five 3D grey-level volumes used in this work. Their characteristics are given in Table 2; PS is the pixel size, in µm, and NA, NB, NC the dimensions of the reconstructed volumes in pixels. In the first column of Fig. 5, 3D renderings of two of these volumes are presented. The study of the anisotropy requires a unique definition of the sample orientation. In this work we used reference axes related to the materials processing: A is along the wall length (red), B is in the extrusion direction (green), and C is normal to the wall (blue). Since sample positioning for the tomographic acquisitions was not precise enough, the volumes have been numerically realigned with the A, B, C axes. Since permeability computations require binary images, the sub-volumes were segmented following the procedure exposed by Bernard et al. [START_REF] Bernard | Constrained sintering of glass films: Microstructure evolution assessed through synchrotron computed microtomography[END_REF]: first, we filtered the grey-level images using an in-house developed anisotropic diffusion filter. The histograms of the filtered images presented two well identified peaks separated by a plateau. The grey-level situated in the middle of the two peaks was used as threshold value for binarization. Finally, considering the very large sizes (in terms of number of pixels) of the sub-volumes from the synchrotron acquisitions, they were sub-sampled by a factor of two.

Table 1 summarises the characteristics of the seven binary sub-volumes (plus the synthetic image presented Fig. 1) that are used for computations: pixel size (PS, in µm), dimensions along the A, B and C axes (NA, NB, NC, in pixels), and porosity fraction (CT). For comparisons, the porosity fraction values obtained by mercury porosimetry (Hg) are also given. Values obtained through the two methods are in good agreement.

Autocorrelation functions

The results detailed in this section were obtained for the binary images (solid/pore space) used for permeability computations. This ensures full and consistent comparison between the methods. It is of importance to notice that the NACFC can be computed from grey-level images as easily as from binary images. The differences between the two approaches are discussed in the last section of this paper.

For each of the eight binary sub-volumes (Table 1) the NACFC was computed and the volumes VI(c) were built. The latter contain all the pixels for which the NACFC is larger than the threshold c. The threshold c needs to be large enough to avoid VI(c) having an irregular shape [START_REF] Bernard | Constrained sintering of glass films: Microstructure evolution assessed through synchrotron computed microtomography[END_REF], but small enough to have a sufficient number of pixels within VI(c). The NACFC vary rapidly for small lags (Fig. 6) and, when pixels are large, the volumes VI(c) might be poorly discretized (composed of a small number of pixels) and subjected to important changes of shape for relatively small variations of c.

Consequences of this discretization effect on the degree of anisotropy are analysed in the corresponding section. To minimize both problems, VI(c) for c between 0.05 and 0.5 were considered.

In Fig. 6 the values of the NACFC are plotted along the principal directions of anisotropy (major axis of the equivalent ellipsoid) of the volumes VI(0.2) presented in the insets. As indicated when presenting the autocorrelation function, the parameters La and Lc are not detectable, except for SiC, a material having a meso-structure similar to a sphere-packing with large overlaps (see Appendix A). The intersections of the curves NACFC(c) with the horizontal line at 0.2 (dashed line Fig. 6) give, along the directions of maximum anisotropy, the lags for which this level of correlation is reached. Qualitatively, these values are correlated with the size of the main features appearing in the CT sections presented in Fig. 4; small features (SiC, Cord4) induce small correlation lags and large features (Cord1, Cord2) larger correlation lags. This qualitative correlation is mainly due to a scale effect between the different images and cannot be used to estimate the dimensions of any specific features (see Appendix A).

From the 3D rendering of VI(0.2) shown in Fig. 6 it is obvious that the degrees of anisotropy are much smaller for these images of real materials than for the synthetic image IEl (Fig. 2). By definition, the volumes VI(c) are discrete objects, and the quality of their discretization varies with the acquisition pixel size and with the size of the features composing the images. Indeed, Cord1, Cord2 and Cord3 comprise large features, but for images acquired with 4 µm pixel size (Cord1 and Cord2) the discretization quality is much lower than for Cord3 that was acquired with 0.434 µm pixel size (sub-sampled at 0.868 µm). Cord4 and SiC were also acquired with 0.434 µm pixel size (sub-sampled at 0.868 µm), but as they are composed of small features, the discretization quality is only marginally better than for acquisitions with 4 µm pixel size.

The parameters through analysis of the volumes VI(c) for the eight cuboidal binary sub-volumes (Table 1) and the c values between 0.05 and 0.5 were used to estimate the anisotropy in terms of amplitude (DA_Ellips(c)) and orientation. The detailed results are exposed in the following sections.

Permeability computation

The closure problem (Eq. 4) completed by the boundary conditions (Eq. 5) was solved for each sub-volume extracted from the CT images. This tensor problem reduces to three vector problems, each vector solution corresponding to Dn (n = A or B or C) a line of the tensor D.

Each vector problem comprises four degrees of freedom: the three components of the line Dn and the corresponding component of the d vector. The total number of unknown is roughly equal to (4NANBNCCT) where NA, NB, NC are the dimensions of the computation domain, and CT the porosity fraction of the binary sub-volume (Table 1). The resulting number of unknowns range from 4.8 10 7 (Cord1) to 1.3 10 9

(IEl), with an average equal to 4.9 10 8 .

These values clearly indicate that calculations are computationally demanding, resulting in CPU times varying from a few hours to a few days using up to 120 threads on a middle-range Linux server.

Each computed vector field Dn (n = A or B or C) is related to the local perturbation caused by the solid phase to the macroscopic Darcy flow induced by a macroscopic pressure gradient in the n direction [START_REF] Bernard | Permeability assessment by 3D interdendritic flow simulations on microtomography mappings of Al-Cu alloys[END_REF][START_REF] Bernard | Using the volume averaging technique to perform the first change of scale for natural random porous media[END_REF]. As such, its 3D representation yields information on the homogeneity of the fluid flow at the sample scale. In Fig. 7 the vector fields are visualised using streamlines and isosurfaces of the module normalised to the maximum component (i.e. D ‖𝐃 ‖ ‖𝐃 ‖ ). For all the pixels enclosed by those isosurfaces the corresponding D is larger than the selected threshold (0.35 in Fig. 7). These regions act like bottlenecks and control the permeability, as they correspond to the zones where the fluid is flowing and where the local perturbation is high.

The features shown in Fig. 7 reflect and complement the observations made on the meso-structures of Fig. 4. In cross-sections or CT slices, the size of the features composing the pore space ("pore elements") are discernible, while the connections between them are not. Through the permeability computations both aspects, i.e. "pore elements" and connection sizes, are evidenced, and the consequences of the differences in pore space morphologies are highlighted by these representations. When large "pore elements" are present (Cord1, Cord2, Cord3), the size of the sample is crucial to determine the representativeness of the investigated volume to calculate the permeability. For Cord1 and Cord2 3D images were acquired using a laboratory XCT with pixels of 4.0 µm. This resulted in a scanned sample much larger than the "pore elements"; consequently, a homogeneous flow distribution in all directions (Fig. 7a andb) is observed. In the C direction (Fig. 7b) the sample thickness is always limited and, when "pore elements" are large, the two sides of the filter wall might be connected through a few "pore elements" constituting channels that control the global flow. Homogeneity is then related to the spatial density of these channels (in the A-B plane). Large scanned samples can then be considered as representative (Fig. 7b), whereas the smaller ones, acquired by synchrotron-CT (Cord3, Fig. 7.c) need to be more closely inspected, because of the very limited number of channels controlling the flow. The representativeness of high resolution images is good for samples having very small "pore elements", as seen in Fig. 7d, showing DC for Cord.4.1. (Note that the vector field for SiC (not shown in Fig. 7) looks similar to that of Cord4, and is homogeneously distributed).

From the computed vector fields Dn (n = A, B, C) the permeability tensor of each sample is obtained using Eq. 6. The resulting symmetric full tensor for sample

Cord3.1 is given by Eq. 7.

D =

0.733 0.013 0.072 0.013 0.842 0.024 0.072 0.024 0.639

In this example, as well as for the others, the diagonal terms are dominant, indicating that the axes (A, B, C) are not very different from the principal directions.

The values of the diagonal terms reported in Table 3 reveal that for all cordierite materials permeability is slightly higher in the extrusion direction B compared to directions A and C. For the synthetic image IEl these axes are not related to any process, and for SiC, it seems that due to the nature of the material (round nearly monodisperse SiC sintered powder particles), the extrusion process does not really influence the mesostructure.

The eigenvalues and eigenvectors of the tensors D can be easily computed, thereby providing a new way to characterise sample anisotropies in amplitude (degree of anisotropy DA) and direction. We define the degree of anisotropy DA_k (k typically symbolises permeability) as the ratio of the larger to the smaller eigenvalues of the permeability tensor. The principal directions of anisotropy correspond to the directions of the eigenvectors of the permeability tensor. For sample Cord3.1, the eigenvalues (i, i=1,3) and the eigenvectors (i, i=1,3) of the tensor given by Eq. 7 are given Table 4.

The eigenvalues and eigenvectors for the eight binary sub-volumes (Table IV

Anisotropy

As stated above, the permeability tensor can be seen as an implicit function of the mesoscale geometry only. The NACFC is also fully determined by the same meso-scale geometry, and it contains most, if not all, the information about this geometry. This 3D real-valued function looks much simpler than the local-scale geometry itself (Fig. 2) and the determination of anisotropy is in general possible. We stress that even if it is highly tempting to use it for extracting relevant meso-structural parameters, some limitations need to be taken into account, as presented in Appendix A. From the NACFC we can estimate the meso-structural anisotropy, i.e., in our case, an averaged characteristic of the spatial distribution of the voids and the solid. This property must be distinguished from the functional anisotropies that are related to effective physical properties determined at the global scale. For instance, in the cases of effective mechanical or thermal properties of multi-materials, the local-scale geometry must be complemented by the local-scale characteristics of the material (local-scale physical properties) that play an essential role, generally amplifying the difference between structural and functional anisotropies. As both normalised autocorrelation function and permeability tensor are fully determined by the meso-scale geometry, a comparison of the anisotropy characteristics derived from them seems relevant.

For a given binary sub-volume (Table 1), the computed permeability tensor gives one set of eigenvalues and eigenvectors. Equivalent information is obtained from VI(c) for each value of c. In this section, anisotropies derived from the permeability tensor and from the properties of VI(c) are compared in amplitude (degree of anisotropy DA) and direction (principal direction of anisotropy). The ratio DA DA_Ellips c /DA_k is introduced to evaluate the agreement between the approaches quantifying the meso-structural and the filtration anisotropy.

Degree of anisotropy

Figure 8 presents the evolutions of DA for all the samples as functions of c, the threshold values used to build VI(c).

The main point to be stressed is that all the values of DA lie between 0.9 and 1.15 for c between 0.05 and 0.5. This shows that the meso-structural anisotropy (induced by the arrangement of the features composing the 3D images) and the filtration one (permeability) are strongly correlated for this family of materials. It also shows that the NACFC gives a good approximation of the permeability degree of anisotropy at a largely reduced computation cost.

The evolutions of DA with c can be analysed in more details to extract some specific information relatively to the considered materials. For that, we first distinguish two groups of materials: the materials showing large "pore elements" in Fig. 4, i.e.

Cord1, Cord2 and Cord3, and the materials showing small "pore elements" in Fig. 4, i.e.

Cord4 and SiC:

 For the first group, there are two regions limited by a c value between 0.2 and 0.3. Below this value DA is almost constant (Cord1, continuous line with diamonds) or smoothly increasing with c (Cord2, dashed line with diamonds, Cord3, dashed lines). Above this value, smooth (Cord3) or abrupt (Cord1, Cord2) changes appeared. The abrupt variations are directly related to the discretization effect previously described. For Cord1 and Cord2 the volumes VI(c) contain a small number of pixels (Fig. 6) and undergo significant changes of shape when c is varying, inducing the abrupt variations visible in Fig. 9. For Cord3, the variations of DA for the two binary sub-volumes (Cord3.1 and Cord3.2) are nearly parallel with a constant shift, and without noticeable discretization effects. In fact, the sub-volume Cord3.1 is extracted between two crossing regions of the filter web, while Cord3.2 is extracted across one of these crossing regions. Consequently, Cord3.2 is more homogeneous and isotropic than Cord3.1 (see Fig. 9a).

 For the second group, variations of DA with c are of two kinds: a smooth decrease for c ≤ 0.3 followed by abrupt variations in the case of Cord4 (continuous lines), an almost constant value for c ≤ 0.25, followed by variations of limited amplitude in the case of SiC (continuous line with triangles). These images are affected by the discretization procedures, because the pixel size is small (0.434 µm, sub-sampled at 0.868 µm), but "pore elements" are also small.

For Cord4, the variations of DA for the two binary sub-volumes are very similar.

This can be explained by the facts that the pore space meso-structure is homogeneous (and therefore all sub-volumes tend to have similar geometric anisotropy), and the two volumes are extracted at similar distances from the crossing points of the filter web (see Fig. 9b).  For the synthetic image IEl, DA is essentially constant, and, for this very homogeneous image, the value of DA effectively quantifies the relation between the meso-structural and the filtration estimates of anisotropy.

From the comments presented above, it appears that, if a single value has to be selected for c, c equal to 0.2 is a good choice. Therefore, this value will be used for DA in the rest of this work (see Table 3).

The NACFC also contains information about the scale, i.e. a given degree of correlation c is reached at a given lag in a specific direction, and DA_Ellips(c) shows the differences in lag according to direction (Fig. 3). The decrease of DA towards 1 with increasing c for Cord4 can then be interpreted, for that material, as an increase of the correlation between meso-structural and filtration anisotropies with an optimum at c=0.3. This corresponds to a lag of about 16 µm in one direction (Fig. 3), and 13.5 µm in the other (DA 1 implying that DA_Ellips = DA_k = 1.18, Table 3). These values seem consistent with the meso-structure presented Fig. 4 (16 µm = 1/19 of the side of the section). Nevertheless, this interpretation cannot be generalised, as for IEL and SiC the correlation between the two types of anisotropy is independent of c, and consequently of the explored scales. The prominent results of this section are that, for these materials, meso-structural anisotropy and filtration anisotropy are clearly correlated. On a case by case basis, some information about the structure can be extracted from the small variations of DA with c.

Directions of anisotropy

The second characteristic after the degree of anisotropy is the direction of anisotropy.

For the cordierite samples the main direction of anisotropy obtained from the permeability tensor is close to the direction of extrusion B (Appendix C). To compare these directions with the ones corresponding to the principal directions given by analysing the morphology of VI(c), the graphic presented Fig. 10 As seen above, for all the cordierite samples the main direction of anisotropy is strongly correlated with the direction of extrusion B. This is not the case for the synthetic image IEL, for which this notion is irrelevant, and for SiC, for which the mesostructure appears as unaffected by the extrusion process. The main directions of permeability and VI(0.2) are reported in Table 5 for the synthetic image and SiC

showing for them also a strong correlation between filtration and meso-structural anisotropy. This difference is a consequence of the form of the closure problems (Eq. 4 and 5 for filtration) that determines the effects of local-geometry on effective properties.

This work focused on NACFC vs. permeability and logically all results were generated using binary images (solid/pore space) as required for permeability computations. It is of importance to notice that NACFC can be computed on the greylevel images as easily as on the binarised ones. The direct use of grey-level images would circumvent the critical segmentation step, which inevitably influences the permeability computation.

To assess the consequences of using a grey-level image instead of a segmented one, the following procedure was followed:

 A sub-volume of 600 x 600 x 540 pixels was extracted from the centre of the original grey-level image of Cord4 (with a pixel size of 0.434 µm).

 This sub-volume was then sub-sampled by factors of 2, 3, 4, 5, 6, 8, and 9.

 For each of these sub-volumes, the NACFC was calculated, and DA * 0.2 evaluated from the corresponding volumes VI(0.2). For a given pixel size, DA * 0.2 is the ratio of DA_Ellips 0.2 of VI(0.2) (for the corresponding subvolume) to DA_k given in Table 3. The permeability and the autocorrelation functions yield relevant information about the anisotropy of their filtration and morphological properties. For cordierite samples, the direction of extrusion is clearly the principal direction of anisotropy. As found in [START_REF] Bruno | Calculation of the Average Coefficient of Thermal Expansion in Oriented Cordierite Polycrystals[END_REF][START_REF] Bruno | Simultaneous determination of high-temperature crystal structure and texture of synthetic porous cordierite[END_REF], the degree of anisotropy of the meso-structure is similar to that of the crystal texture, which in turn induces anisotropy of the coefficient of thermal expansion and Young's modulus (since the properties themselves are anisotropic). This correlation is strongly influenced by micro-cracking in DPF materials [START_REF] Bruno | Microcrack orientation in porous aluminum titanate[END_REF]. In the present case, however, pore space is dominant over micro-cracking (see also the analysis in [START_REF] Kupsch | Evaluating Porosity in Cordierite Diesel Particulate Filter Materials, Part 1-X-ray Refraction[END_REF][START_REF] Onel | Evaluating Porosity in Cordierite Diesel Particulate Filter Materials, Part 2 Statistical Analysis of Computed Tomography Data[END_REF]), because microcracks have negligible dimensions. In fact, Table IV.4.1 shows that an increase in meso-structural anisotropy (compare Cord 1 to Cord 2 or Cord3) corresponds to an increase in permeability anisotropy. It is interesting to note that for SiC the meso-structure is almost isotropic (DA_Ellips(0.2) = 1.08), but DA_Ellips(0.2)/DA_k = 0.96, which implies that the filtration properties of SiC are more textured than its physical and mechanical properties [START_REF] Bruno | Thermal and Mechanical Response of Industrial Porous Ceramics[END_REF]. It is generally confirmed that cordierite materials possess relatively weak morphological texture, and that increasing porosity fraction (and decreasing the size of the pore space features, i.e.

smaller but more pore space features) actually decreases the degree of anisotropy. In fact, the thermal and physical properties also become less anisotropic [START_REF] Bruno | Connecting the macro and microstrain responses in technical porous ceramics. Part II: microcracking[END_REF].

Conclusions

To study quantitatively the anisotropy of both meso-structure and fluid flow in Diesel Particulate Filter (DPF) materials we used the normalised autocorrelation function of the centred image (NACFC), and permeability calculations for 3D CT data reconstructions.

For large 3D images such as the ones considered here, permeability calculations are computationally demanding, therefore requiring advanced numerical tools. From the resulting permeability tensor, the principal direction and the degree of anisotropy are easily deduced.

On the contrary, the NACFC is easy to calculate, but the resulting 3D realvalued function is difficult to interpret. For that, we first clarified, by a theoretical analysis, the nature of the information that can be extracted (Appendix A), and we developed a new method to quantify the anisotropy based on the morphological properties of the volume VI(c) constituted by all the pixels for which NACFC is larger than a certain threshold c. Examining the variations of these properties for c varying between 0.05 and 0.5, we shown that, for all the considered samples, the values provided by both approaches are in very good agreement. If a single value of c has to be used, c equal 0.2 is a good choice.

We focused our attention on Cordierite and SiC, two widely used materials for DPF. In general, meso-structural anisotropy directly affects physical properties, but the two may only be loosely linked. In this work we found that meso-structural and filtration anisotropies are strongly correlated for both the degree of anisotropy and the orientation. In line with the few other studies available in the open literature, it appears that cordierite materials are weakly anisotropic and clearly influenced by the extrusion process that determines the main direction of anisotropy. A slight dependence of anisotropy on the feature size of the meso-structure as seen on 2D images was put into evidence for a family of cordierite materials (all having porosity fraction around 0.5):

the finer the grain and the "pore elements" size, the more anisotropic the meso-structure and the permeability. Since the meso-structure in such materials is dictated by the raw powder and by the pore-former particle size, we deduce that controlling those parameters would allow controlling the degree of anisotropy of both the filtration and of the physical properties of such materials.

The autocorrelation function can yield several insights on the meso-structure, such as the dependence of the degree of orientation on the length range of the mesostructure considered in the analysis: homogeneous structures such as SiC (and ideal materials) show the same degree of anisotropy at all length scales.

The approach based on the analysis of the NACFC that we proposed appears as precise and efficient even for materials having a small degree of anisotropy. Finally, we showed that the NACFC applied on grey-level images can give a good estimate of the degree of anisotropy even with limited resolution.

Similarly, it can be shown that the NACFCs are related by:

C 𝐯 C 𝐯 (12)
Consequently, any characteristic length obtained from C 𝐯 and associated to an object (the grains for instance) can also be obtained from C 𝐯 and now associated to the background of I(u). Accordingly, NACFCI(v) cannot be used to determine object dimensions.

Single items or, more generally, loose aggregates of items constitute exceptions to this rule. Indeed, when the distance between the different items is large enough, there is a lag interval for which C 𝐯 is well approximated by the function corresponding to a single item.

To support this analysis we computed C 𝐯 for different 3D images (750 3 pixels) representing spheres aggregates of different densities, i.e. different number of spheres per unit volume. The spheres diameters equal 20 pixels and Nsph, the number of spheres within the discretized volume, varies from 100 to 100000 (Fig. 12). To insure a random and homogeneous spatial distribution of the spheres, centres positions were computed using a random close packing generator based on [START_REF] Zinchenko | Algorithm for random close packing of spheres with periodic boundary conditions[END_REF]. This generator works as following:

1. Initialization: Nsph points are randomly placed within a cube. These points will be the centres of the Nsph spheres composing the packing, and the distance between the nearest centres will determine the spheres diameter. At this stage, due 〈 𝐮 〉 〈 𝐮 〉 〈 𝐮 〉 [START_REF] Saha | A robust method for measuring trabecular bone orientation anisotropy at in vivo resolution using tensor scale[END_REF] which corresponds to the value of the NACFC for v >  in the case of a single sphere.

This means that each sphere is surrounded by an empty spherical domain having a radius larger than 2. Table 6 gives, for the different values of Nsph,  the volume fraction of the background in the corresponding three-dimensional discrete images, and MinDC, the minimum of the distances between sphere centres. For Nsph > 7500,

MinDC is larger than 2, and interferences between neighbouring spheres begins. It can be noted that this occurs for very loose packing ( larger than 90%). Looking at Fig. 13 we observe that using any characteristic value of the NACFC (for instance 0.5 in [START_REF] Heilbronner | The autocorrelation function: an image processing tool for fabric analysis[END_REF]) to estimate the grains diameter is illusory. Indeed, the short-lag part of all the NACFC curves varies continuously with the sphere number. This is true for non-overlapping spheres aggregates (Nsph <= 64119, the value for having a random close packing) or for overlapping spheres aggregates.

La and Lc are characteristic distances related to the entire image and it is a priori not possible to distinguish the effects of its different components. This is unequivocally illustrated by the results of the series of tests summarized in Fig. 14. Seven discretized grey-level images (750 3 pixels) of aggregates comprising 25000 equal spheres were generated. In these aggregates the positions of the spheres centres were the same, but the spheres diameter varied between =20 pixels and =32 pixels by steps of 2 pixels.

On Fig. 13a are represented C v. 𝐗 for the 7 values of . La and Lc increase with , while both the anti-correlation and correlation levels decrease. La corresponds to the lag for which the average intersection between the spheres and the background is maximum. The positions of the sphere centres being constant (Fig. 14b), the variations of La can only be attributed to small variations of the distance between the spheres and the larger zones of the background. Lc represents the lag for which both background regions and the spheres reach maximum overlap. The increase of Lc is therefore also due to the variation of the distance between spheres and background regions, and the

Fig. 1 .

 1 Fig. 1.

Figure 1 :

 1 Figure 1: About 1500 of the ellipsoids represented in b) were randomly distributed in a

Figure 2 :

 2 Figure 2: Three different representations giving a partial view of CIEl(v): a) Section in

Figure 3 :

 3 Figure 3: Evolutions of NACFCIEl along the three directions of anisotropy (from left to right: 1, 2, 3).Well defined minima and maxima are visible for the three directions. The lags corresponding to the minima (La) indicate the offset for which the level of anticorrelation is maximum, and the lags corresponding to the maxima (Lc) indicate the offset for which the level of correlation is maximum. La and Lc are clearly different for the three directions of anisotropy. Nonetheless, they cannot be interpreted in terms of characteristic dimensions of the image components (grains size or grains distances, see Appendix A), and we must refrain from using their values for anisotropy characterisation. Furthermore, in most real cases La and Lc are not even detectable, as exemplified by the materials studied in this work.Instead, the use of the properties of VIEl(c) should be favoured. The principal directions of anisotropy are the principal directions of its tensor of inertia, and we propose to estimate the degree of anisotropy DA as the ratio of the larger to the smaller moment of inertia with respect to the directions of anisotropy (DA_Inertia), or the ratio of the major to the minor length axis of the ellipsoid (DA_Ellips) having the same tensor of inertia as VIEl(c) (see Appendix B).

  Samples Cord3, Cord4, and SiC were scanned on the synchrotron beamline BAMline, BESSY II, HZB Berlin, Germany; 2D slices of their corresponding reconstructions are shown in Fig. 4. Cordierite samples Cord1 and Cord2 were scanned in a laboratory CT system at BAM (see section on computed tomography), and their image size (as well as the field-of view) is much larger (see caption of Figure 4). For comparison purposes, slices of Cord1 and Cord2 carry an inset with the same field of view as slices of Cord3, Cord4, and SiC.

Figure 4 :

 4 Figure 4: Overview of DPF samples under investigation. (a) IEl: Image size 1000 x 1000

For

  permeability and NACFC calculations domains fully immersed in the porous sample are necessary. They were obtained by extracting cuboid sub-volumes with boundaries near the interfaces between the porous material and the free fluid zones. In the middle column of Fig.5the boundaries of the sub-volumes (in black) are visible through the pore space of Cord2 (top) and Cord3 (bottom). Due to the peculiar shape of samples Cord3 (Fig.5) and Cord4, it was possible to extract two overlapping sub-volumes with the zone corresponding to the intersection of two walls in common.These sub-volumes are labelled Cord3.1, Cord3.2, Cord4.1, and Cord4.2 hereafter.

Figure 5 :

 5 Figure 5: Visualization of process going from the initial reconstructed CT data to the

Figure 6 :

 6 Figure 6: The correlation function CI(c) along the direction of maximum anisotropy for

Figure 7 :

 7 Figure 7: 3D representations of computed vector fields Dn as streamlines (in blue) and isosurfaces 𝐷 0.35 (in red) of the normalized vector modules for Cord1 in the

  Two different estimates of the degree of anisotropy based on the morphology of the volumes VI(c) are proposed; DA_Inertia(c), directly derived from the tensor of inertia, and DA_Ellips(c), defined as the ratio of the major to the minor axis of the ellipsoid having the same tensor of inertia (see Appendix B). For the synthetic 3D image IEl these two estimates are very different; DA_Ellips(0.2) = 8.83 and DA_Inertia(0.2) = 4.62.DA_k obtained from the permeability tensor computed for this image being 9.53, DA_Ellips(c) appears to agree much better with DA_k. In the case of all the other samples DA_Ellips(c) and DA_Inertia(c) are very similar (Table3), indicating that the volumes VI(c) are nearly spherical (see Appendix B). We therefore choose to favour the use of DA_Ellips(c) for the remainder of the paper.

Figure 8 :

 8 Figure 8: 𝐷𝐴 (the ratio between the anisotropy factors extracted from the NACFC and

Figure 9 :

 9 Figure 9: (a) Location of Cord3.1 and Cord3.2 sub-volumes; (b) Location of Cord4.1

  is used. It shows the projection on the plane (A, C) of the surface of a unit sphere with centre at the origin of the coordinate system (A, B, C). All points represent the intersections of the principal direction lines (drawn through the origin) and the unit sphere. The angle between a given line and the B axis is equal to 90 degrees minus the latitude of the intersection point on the surface of the sphere (circles corresponding to some values of latitude are drawn in light grey). For sample Cord3, points corresponding to permeability (triangles) or NACFC (squares) are confined in a limited zone (Fig. 10 central image) for both of the binary sub-samples (Cord3.1 in white, Cord3.2 in grey). Focussing on the central region of Fig. 10 (upper inset), it appears that the intersection points of the main direction of VCord3.1(c) and VCord3.2(c) tend towards the intersection points of the principal direction of the permeability tensor (triangles) when c increases. On the lower inset of Fig. 10 the pairs of intersection points corresponding to the main direction of VI(0.2) and of the permeability principal direction (symbols with a black dots) are displayed for each of the six cordierite binary sub-samples. All these directions have a maximum misalignment of 10 degrees with the axis B.

Figure 10 :

 10 Figure 10: Intersections between the principal directions of anisotropy (calculated from

Figure 11 Figure 11 :

 1111 Figure 11 shows a section of the high resolution sub-volume (a), and approximately the same section sub-sampled by a factor of 8 (b). The representation of

Figure 13 :

 13 Figure 13: Evolution of the shape of NACFC with the number of spheres in the 3D



  (X,Y,Z) is the coordinate system based on the principal directions of a moment of inertia tensor, the principal semi-axes of the ellipsoid having the same moment of inertia with respect of the directions X, Y and Z are given by the following expressions: autocorrelation function, two definitions were introduced for the degree of anisotropy; DA_Inertia and DA_Ellips. Considering an ellipsoid described by Eq. 12 for which a ≥ b ≥ c and using the relations 13 and 14, the following expressions are obtained: is larger than 1, and depending on its value, DA_Inertia is larger or smaller than DA_Ellips. If we take the case of needle (e.g. a = 10, and b = c = 1), we obtain DA_Ellips = 10 and DA_Inertia = 50.5, and in the case of a disc (e.g. a = b = 10, and c= 1), we get DA_Ellips = 10 and IZZ/IXX ~ 1.98. This implies that the objects in Fig. 1 have the shape

Table 1 :

 1 total of five materials were investigated, four are cordierites (named Cord1 to Cord4) adding different amounts of pore formers. Sintering was made at temperatures in excess of 1300 o C, with successive cooling at rates below 1 o C/min in N2-enriched air. SiC filter materials were commercially available. Porosity fraction was determined by mercury intrusion with an Autopore 6520 Micromeritics, Norcross, USA.Porosity fraction values are reported in Table1, together with the results of image analysis of computed tomography data (see below). Characteristics of the 3D binary sub-volumes. Pixel size (PS, in µm),

	and one is SiC. Cordierite materials were provided by Corning Incorporated (Corning,
	NY, USA) and Oak Ridge National Laboratory (Oak Ridge, TN, USA). All of them are
	extracted from larger honeycomb filters, all produced by the extrusion process

mentioned above. In the case of cordierite, raw materials consisted of SiO2, MgO, hydrated Al2O3, and Al2 Si2O5(OH)4 (Caolin). Different porosity fractions were achieved by the CT images (CT) of the sub-volumes, and porosity obtained by mercury porosimetry (Hg) on samples of the same materials.

Table 2 :

 2 Characteristics of the 3D reconstructed volumes. PS is the pixel size, and NA, NB, NC are the dimensions (in pixel) of the volumes along the three directions.

	Sample	Cord1	Cord2	Cord3	Cord4	SiC
	PS (µm)	4.00	4.00	0.434	0.434	0.434
	NA	1000	860	3201	3001	2501
	NB	800	820	2701	2101	2701
	NC	200	240	2672	2672	2672

Table 3 :

 3 .1.2) are listed in Appendix C. Values of the permeability tensor diagonal entries and of the degree of anisotropy calculated by the three methods proposed in the text.

	Sample	IEl	Cord1 Cord2 Cord3.1 Cord3.2 Cord4.1 Cord4.2	SiC
	kAA	3.21	6.01	2.45	0.73	0.85	1.91	1.92	0.70
	kBB	1.53	6.70	3.31	0.84	1.04	2.12	2.13	0.65
	kCC	0.52	5.93	2.64	0.64	0.76	1.80	1.81	0.63
	DA_k	9.53	1.14	1.36	1.42	1.44	1.18	1.18	1.12
	DA_Ellips (0.20)	8.83	1.18	1.37	1.56	1.43	1.26	1.26	1.08
	DA_Inertia (0.20) 4.62	1.19	1.42	1.54	1.43	1.28	1.27	1.08

Table 4 :

 4 Eigenvalues and Eigenvectors for sample Cord3.1

		Eigenvalues		A	Eigenvectors B	C
	 1	0.845	 1	4.800 E-2	0.99400	-9.824 E-2
	 2	0.772	 2	0.88500		3.282 E-3	0.46558
	 3	0.597	 3	-0.46312		0.10929	0.87953

Table 5 :

 5 Components of the principal directions of the permeability tensor and of the volume VI(0.2) for the synthetic and SiC samples..

	Sample		IEl			S i C
	Axes	A	B	C	A	B	C
	Permeability	0.99998	-0.00276	-0.00450	0.97144	0.22907	-0.06196
	VI(0.2)	0.99997	0.00044	0.00047	0.84866	0.50899	-0.14393

Table 6 :

 6 Evolution with Nsph, the number of spheres in the volume, of , the porosity in %, and MinDC, the minimum of the distances between sphere centres.

	Nsph	100	1000	2500	5000	7500	10000	25000	50000	64119	75000	100000
	 (%)	99.90	99.01	97.53	95.05	92.56	90.08	75.59	50.37	36.35	26.47	11.12
	MinDC	371.57 172.46	80.05	46.81	40.90	37.16	27.38	21.73	20.00	18.98	17.25

Acknowledgments

DB acknowledges financial support from BAM, in the frame of the 'Guest Scientists' programme.

The authors acknowledge the anonymous reviewer for his/her valuable work.

Data availability statement

Data used in this work are available from the corresponding author upon reasonable request and with the permissions of BAM and CNRS.

Disclosure statement

The authors have no competing interests.

Appendix A: Why NACFC I (v) of an image cannot be used for grain size estimation

Attempts to characterize rock meso-structure using CI(v), the NACFCI(v) of image I, as defined by Eq. 2, are already dated [START_REF] Heilbronner | The autocorrelation function: an image processing tool for fabric analysis[END_REF][START_REF] Lin | Microgeometry I: Autocorrelation and rock microstructure[END_REF], but the transition to quantification of grains sizes, even proposed as modules in some software [START_REF] Heilbronner | Image Analysis in Earth Sciences. Microstructures and Textures of Earth Materials[END_REF], generally leads to poor approximations for dense aggregates [START_REF] Thissen | An autocorrelation method for threedimensional strain analysis[END_REF]. The main reason is linked to a property of the function CI(v) that we expose in this appendix.

If I(u) is a binary image (0/1), IInv(u) is also binary, with the pixels with grey value = 0 corresponding to the pixels of I(u) with grey value = 1, and vice-versa. The background of I(u) is now the object of IInv(u) and vice versa.

Using the definition of the autocorrelation function given by Eq. 1, taking into account the linearity of the averaging operator < > and the fact that the average of a constant is equal to this constant, the following relation between the ACFs of I(u) and IInv(u) is obtained after a little bit of algebra:

to the unconstrained random placement, the centres will be very heterogeneously distributed within the cube and the resulting spheres diameter will be very small.

2. Homogenisation phase: At each iteration, the two nearest spheres are symmetrically moved away along their centres line to a distance DI. During this phase, a large value is given to DI creating an efficient virtual shacking of the packing that will homogenise the distribution of the centres within the cube.

3. Densification phase: During this phase, DI is slowly decreased at each iteration, and the spheres diameter will gradually increase until convergence to a value corresponding to a dense random packing.

For more details on the parameters to be used during the different phases to effectively converge to a dense random packing, refer to [START_REF] Zinchenko | Algorithm for random close packing of spheres with periodic boundary conditions[END_REF].

Appendix B: Tensor of inertia and ellipsoid

The general equation of an ellipsoid centred at (0,0,0) and aligned on the X, Y, Z axes is:

where a, b, and c are the principal semi-axes.

Assuming that the density of the ellipsoid is uniform, its moments of inertia with respect to the directions X, Y and Z are given by the following expressions:

of discs rather than that of needles, as expected from their definition. Trivially, in the case of a sphere (i.e., a = b = c), DA_Ellips = DA_Inertia = 1.

Appendix c: Permeability eigenvalues and eigenvectors

Sub-sample I EL

Eigenvalues


Dr Nathalie Daro

Tony Vaudel

Dr Luc Afindouli

Dr Samuel Marre

Dr Cyril Aymonier
email: cyril.aymonier@icmcb.cnrs.fr

Guillaume Chastanet
email: guillaume.chastanet@icmcb.cnrs.fr

One-Step Synthesis of Spin Crossover Nanoparticles Using Flow Chemistry and Supercritical CO2

Switchable materials are increasingly considered for implementation in devices or multifunctional composites leading to a strong need in terms of reliable synthetic productions of well-defined objects. We report here an innovative and robust template-free continuous process to synthesize nanoparticles of a switchable coordination polymer, including the use of supercritical CO2, aiming at both quenching the particle growth and drying the powder. This all-inone process offered a 12-fold size reduction in few minutes, maintaining the switching properties of the selected spin crossover coordination polymer.

Molecular switches are increasingly implemented in functional materials for sensing applications 1 or in molecular electronics 2 . Among the switchable systems under study, spin crossover (SCO) materials are widely investigated in regard to their ability to change their electronic configuration in response to external stimuli (temperature, pressure, magnetic field, light, adsorption of molecules) [3][4][5] . The vast majority of SCO compounds reported so far concerns Fe(II) and Fe(III) complexes and especially the 1D coordination polymers based on 4-R-1,2,4triazole ligands and iron(II) salts (SCO-triazole family). Indeed they exhibit a room temperature (or close to) thermochromism [6][7][8] , related to the switching between high-spin (HS, paramagnetic) and low-spin (LS, diamagnetic) states, which can result in memory effects of interest in displays 9 and sensors 1 .

Over the years, such systems have become more and more relevant to applications 1,2,10 . Such increasing interest is faced with the need for an upscaled and reliable production of particles with well-defined and tunable size, morphology and properties. Currently, two main synthetic routes are followed to obtain these particles 11 : (i) direct fabrication based on batch mode synthesis, leading to a poor control of size, morphology and size distribution; (ii) a templating approach based on soft or hard organic or inorganic templates, offering excellent control of the nanoparticle characteristics. The templating approach is widely used for the synthesis of SCO nanoparticles, but is mitigated by two main issues. The first is that reproducibility is not always fully achieved, especially at a very small scale (below 100 nm) [12] . The second issue concerns the template removal, which can be difficult or impossible in the case of hard templates (such as porous silica or polymers) or charged surfactants. Therefore, the synthesis of such materials in large quantities, with a high reproducibility, and using template-free methods able to tune the size and sizedistribution of the particles, is still a challenging aim. To achieve this goal, direct precipitation is the most trivial method, but does not provide good control over the size and morphology [13] . The authors have explored other techniques, such as spray drying [14] or microfluidics [15] . While spray drying offered interesting morphologies and phases, it appeared difficult to control in the case of the SCO-triazole family. Regarding the microfluidic approach, a drastic downsizing effect was observed using droplet microfluidics in which each droplet of water containing the reactants is carried by a flow of oil. Even if, strictly speaking, no surfactants are used in this method, the oil must still be removed afterwards. We have therefore turned our attention to the use of flow chemistry allowing a fine control of process parameters, which is an alternative and original route to the aforementioned approaches.

Flow chemistry has already been successfully applied to different kinds of materials 16,17 such as luminescent quantum dots [18][19][20][21][22] , metal nanoparticles 23,24 , polymers 25,26 , metal-organic frameworks 27,28 , organometallic species 29 and molecules [30][31][32][33] . Several advantages of moving from batch to flow mode have been recognized, such as facile automation and feedback optimization, increased reproducibility, and improved safety and process reliability. Indeed, with continuous flow processes, stable reaction parameters (temperature, time, amount of reagents and solvents, efficient mixing, etc.) can be ensured 33 . Moreover, these approaches are easily scalable to reach the quantities needed for development and optimization. However, most of the conventional flow processes are still limited to normal conditions of pressure and temperature, thus limiting the process parameter space and the considered solvent systems. On the other hand, high pressure systems can easily handle high flowrates, providing both intense mixing 34 , favorable for homogenous nucleation, and particle size control down to the nanometer range 35 , along with higher production rates.

Continuous flow synthesis of SCO nanoparticles (NPs) has been recently demonstrated using a milli-scale segmented flow crystallizer reactor 36 . The reported synthesis was based on a liquid-liquid droplet strategy, therefore requiring similar posttreatment as for the conventional batch mode synthesis. However, it does not provide a way to size reduction -mostly in the micrometer range -nor does it allow access to a fast, continuous synthesis process for further scale up, as the reported synthesis time (~ 20 min) was not improved. Therefore, being able to develop an all-in-one multi-step process for synthesizing SCO NPs could provide a way for realizing the promises of these materials in a fast, reliable and robust way.

We here demonstrate a template-free continuous process to synthesize SCO NPs, including the use of supercritical CO2 (scCO2). We can reach improvements both in the synthesis efficiency and in access to nanosized SCO particles (by a factor of at least 10 compared to the literature), thanks to the role of scCO2. It permits not only the quenching of the particle growth during the precipitation, but also the recovery of dry powder with no need for additional post treatment. For this purpose, we choose a model compound, [Fe(Htrz)2(trz)](BF4) 37,38 , which has been largely investigated using conventional methods. It is a polymeric compound, known to be the most stable of the triazole family; it does not contain solvent molecules and shows reproducible SCO hysteresis features, even upon size reduction 12,39 . The characteristics of the materials (size, phase, magnetic properties) obtained with the reference conventional batch synthesis are compared with those obtained through this new all-in-one continuous synthesis approach, coupling a flow reactor for precipitation with a supercritical CO2 drying step.

The polymeric [Fe(Htrz)2(trz)](BF4) material was prepared using three different methods (Table 1), while keeping the reactant concentrations constant (Fe(BF4)2.6H2O (10 mmol) and Htrz (33 mmol), with both dissolved in 10 mL water, see SI for details):

(1) A batch method, consisting of four different steps 1top): (i) dropwise addition (at 1 mL/min) via a syringe pump of an aqueous solution of 1,2,4-1H-triazole to an aqueous solution of Fe(BF4)2.6H2O containing ascorbic acid at 80 °C. The resulting solution was stirred for 15 min. (ii) aging of the solution for 1h at low temperature (5 °C). (iii) centrifugation of the suspension followed by three washing + centrifugation steps with ethanol (EtOH). (iv) sample drying overnight at ambient atmosphere 30 . This method led to sample 1 (Figure 2, top) displaying micrometric size rod-shaped particles, whose mean length and width of 2116 ± 382 nm and 487 ± 232 nm, respectively (Figures 3 andSI1).

(2) A continuous reaction process including a washing step with EtOH (Figure 1, bottom, with the switch "off"). The reactor consisted of a 1/16" stainless steel coiled tubing (internal diameter = 1 mm, length = 1.1 m) immersed in an oil bath used to control the temperature. This flow reactor is fed with the two reactant solutionsdelivered at constant flowrate (1mL/min)by conventional syringe pumps (KDS100, KD Scientific) which then mix in a Tee mixer, with a residence time in the reaction zone of 15 s. Downstream of the reaction zone, EtOH was introduced with an HPLC pump, to prepare the further washing step. The particles were collected in a beaker at the outlet of the process. The particles in suspension were stirred for 1 min at 80 °C before undergoing a double washing + centrifugation step with EtOH, and were later let to dry overnight at ambient atmosphere, similarly to sample 1. The obtained particles (sample 2) exhibit a rod-shaped submicrometric morphology (Figure 2, middle) with a mean length and width of 277 ± 32 nm and 116 ± 25 nm, respectively (Figures 3 andSI1). (3) An all-in-one process including a drying step with supercritical CO2 (Figure 1, bottom, with the switch "on"). This approach was performed under pressure (p = 100 bar) in order to implement the scCO2 drying step at the very end of the process. The overall principle can be detailed as follows (Fig. 1 bottom with the supercritical box switched on): first, two high pressure syringe pumps (PhD 2000, Harvard apparatus) were used to separately deliver the two reagent solutions, which mixed in a Tee before entering the reactor (similar to the second approach). Then, EtOH was continuously injected with an HPLC pump as explained in (ii). The particles were captured within a high pressure collection vessel equipped with a filter. An automated back pressure regulator (BP2080, Jasco) was placed downstream of the vessel to control the pressure over the entire process. The EtOH flow was used to keep the process conditions similar to the first two approaches and also to make the scCO2-based drying efficient. The scCO2 was eventually injected through the vessel thanks to a continuous high pressure membrane pump (Dosapro, Milton Roy) for 1 hour under a constant CO2 flowrate (12 mL/min). In the considered conditions (p = 100 bar, T = 80°C), the EtOH / CO2 mixture was fully homogeneous. The obtained pink-violet powder was recovered in the filter upon depressurization of the system. The particles of sample 3 presented well-defined nanorod shapes (Figure 2, bottom), with mean length and width of 169 ± 45 nm and 48 ± 16 nm, respectively (Figures 3 andSI1).

T y p i c a l b a t c h s y n t h e s i s Flow synthesis with the scCO 2 OFF (3 steps) or ON (1 step)

All the characteristic sizes of the obtained rods are summarized in Table 1 along with the reaction yields. The latter are very similar for all the processes, around 30%. This validates the possibility to significantly reduce the reaction time by switching from batch to continuous modes.

When working with the [Fe(Htrz)2(trz)](BF4) compound, one must check the purity and the phase obtained 37 . The chemical analysis demonstrated a good agreement with expected compound (see SI). The X-Ray diffraction patterns (Figure 4) showed that, whatever the synthesis protocol, the crystalline phase of the obtained powders are the same. From these patterns, an enlargement of the Bragg peaks can be noticed for samples 2 and 3 compared to sample 1. This is in accordance with the particle size reduction observed in Figures 2 and3, previously reported observations [39][40][41][42] .

Regarding the switching properties, we have recorded the molar magnetic susceptibility (χM) as a function of the temperature for samples 1, 2 and 3 (Figure 5). Sample 1 presents the expected SCO behaviour 37 : T1/2up = 386 K and T1/2down = 353 K, describing a large hysteresis loop of 43 K. Sample 2 presents the same features between 349 K and 377 K, with a hysteresis loop of 28 K. Sample 3 exhibits a transition at T1/2 up = 369 K and T1/2down = 346 K and a hysteresis loop width of 23 K. The differences in the SCO behaviours (transition temperature and hysteresis width) has to be linked to the size of the particles. Indeed, this trend has already been described for rods of the same compound obtained with reverse-micellar synthesis 42 .

The use of the continuous reaction process combined with a supercritical CO2 treatment clearly allows a significant downscaling (more than 10 times in both dimensions of the rods) of the particle size compared to the batch synthesis. This process also affords well-defined particles in a reproducible and reliable way with the same yield as the batch synthesis.

The first main difference between the batch and the continuous modes concerns the hydrodynamics, especially the mixing conditions. This mixing in batch mode is much weaker than in continuous flow 43 . The direct consequence is a lower nucleation rate in batch mode compared to the continuous one, which could be an explanation for the size reduction between samples 1 and 2. Another important step in the preparation of SCO materials is the aging step. During this step, the formed particles are still in contact with unreacted precursors, promoting further growth of the particles. Beyond the continuous reaction process, an additional advantage of the all-in-one process is the removal of the aging step. Indeed, the addition of scCO2 not only permits the drying of the sample but also avoids the additional growth of the rods, opening the door to nano-sized materials, as evidenced by the difference in size between compounds 2 and 3.

The demonstrated all-in-one template-free process provides a robust way for synthesizing switchable SCO nanoparticles with a significant reduction of preparation time. The introduction of a supercritical drying step affords a one-step process for the production of dried and well-defined nanoparticles at large scale. This unique process opens routes towards process intensification and offers new opportunities for molecular materials.

Figure 1 :

 1 Figure 1: Schematic descriptions and comparison of the conventional batch synthesis process (left) and the continuous scCO2-assisted flow synthesis (right).

Figure 2 :

 2 Figure 2: SEM (left) and TEM (right) images of the samples obtained from batch mode (1), continuous reaction process (2) and the all-in-one process (3).

Figure 4 :

 4 Figure 4: X-ray diffraction patterns for samples 1-3.

Figure 3 :

 3 Figure 3: Histograms of the lengths of samples 1-3 and their corresponding statistical distributions obtained by TEM, on at least 100 particles (using the standard deviation from the Gaussian distribution fit).

Figure 5 :

 5 Figure 5: χMT versus T plot for samples 1 to 3, recorded after a first heating at 395 K. 320 330 340 350 360 370 380 390 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Table 1 :

 1 Comparison of the experimental features and particle sizes of the different synthetic processes (batch synthesis, flow chemistry and flow chemistry with scCO2) for the same switchable material with the same concentration of reactants and reaction temperature

		Batch	Continuous	All-in-one
		synthesis	process	process
	Reaction yield	30 %	23 %	34 %
	production (g)	1.1	0.8	1.2
	Steps	4	3	1
	Reaction	1h15	15 s	15 s
	(residence) time
	Rod size	~ 2 100 nm	~ 280 nm	~ 170 nm
	(length x width)	x 490 nm	x 120 nm	x 50 nm

Acknowledgements

This work was supported by the University of Bordeaux, the CNRS, the ANR FlowSwitch (2019-CE07-0022-01), and by the Region Nouvelle Aquitaine and the ADEME agency through the ISOCEL project. The UMS PLACAMAT is acknowledged for the microprobe measurements. Marion Gayot and Sonia Buffière are acknowledged for the help for TEM and SEM images.

Keywords: spin crossover • nanoparticles • flow chemistry • supercritical CO2 •

Entry for the Table of Contents

We report here an innovative and robust template-free continuous process to synthesize nanoparticles of a switchable coordination polymer, including the use of supercritical CO2, aiming at both quenching the particle growth and drying the powder.