François Hild
email: francois.hild@lmt.ens-cachan.fr

Eric Maire
email: eric.maire@insa-lyon.fr

Stéphane Roux
email: stephane.roux@lmt.ens-cachan.fr

Jean-François Witz
email: jean-francois.witz@lmt.ens-cachan.fr

Three dimensional analysis of a compression test on stone wool

Keywords: Cellular materials, compression test, deformation heterogeneities, microtomography, 3D image correlation

A full three dimensional study of a compression test on a sample made of stone wool is presented. The analysis combines different tools, namely, X-ray microtomography of an in situ experiment, image acquisition and treatment and 3D volume correlation. This set of tools allows for the measurement of three dimensional displacement fields and to evaluate the Poisson's ratio of this type of material. The strain fields are used to analyze localization phenomena. The uncertainty of the analysis is evaluated by actually moving the sample in its undeformed and deformed states, qualifying both image acquisition and correlation steps.

1 Introduction X-ray computed microtomography (XCMT) is increasingly used to visualize the complete microstructure of various materials. One of its main advantages lies in the non destructive way of obtaining 3D views of various materials [START_REF] Baruchel | X-Ray Tomography in Material Sciences[END_REF].

By analyzing 3D reconstructed pictures, one has access, for instance, to the structure of biological [START_REF] Ambrose | [END_REF]3] or cellular (either metallic or polymeric foams [START_REF] Bernard | 1st Conference on 3D-Imaging of Materials and Systems 2008[END_REF]) materials, and sometimes to the way they deform by using in situ experiments [START_REF] Maire | [END_REF] analyzed by 3D correlation algorithms [6][7][8][9]. This procedure is to be distinguished from X-ray diffraction which may be used to evaluate elastic strains [10] for crystalline materials. In the present study, the reconstructed scans themselves are used to evaluate full displacement fields.

To measure 3D displacements in the bulk of a scanned material, different routes were followed. First, marker (e.g., W particles) tracking was used to analyze permanent strains in a compression test on aluminum [11]. Second, local correlation techniques were utilized [9,12,13], in which small interrogation volumes in two scans are registered. The analysis of the deformation of biomechanical tissues was reported [7,14]. Strain fields were also studied in heterogeneous materials [13,15]. Third, global (i.e., Galerkin) correlation techniques are an alternative to local approaches. Strain localization in a solid foam was studied with such a technique [9]. When the chosen kinematics is enriched to account for displacement discontinuities, it is possible to measure crack opening displacement fields [16] and profiles of stress intensity factors along the whole crack front [START_REF] Rannou | Submitted for publication[END_REF]. This information is useful, for instance, to model 3D crack propagation.

In the case of entangled materials such as mineral wools, there are very few results on the details of the microstructure and more importantly on the way these materials deform. 2D measurements were performed on the surface of mineral wools [START_REF] Hild | [END_REF]19]. Heterogeneous deformation modes of high density ma-terials occur in uniaxial compression and were mainly controlled by local details of the microstructure such as density [START_REF] Hild | [END_REF] and fiber orientation [20]. The latter is crucial when describing the mechanical behavior of high density materials [21]. One of the unanswered questions concerns the representativeness of surface measurements when bulk properties are sought. XCMT provides a unique way of eventually addressing this issue.

The aim of the present paper is to analyze a compression test on a stone wool sample by using XCMT performed with a lab tomograph to analyze the way these materials deform internally. The standardized mechanical performance of those materials refer to a strain level well above the "elastic" limit [START_REF]Thermal insulating products for building applications -Determination of compression behaviour[END_REF].

In that case, the strain field appears to be generically non uniform, and it is important to characterize the regions that concentrate strains, and possibly identify morphological features that are responsible for such strain concentrations as they appear to be stress (and thus performance) limiting. Full three dimensional strain maps and morphology are therefore extremely valuable. A finite-element based approach to digital image correlation [9] is used to measure displacement and evaluate strain fields during a compression test on stone wool. In Section 2, the studied material, the experimental configuration and the imaging system are presented. The correlation procedure used herein is briefly recalled in Section 3 and its performance is evaluated. The analysis of several images of the same unstrained sample reveals resolution limits resulting from the computed reconstruction. The experimental results are finally analyzed in Section 4 and discussed in Section 5.

Experimental configuration

Material

Stone wool is an insulating material made of fibers with basalt-like composition. Figure 1a shows a large scale SEM view of the studied material. The production process of stone wool, called REX, involves (among other mechanisms) a centrifugal instability of a molten rock layer deposited on rapidly spinning wheels. The destabilization of the viscous layer forms a tadpole whose bulky head is drawn by a hot gas burner and whose tail connected to the spinning wheel produces fibers. In the final product, as can be seen in Figure 1a, the fibers are mixed with a fraction of so called "shot" particles (the initial head of the tadpole). These shot particles are a signature of the production process. They increase the density of the final material, and play no role for insulation. The size of the fibers is mostly in the range 1-10 µm, while the shot particles are of the order of 50 to 100 µm in diameter.

Figure 1b shows a typical normalized load vs. macroscopic compressive strain.

The latter is evaluated by measuring the relative displacement of the two platens divided by the initial sample height. After an initial non-linear part associated with a gradual contact of the sample surface with both platens, there is an elastic regime followed by another non-linear part, presumably due to frictional slip between fibers. Upon unloading the material in the nonlinear regime, the response is strongly hysteretic as one could expect when solid friction and contact opening occur.

XRCT Setup

Standard laboratory X-Ray microtomography enables the experimentalist to get 3D pictures of the local density of solids by exploiting the differential attenuation of a polychromatic X-ray beam while it traverses the analyzed sample [START_REF] Maire | [END_REF]. The tomograph used in the present study is made by Phoenix X-ray. It contains two principal components:

• The source is an open transmission nanofocus X-ray tube operated at 90 kV and 140 mA. The thin transmission target bombed by focused electrons is made of tungsten and the focus size is about 6 µm.

• The X-ray detector is a Paxscan TM amorphous silicon flat panel initially developed for medical applications. It is composed of about 1900 rows and 1500 lines of sensitive pixels, the size of which is 127 × 127 µm 2 . The pixels are binned to accelerate the acquisition time during the present experiment (final size of the detector: 950 × 750 pixels, pixel size: 254 × 254 µm 2 ).

The source/detector distance is fixed to 80 cm, but the resolution (defined in the present paper as the voxel size in the reconstructed image) can be varied by changing the position of the rotating object between the source and the detector. A good trade-off between sample size and resolution was chosen to be 13.5 µm in the present case.

Reconstructed volumes

Image and/or volume correlations require a contrasted texture to measure displacements. The shot particles are extremely helpful for image correlation, as they constitute a random marking with a high contrast evenly distributed within the specimen. As can be seen in Figure 2, they appear as small white clusters with a well defined morphology as compared to the fibers that can-not be distinguished individually at the resolution used for this study. The dynamic range of the reconstructed pictures (or scans) is equal to 16 bits.

The reconstructed scans are subsequently recoded as 8 bit data. During this recoding, 0.1 % of the black and of the white pixels are intentionally saturated to preserve a good dynamic range, as seen on the histogram of Figure 2. The mean gray level is 43 and the corresponding standard deviation is 27. The size of the reconstructed volumes is 750 × 750 × 750 voxels, and the analyzed region of interest (ROI) is centered and has a size of 128 × 128 × 128 voxels because of memory limitations (all the results presented herein are performed on a standard PC with a dual core CPU). The computation time per couple of analyzed pictures lasted on average ten minutes.

In situ compression test

A testing machine (described in details in Ref. [23]) is clamped on the rotating stage of the tomograph. It allows one to take 3D pictures of the specimen during a compression test. The sample was compressed between two parallel platens with no lubrication. Six scans were acquired for the analyzed test.

First, two scans were performed with no applied load. They will be used in the uncertainty analysis (see Section 3.2). Second, four loading steps were considered with a targeted constant displacement between each step. For the last deformation step, the whole experimental setup (sample + compression rig) was moved upwards at constant load and a subsequent acquisition was performed. These two scans will also be used for the uncertainty analysis.

The displacement speed was kept rather low (1 µm s -1 ) leading to a strain rate of about 10 -4 s -1 . Due to experimental difficulties, the load could not be measured during this in situ experiment. This will not modify the conclusions of the present paper.

3D Digital Image Correlation

In this section, the correlation procedure is first presented. Its performance is evaluated by using two scans of the same sample for which a rigid body motion is applied between the two acquisitions.

Finite-element approach to DIC

Image correlation consists in matching the texture of two pictures (or scans in the present case) with the help of a displacement field to be determined.

The passive advection of the texture between the reference f and deformed g images (e.g., see Figure 2) reads from a Lagrangian point of view

f (X) = g(X + U(X)) (1)
where X is the position vector in the reference configuration (i.e., Lagrangian coordinate), and U the displacement vector. Equation (1) corresponds to the conservation of the optical flow written at a local level, in the reference configuration. In practice, the optical flow conservation hypothesis is not strictly satisfied. This is even more relevant in the case of XCMT for which the 3D scans are the results of a reconstruction procedure that induces some artefacts [START_REF] Baruchel | X-Ray Tomography in Material Sciences[END_REF]. Consequently, correlation residuals arise since the difference f (X)g(X + U(X))

does not vanish. From the knowledge of f and g, the measurement problem consists in identifying U as accurately as possible. To estimate U, the

quadratic difference ϕ 2 = [f (X) -g(X + U(X))] 2 is integrated over the stud- ied domain in its reference configuration Ω Φ 2 = Ω ϕ 2 dX (2)
and minimized with respect to the degrees of freedom of the measured displacement field. In the present case, a 3D finite element kinematics is chosen [START_REF] Zienkievicz | The Finite Element Method[END_REF] for the searched fields U, and the simplest shape functions are used, namely, trilinear polynomials associated with 8-node cube elements (or C8-DIC [9]).

The functional Φ 2 is minimized with respect to the set of unknown degrees of freedom. To capture large scale displacements, it is important to perform a first determination of the displacement field based on low-pass filtered images where small scale details are erased. After correction of the deformed picture by this displacement, finer details of the images are restored and the residual displacement computed. This procedure is repeated until the unfiltered images are considered. A very crude filtering is used here because of its simplicity and efficiency, namely, a filtered image is constructed by gathering each cube of 2 × 2 × 2 voxels into one "super-voxel" and summing their gray level values. A higher degree of filtering is obtained by iterating this procedure. This multiresolution procedure has the additional benefit of decreasing drastically the size of the images to be studied.

Uncertainty analyses

To assess the quality of a correlation result, the correlation residuals are the only data available when the measured displacements are not known. In the following, the normalized correlation residual is considered

η(X) = |ϕ(X)| max Ω (f ) -min Ω (f ) (3)
and its mean value η is computed over the whole correlation volume Ω.

The uncertainty is conventionally evaluated by the standard deviation of the displacement field σ(U ). Last, to quantify the strain error, the mean principal strains ǫ i are estimated over the entire region of interest for each rigid body translation.

Displacement perpendicular to the rotation axis

Two scans were experimentally acquired for the same (unloaded) state. Yet, the sample was slightly moved by hand (i.e., U x ≈ 3.1 voxels, U y ≈ 13.0 voxels, and U z ≈ 0.3 voxel, with 1 voxel ↔ 13.5 µm) between the two acquisitions to assess the minimum achievable correlation residual and the corresponding performance in terms of strain measurements by correlating the two scans. This procedure allows one to determine the cumulated effect of the correlation algorithm and of the acquisition / reconstruction procedure on the measurement uncertainties. This displacement was mostly performed in the horizontal (X-Y) plane of the tomograph, perpendicular to the rotation axis i.e., parallel to the platens of the compression rig. The rotation axis Z during the tomographic scanning is also the compression axis of the in situ tensile rig.

The element size was changed to assess its influence on the overall performance of the correlation algorithm. Figure 3 shows that the larger the element size, the smaller the standard displacement uncertainty. The mean strain levels remain very small and less than 0.05 % in absolute value for all the analyzed cases. The displacement uncertainties level off for element sizes greater than 16 voxels (or 216 µm). In the second regime where a constant level of the displacement uncertainty is observed, it is believed that the saturation is caused by reconstruction artefacts. Figure 4 shows cuts of the residual volume that clearly indicate two ring artefacts when a transverse displacement is performed. The presence of these rings is likely to affect the correlation procedure. This is further confirmed by the fact that the in-plane displacement uncertainties are the only ones that level off.

Displacement parallel to the rotation axis

Two scans were also acquired to reproduce the previous analysis for one of the deformed states of the sample. Contrary to the previous case, the translation was performed along the loading axis by moving the whole loading setup.

Figure 3 shows that the reconstruction procedure clearly impacts differently measurement uncertainties when actual displacements are performed along and perpendicular to the rotation axis. A displacement along the latter leads to a single set of rings in the correlation residuals (Figure 4) and then to a smaller measurement error.

Last, the levels of the correlation residuals with the present analyses constitute minimum levels to be expected when two actual scans are correlated. The fact that they are virtually independent of the element size is all the more important since there will be a single reference level η = 3 %.

First analysis of the compression test

The element size is equal to 16 voxels (or 216 µm) for the first analyses, which is a compromise between the uncertainty level, the spatial resolution (i.e., the element size), and the computation time. Their size is subsequently decreased to enable for a better description of finer displacement details induced by strain heterogeneities. The reference volume is always the same and four loading steps could be analyzed with a reasonable convergence of the algorithm. In Figure 1a, over the SEM view, a box is drawn to show the size of an ℓ = 16 voxel element (216 µm) in comparison with the microstructure. Each single fiber is not followed; but rather a small bundle of fibers whose kinematics will be assessed.

Typical results obtained from the first deformation step

Prior to the C8 procedure itself, rigid body translations are evaluated by using the cross correlation product calculated via FFTs. The C8 procedure is then run recursively at different scales (i.e., by resorting to the coarsegraining procedure discussed above; the latter allows one to capture large displacements and strains [9,[START_REF] Hild | [END_REF]). Figure 5 shows three residual maps. The first one corresponds to the initial picture difference when no corrections are made. There is a clear mismatch between the two analyzed states. When an initial correction estimated as a rigid body translation is performed, it leads to the second map that shows the benefit of this first correction. However, there are still zones in which the residuals are high. Last, the third map shows the residuals at convergence. Except for a few points, the residuals are very low everywhere. This additional test is important, since it is only after having checked that the residual field is small in the whole ROI that the results are definitely deemed trustworthy.

This first deformation step deals with moderate strain levels (ca. -4.5 %)

in the elastic regime (Figure 1b). Four different element sizes are considered, namely, ℓ = 8 voxels, 10 voxels, 12 voxels, 16 voxels. The mean correlation residuals (see Table 1) are very close to the value observed in the uncertainty analysis ( η = 3 %). The strain levels in the transverse directions remain very small compared with those along the loading direction. In particular, when the element size is less than 16 voxels, the mean transverse strain is less than 0.02 %.

Second loading step

The mean strain level is higher in absolute value (ca. -8.1 %) for 16-voxel elements. From this first evaluation, one may want to decrease the element size to capture more accurately local displacement variations. This is made possible by running a further analysis and using the converged solution when the element size is equal to 16 pixels as an initialization. Three elements sizes are considered, namely, ℓ = 12, 10, and 8 voxels. No coarse (filtered) image and less than 10 additional iterations are needed. Table 1 summarizes the results. As the element size decreases, the mean correlation residual decreases as well. In the present case, it is believed that more degrees of freedom are needed to capture strain heterogeneities. When compared with the reference value (i.e., η = 3 %), it is concluded that the correlation results with the smallest element size (i.e., η = 3.4 %) are likely to be closer to the actual displacement field. One possible explanation is that it is the signature of strain heterogeneities requiring a fine mesh to capture more accurately the displacement heterogeneities. The strain heterogeneities might be understood as the consequence of the non-linear regime (Figure 1b) of the material. This point will be further discussed in Section 5.

Third and fourth loading steps

The third and fourth scans correspond to strain levels for which the nonlinear regime of the material is completely established (Figure 1b). For the third scan, even larger strains occur (ca. -11 %). This level is close to that considered in standardized tests, namely -10 % [START_REF]Thermal insulating products for building applications -Determination of compression behaviour[END_REF]. In terms of correlation residuals, there is a small degradation of their levels when compared with those achieved for the first loading step (see Table 1). However, the same trend is observed when the element size decreases, namely, η decreases as well. When compared with the reference value (i.e., η = 3 %), it is concluded that the correlation results with the smallest element size (i.e., η = 3.8 %) are still acceptable.

Last, the fourth scan is analyzed. Mean strain levels of the order of -14 % are reached, and many iterations are needed. The correlation residuals are still acceptable, especially for small element sizes (see Table 1), even though the computation time is significantly larger. This is an indication that at this level, the reference scan should be updated when additional load levels are considered, namely, incremental displacement fields between different load levels should be computed from image correlation, and then combined together to compute the total displacement.

Analysis of the whole sequence

It is of particular importance to note that in the present case no updating was necessary to analyze the four deformation steps and this despite the fact that the mean strain level was rather large (i.e., -14% for the last step). This good performance of the correlation procedure is partly due to the fact that only a small ROI is considered, and more importantly, to the multi-resolution procedure. All correlation results converged for mean average displacements between two iterations less than 0.1 mpixel, and mean correlation residuals less than 5 %. From the analyzed displacement field, the mean strains are determined over the whole ROI. In the present case, the infinitesimal strain assumption cannot be made. Consequently, a large transformation framework is used. The deformation gradient tensor F is considered. The latter is related to the displacement U by

F = 1 + ∇U (4)
where 1 denotes the second order unit tensor. A polar decomposition of F is used [START_REF] Truesdell | The Non-Linear Field Theories of Mechanics[END_REF]

F = RS ( 5
)
where R is an orthogonal tensor (R -1 = R t ) describing the rotations, and S the right stretch tensor (S = S t ). From the latter, the right Cauchy Green tensor (C = F t F = S 2 ) allows one to define, for instance, the following strain

measures E m E m = 1 2m (C m -1) (6)
when m = 0. In the following, the stretch tensor S itself is considered, and the corresponding nominal (or Cauchy-Biot) strain E 1/2 = S -1 denoted by ǫ. When averaged over the whole region of interest, the mean strain is defined as

ǫ = S -1 (7)
where S is the main stretch tensor defined via the polar decomposition of the mean deformation gradient

F F = 1 + 1 V ∂Ω U ⊗ N dS (8)
and N denotes the outward normal vector of the external surface ∂Ω of the ROI Ω, of volume V . The principal strains are subsequently determined, and the apparent Poisson's ratio ν is defined as

ν = - ǫ 2 + ǫ 3 2ǫ 1 (9)
where ǫ 1 is the contraction strain (i.e., |ǫ

1 | > |ǫ 2 |, |ǫ 3 |).

Global analysis

The first step leads to strain levels ǫ 1 of the order of -4.2 % when 8-voxels elements are used. Since the strains ǫ 2 and ǫ 3 remain very small (compared with ǫ 1 ), the evaluation of Poisson's ratio may be difficult. The values of the latter (Table 1) are on average equal to 0.00 with small variations when using large element sizes. This level is to be expected from such type of material for which the entangled and porous texture does not allow for large transverse strain.

This result is also observed when a mechanical identification of local elastic parameters is performed on glass wool by using surface measurements only [21] for the same order of magnitude of average strain. The entangled nature of the material explains that it firstly densifies before expanding laterally. Poisson's ratio is reasonably independent of the element size, even if the correlation residuals decrease as the element size decreases (Table 1).

For the second step, the mean strain ǫ 1 is already large in absolute value (ca. -7.6 % for 8-voxel elements). This level lies above the apparent elastic regime of the material (Figure 1b), yet Poisson's ratio remains very small (0.01). This value is also found for the third step when 8-voxel elements are used. The strain ǫ 1 increases again in absolute value (ca. -13.9 % for 8-voxel elements) when the fourth step is analyzed. Poisson's ratio varies slightly (0.02) when compared with the previous load levels.

Figure 6 shows the longitudinal displacement maps for the whole sequence when a cut of the analyzed ROI is performed by a plane containing the axis of reconstruction. The fours maps are identical in terms of heterogeneity distributions. Only the overall levels vary from step to step. From the first step on, the heterogeneity of the displacement is virtually identical. It is worth noting that the displacement amplitudes are at least two orders of magnitude greater than the measurement uncertainties (Figure 3).

Strain heterogeneities

The previous displacement fields are analyzed in terms of the component F zz -

1 = U z,z .
If only small rotations occur, as is likely in the present case at the level of each considered element (Figure 1a), this quantity corresponds to the longitudinal nominal strain ǫ zz (see Equation ( 7)). In the following, the values ǫ e zz are estimated as averages per element of F zz -1. This is made possible by the use of Equation ( 8) applied element-wise. This value is assigned to the middle point of each element and the fields shown hereafter correspond to trilinear interpolations of these "nodal" quantities.

Figure 7 shows the maps of ǫ e zz for the four steps. Strain heterogeneities are well marked since the beginning of the experiment and the overall distribution remains identical throughout. Figure 8 shows the incremental strain maps ∆ǫ e zz for the same sequence. This last information allows one to visualize the "active" parts of the deformation process. In the present case, the third and fourth steps lead to more uniform fields compared with the first and second steps, thereby confirming the conclusions drawn from the analysis of the total strain and displacement maps.

Stone wool, akin to many cellular solids, exhibits a strongly non-linear mechanical behavior in compression. As it densifies, solid friction between fibers would lead to a plateau stress, if it was not balanced by the hardening effect due to the multiplicity of inner contacts created in compression (Figure 1b).

Consequently, the fiber length between contacts is reduced, and hence the stiffness (directly related to this length) increases rapidly. Therefore, any density heterogeneity present in the unloaded state is expected to lead to significant strain heterogeneity. Conversely, in compression, it is expected that density heterogeneities are to progressively vanish. This is consistent with the previous observations, where incremental strains tend to become more uniform under load (Figure 8), while the total strain seems to saturate to a state of fixed contrast (Figure 7).

It is to be noted that the previous argument is only a "mean-field" view of the problem. In reality, the volume elements are not all subjected to the same stress, and the latter itself is dependent both on the local properties of the element, but also on its surrounding. The strain compatibility is a global constraint that makes the analysis more complex. In particular, Figure 7 shows that large strain elements form a pattern that has to span through the entire sample.

To better understand the spatial distribution of this pattern, a 3D visualisation showing the clustering of large strains in 3D, after thresholding ǫ zz is shown in Figure 9-a. The high strain regions are more or less oriented with an oblique angle compared to the compression direction, which is vertical in this figure.

Figure 9-b shows for the same sample, oriented in the same way as in Figure 9a, the outline of the regions where the density in the initial sample (i.e., before deformation) are low (the threshold is here applied to capture the darkest regions of the tomographic scan). Given the origin of the contrast in these images, low gray levels indicate low densities. The spatial pattern is again complex. The regions do not correspond perfectly but the specific oblique angle is the same in both figures.

Let us stress the difference with the phenomenon of localization in metals and alloys, which also reflects an uneven strain distribution but that usually reveals an instability. For stone wool, the reverse is true, namely, heterogeneity is progressively erased under load and incremental strains tend to be increasingly homogeneous. To investigate this question further, a correlation between density and (axial) strain is searched for. By averaging strains for elements of mean density belonging to successive intervals (Figure 10), a strong decrease of compressive strain with density is observed. As the deformation level increases (in absolute value), the same trend is observed, yet shifted in the log-log plot.

The same type of conclusion was drawn when pictures of the surface of light density glass wool were analyzed [START_REF] Hild | [END_REF]. Note that the gray levels give a rather direct indication of absorption, and hence density, but absolute densities are scaled and offset to provide a slightly saturated histogram (Figure 2b) at the reconstruction stage.

Unfortunately, it is not possible to be more quantitative without performing a full numerical modeling of the material. Even in a statistical sense, for a heterogeneous elastic material, there is no unique relationship between local strain and local elastic modulus. The microstructure and stiffness spatial correlations affect crucially such a link. However, the fact that lighter volumes are more compressed than denser ones is likely, and supports the observation that incremental strains tend to become increasingly uniform under increasing load.

Summary

It was shown that in situ mechanical tests can be performed in a laboratory tomograph, and processed from digital image correlation techniques to evaluate quantitatively global mechanical properties, such as Poisson's ratio.

The uncertainty quantification methodology and results are presented to make explicit the choice of the correlation parameters to be used (such as the element size), and to decide when the correlation procedure can be trusted.

These procedures are fairly general and applicable to a wide variety of materials. Furthermore, strain heterogeneities could be captured even for large strain levels with a good confidence in the results thanks to the correlation residuals.

For stone wool, Poisson's ratio evaluations at different load levels were performed, and remain consistently equal to 0 (at the scale of the accuracy of the analysis) for compressive strains of the order of 10 Correlation residuals |f -g| expressed in gray levels, when no correlation is performed (a), when only rigid body translation is accounted for (b), and at convergence (c) after the whole correlation procedure was carried out. This last residual shows that the registration was successful when comparing it to the deformed scan (d) and the reference scan shown in Figure 2.

All positions are for the same voxel frame as that used in

Figure 2 . 29 6

 26 Figure 2. 29

Fig. 1 .Fig. 2 .

 12 Fig. 1. (a) Large scale SEM view of the analyzed sample showing the fibers and coarse shot particles. The box drawn on the right hand top corner shows the size of a 16-voxel element (where the voxel size refers to the tomographic image shown in Figure 2a) as a reference. (b) Typical normalized load vs. macroscopic compressive strain of the studied stone wool. The arrows show the mean strain levels at which the DIC analysis was performed.

Fig. 3 .

 3 Fig. 3. Standard displacement uncertainty σ(U ) as a function of the element size when transverse and longitudinal displacements are prescribed (1 voxel ↔ 13.5 µm).

Fig. 5 .

 5 Fig. 5. Correlation residuals |f -g| expressed in gray levels, when no correlation is performed (a), when only rigid body translation is accounted for (b), and at convergence (c) after the whole correlation procedure was carried out. This last residual shows that the registration was successful when comparing it to the deformed scan (d) and the reference scan shown in Figure 2. All positions are for the same voxel frame as that used in Figure 2.

Fig. 6 .

 6 Fig. 6. Longitudinal displacement maps U z expressed in voxels (1 voxel ↔ 13.5 µm)for the first four loading steps when 8-voxel elements are used.

Fig. 10 .

 10 Fig.10. Log-log plot of the average axial compressive strain vs. initial average density (in gray levels) for the first (•), second (×), third (× +) and fourth (•) deformation steps. A dotted line of slope -1 is shown as a guide to the eye.

Table 1

 1 Effect of the element size on the correlation results for the four deformation steps.

	ℓ (voxels)	η	ǫ 1	ǫ 2	ǫ 3	ν
	16	3.5 % -4.5 % 0.0 % 0.1 % 0.01
	12	3.3 % -4.3 % 0.0 % 0.0 % 0.00
	10	3.2 % -4.2 % 0.0 % 0.0 % 0.00
	8	3.1 % -4.2 % 0.0 % 0.0 % 0.00
	16	4.0 % -8.1 % 0.0 % 0.3 % 0.02
	8	3.4 % -7.6 % 0.0 % 0.1 % 0.01
	16	4.6 % -11.4 % 0.0 % 0.6 % 0.03
	8	3.8 % -10.7 % 0.0 % 0.3 % 0.01
	16	4.9 % -14.5 % 0.1 % 1.0 % 0.04
	8	3.9 % -13.8 % 0.1 % 0.5 % 0.02

Acknowledgments

This work was part of the project PHOTOFIT funded by "Agence Nationale de la Recherche." The stone wool sample was kindly provided by Jean-Baptiste Rieunier (Saint-Gobain Isover).

List of Tables

Effect of the element size on the correlation results for the four deformation steps. 22

List of Figures -a--b- with the smaller density (b). The strains were obtained by using the sample in its first deformed state, and the gray level for the sample in its initial state (i.e., before deformation).


Yohan Biecher

Anthony Baux

Francois Fauth

Claude Delmas

Gillian R Goward

Dany Carlier
email: dany.carlier@icmcb.cnrs.fr

Structure and electronic structure evolution of P2-Na x CoO 2 phases from XRD and 23 Na MAS NMR

 Na MAS NMR we discuss the structural changes occurring due to the deintercalation of Na + ions from the interlayer slabs and the change in the electronic structure and magnetic properties. The XRD study allows discussing the general evolution in relation with previous works. The novelty lies here in the observation for the first time of an ordered phase for x=1/3 appearing between above 4.3 V followed by a disordering in the slabs stacking for higher voltages. The combination of the data obtained by the different techniques allowed the interpretation of the NMR shift and shape evolution versus the Na content. This study reveals a complex behavior due to the presence of localized and delocalized electrons whose relative proportions is changing versus Na content.

Introduction

Layered colbaltates oxides have been investigated for their application as positive electrode in Liion or Na-ion batteries [START_REF] Fouassier | Sur de Nouveaux Bronzes Oxygénés de Formule Na x CoO 2 (X < 1). Le Systeme Cobalt-Oxygene-Sodium[END_REF][START_REF] Mizushima | x CoO 2 (0< x<1): A New Cathode Material for Batteries of High Energy Density[END_REF][START_REF] Braconnier | Comportement Electrochimique Des Phases Na x CoO 2[END_REF][START_REF] Delmas | Electrochemical Intercalation of Sodium in Na x CoO 2 Bronzes[END_REF][START_REF] Nitta | Li-Ion Battery Materials: Present and Future[END_REF][START_REF] Delmas | Sodium and Sodium-Ion Batteries: 50 Years of Research[END_REF] and are often considered as model compounds. These lamellar oxides consist of a stacking of edge-sharing MO 6 octahedra layers forming MO 2 slabs between which alkali ions are intercalated. Whereas the lithium ions are in octahedral sites, the sodium ones can be found either in octahedral sites, like in O3-NaCoO 2 [START_REF] Lei | Synthesis and Stoichiometry of Different Layered Sodium Cobalt Oxides[END_REF] or in prismatic sites for lower sodium contents, like in P2-Na x CoO 2 , P3-Na x CoO 2 [START_REF] Delmas | Sodium and Sodium-Ion Batteries: 50 Years of Research[END_REF] and P'3-Na 2/3 Mn 1-y Fe y O 2 [START_REF] Baskar | Layered P2-Na x CoO 2 and Na x FeO 2 as Cathode Materials for Potassium-Ion Batteries[END_REF][START_REF] Yoncheva | Structure and Reversible Lithium Intercalation in a New P′3-Phase: Na 2/3 Mn 1-y Fe y O 2 (Y = 0, 1/3, 2/3)[END_REF] . Following the nomenclature developed by Delmas et al. [START_REF] Delmas | Structural Classification and Properties of the Layered Oxides[END_REF] , the P2 stacking of Na x CoO 2 phases consists in an alternation of CoO 2 slabs formed by edge-sharing CoO 6 octahedra and interslab spaces in which

x Na + ions can be located in two prismatic sites (Figure 1a) : one sharing edges (Na e ) and one sharing faces (Na f ) with the surrounding CoO 6 octahedra (Figure 1b). Due to large Na + -Na + electrostatic in-plane repulsion, the sodium ions tend to order in the interslab space, yielding a very complex phase diagram as investigated by some of us using in-situ X-ray Diffraction (XRD) for 0.45 < x < 0.9 [START_REF] Berthelot | Electrochemical Investigation of the P2-Na x CoO 2 Phase Diagram[END_REF] .

Figure 1: (a) Representation of the structure of layered P2-Na x CoO 2 . (b) local coordination environment of the Na e (in blue) and Na f (in green)

The P2-Na x CoO 2 layered phases were also investigated extensively in literature for their physical properties as thermoelectric and superconductivity performances (if hydrated) [START_REF] Terasaki | Large Thermoelectric Power in NaCo 2 O 4 Single Crystals[END_REF][START_REF] Takada | Superconductivity in Two-Dimensional CoO 2 Layers[END_REF][START_REF] Chou | Thermodynamic and Transport Measurements of Superconducting Na 0.3 CoO 2 •1.3H 2 O Single Crystals Prepared by Electrochemical Deintercalation[END_REF][START_REF] Landron | Ab Initio Evaluation of the Local Effective Interactions in the Superconducting Compound Na 0[END_REF][START_REF] Lee | Large Enhancement of the Thermopower in Na x CoO 2 at High Na Doping[END_REF][START_REF] Lee | Enhancement of the Thermopower in in the Large-X Regime (0 > 0.75)[END_REF][START_REF] Huang | Low Temperature Phase Transitions and Crystal Structure of Na 0.5 CoO 2[END_REF] . One interesting aspect in such layered cobalates is the co-existence of localized and itinerant electrons for some compositions. Foo et al. [START_REF] Foo | Commensurability, and Metallicity in the Phase Diagram of the Layered Na x CoO 2[END_REF] reported the magnetic phases diagram of the P2-Na x CoO 2 system and proposed a change of the magnetic regime from a "Curie-Weiss metal" regime for x > 0.5, with the co-existence of localized and itinerant electrons to a "Pauli metal" regime for x < 0.5 with only itinerant electrons. This was interpreted by strong electron correlations in the CoO 2 layers for x > 0.5, with on the triangular lattice, a fraction (1-x) of the sites occupied by itinerant spin-1/2 holes (Co 4+ ) hopping in a diamagnetic background of Co 3+ ions [START_REF] Wang | Spin Entropy as the Likely Source of Enhanced Thermopower in Na x Co 2 O 4[END_REF] . In order to interpret those physical properties, Landron et al. [START_REF] Landron | Importance of t 2g -E g Hybridization in Transition Metal Oxides[END_REF] suggested that the trigonal distortion of CoO 6 octahedra along the c direction leads to the lift of degeneracy of the t 2g orbitals into two e g ' orbitals and one a 1g orbital as schematized on Figure 2. Therefore the band diagram considering this D 3d distortion has a narrow band at the Fermi level for a 1g and a broad brand for the e g ' levels providing respectively "heavy" charge carriers (localized electrons) and "light" charge carriers (delocalized electrons) [START_REF] Singh | Electronic Structure of NaCo 2 O 4[END_REF] . paramagnetic compounds, as in the case of most of the positive electrode materials, the Fermi contact interactions mainly govern the signal position, according to:

eq(1)
where S is the spin quantum number of the paramagnetic ion, ρ i (0) is the spin density on the i nucleus resulting from a spin transfer from the paramagnetic ions and the molar magnetic susceptibility (at the temperature of the NMR measurement). The spin transfer from the transition metal ion to the probed nucleus implies the local chemical bonds either by a delocalization (through orbital hybridization) or polarization mechanisms [START_REF] Carlier | Understanding the NMR Shifts in Paramagnetic Transition Metal Oxides Using Density Functional Theory Calculations[END_REF] . In metallic compounds, the spectra are dominated by Knight interactions leading to the following shift: eq

where is the Pauli susceptibility of conduction bands electrons at Fermi level and is the contribution of "s" orbitals at Fermi level of probed nucleus.

In recent studies [START_REF] Berthelot | Reinvestigation of the OP4-(Li/Na)CoO 2 -Layered System and First Evidence of the (Li/Na/Na)CoO 2 Phase with OPP9 Oxygen Stacking[END_REF][START_REF] Biecher | Original Layered OP4-(Li,Na) x CoO 2 Phase: Insights on Its Structure, Electronic Structure, and Dynamics from Solid State NMR[END_REF] , we used [START_REF] Lei | Synthesis and Stoichiometry of Different Layered Sodium Cobalt Oxides[END_REF] Li and [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na solid state NMR to investigate the local structure, electronic structure and the dynamics in an original layered cobalt oxide: OP4-Li 0.42 Na 0.38 CoO 2 , that exhibits Na and Li layers alternating along the c axis [START_REF] Balsys | The Structure of Li 0.43 Na 0.36 CoO 1.96 Using Neutron Powder Diffraction[END_REF] . In particular, variable temperature [START_REF] Lei | Synthesis and Stoichiometry of Different Layered Sodium Cobalt Oxides[END_REF] Li MAS NMR showed that NMR shifts result from two contributions: the Fermi contact and the Knight interaction due to the presence of both localized and delocalized electrons in the material [START_REF] Grey | NMR Studies of Cathode Materials for Lithium-Ion Rechargeable Batteries[END_REF][START_REF] Pecher | Methods: NMR in Battery Research[END_REF] .

In this paper, we aim to have a better insight of the mechanisms involved during cycling of P2-Na x CoO 2 is used as positive electrode material in a Na cell. We thus studied the material evolution from the point of view of structure at different scales and of electronic properties upon charge up to high voltage (4.6 V) using a combination of ex-situ and operando XRD and ex situ [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR. Especially, the evolution of the NMR spectra (signal shape and shift) is discussed as a function of the local structure and electronic structure evolution. This study reveals a complex behavior due to the presence of localized and delocalized electrons in the electronic structure of the material resulting in a Fermi contact or Knight interactions respectively.

Experimental

Synthesis of the pristine phase

The synthesis of the pristine phase, P2-CoO 2 , was performed using two precursors: Na 2 CO 3 (Sigma Aldrich, 99.9 %, dried in an oven before the synthesis) and Co 3 O 4 (home-made from cobalt nitrate and heat treated under air at 450 °C). Both precursors were mixed and ground in a glovebox filled with argon. Moreover, an excess of 5 % wt of Na 2 CO 3 was added to the stoichiometric proportion because of the sodium high volatility at high temperature. The mixture was then heated in a furnace under O 2 at 900 °C for 12 h. The heating rate was set at 2 °C.min -1 .

The product was then quenched to room temperature and entered directly in a Ar-filled glovebox.

The XRD structural characterization of the pristine material can be found in SI-1 for P2-. SI-2 shows the Scanning Electron Microscopy images obtained on the pristine phase.

Synthesis of the deintercalated phases

A floating technique was performed in order to synthetize the deintercalated phases with a precise control of the voltage to ensure the reproducibility of the results. These syntheses were performed in an airtight electrochemical cell using Na(s) / NaPF 6 in 1M Propylene Carbonate (PC) + 2% Fluoroethylene Carbonate (FEC) / P2-CoO 2 batteries. FEC was added to the electrolyte to limit its decomposition at high voltage [START_REF] Komaba | Fluorinated Ethylene Carbonate as Electrolyte Additive for Rechargeable Na Batteries[END_REF] . The positive electrode, negative electrode and the electrolyte were assembled in Swagelok type cells in an Ar-filled glovebox. The positive electrodes consist in a mixture of 90 mg of active material and 10 mg of black carbon with no binder. The mixture was pressed into pellets in the glovebox to form a dense positive electrode of about 100 mg. The electrochemical syntheses were realized in two steps as described on Figure SI-3. First, a galvanostatic step was performed at C/20 to the targeted voltage. Then, a potentiostatic step was used until the current reaches negligible values. The cell was then disconnected and entered in the glovebox where its voltage was measured before dissembling to be insured that the material is stable. Finally, the positive electrode powder was washed with DiMethyl Carbonate to remove any trace of electrolyte. Ex-situ XRD and [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR analyses were performed on the same Na x CoO 2 sample.

Characterizations ICP-OES

The chemical analysis of the Na and Co proportion of the pristine phase was realized by inductively coupled plasma-optical emission spectroscopy (ICP-OES) using a Varian Model 720-ES spectrometer. The powder was dissolved into a concentrated hydrochloric acid solution before analysis. The measured Na/Co ratio confirms the composition of Na 0.70(8) CoO 2 for the pristine phase.

Electrochemical measurements

The positive electrode was composed of P2-Na 0.70 CoO 2 as active material mixed with Polytetrafluoroethylene (PTFE) as binder and a 50/50 wt mixture graphite/carbon black to improve the electronic conductivity in the electrode with a respective 88:2:10 mass ratio. The battery was assembled in a coin cell configuration in a dry Ar-filled glovebox. The galvanostatic measurements were performed at 25 °C using a C/20 cycling rate between either 1.5 V and 4.3 V or between 1.5 V and 4.6 V.

XRD

The XRD characterization of the pristine sample was performed on a PANalytical X'Pert Pro diffractometer and equipped with Co radiations. The sample was enclosed in the Ar glovebox in a hermetic aluminum sample holder with a Kapton window on the top. The measurements were performed between 5° and 135°. The profile refinement using the Rietveld method was realized with the software FullProf developed by J. Rodriguez-Carvajal 31 for 2 from 15° to 85°.

The XRD analyses of the phases synthesized by electrochemistry were performed on a PANalytical X'Pert3 Powder diffractometer with a X'Celerator detector and equipped with Cu radiations. The samples were enclosed in 0.3 mm-diameter glass capillaries sealed in the Ar glovebox. The measurements were performed between 10° and 80°. The profile refinements and structure refinements using the Rietveld method were realized with the FullProf software [START_REF] Rietveld | A Profile Refinement Method for Nuclear and Magnetic Structures[END_REF] . Due to the poor quality of the XRD acquisitions (Co fluorescence), we performed the Rietveld refinements of the XRD patterns by fixing the isotropic thermal factor (B iso values) to 0.5 Ų for the Co and O ions and 1.0 Ų for the Na + ions. Moreover, the total Na + content was also fixed based on the electrochemical experiments. Therefore, we only refined the z position of the oxygen ions and the repartition of the occupancies between the two Na e and Na f sites.

In-situ Synchrotron XRD

High-angular resolution Synchrotron X-ray powder diffraction (SXRD) experiments were performed on the BL04-MSPD beamline of the ALBA Synchrotron [START_REF] Fauth | The New Material Science Powder Diffraction Beamline at ALBA Synchrotron[END_REF] (Cerdanyola del all s, Spain) on Na(s) / NaPF 6 in 1M Propylene Carbonate + 2% FEC / P2-Na 0.70 CoO 2 cells and at room temperature. A λ = 0.8261 Å wavelength was used and data were recorded from 2 = 1.3° to 2 = 43.2°. We used a specific home-made coin cell with 3 mm holes and pasted Kapton windows. Moreover, a thin aluminum foil between the Kapton window and the electrode was used as current collector on both sides of the coin cell. SXRD acquisitions were performed operando during the charge and the discharge of the battery at C/10 from 1.5 V to 4.6 V.

A succession of potentiostatic steps was performed from 4.1 V to 4.8 V with a step of 0.05 V and at a rate of C/10 in order to deeply charge the cell. The SXRD patterns were collected at the end of the different relaxations.

Na MAS NMR

The samples were introduced in a 2.5 mm zirconia rotors inside a glovebox. [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR was performed on a Bruker Avance spectrometer at 79.4 MHz (7 .05 T magnet). As the [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na nucleus is a quadrupolar nucleus with I = 3/2, a short pulse length of 1 µs corresponding to a π/8 pulse determined using an aqueous 1 M NaCl solution was employed. In these conditions, the main signal observed is due to the -1/2  +1/2 central transition. The spectral width was set to 1 MHz, and the recycle time D 0 = 0.2 s, is long enough to avoid the T1 saturation effects. The baseline distortions resulting from the spectrometer dead time (8 µs) were computationally removed using a polynomial baseline correction routine with the software DMFit [START_REF] Massiot | Modelling One-and Two-Dimensional Solid-State NMR Spectra[END_REF] .

Results and Discussion

Electrochemical cycling properties up to 4.6V vs Na + /Na

Figure 3 shows the evolution of the voltage as a function of the theoretical Na + content (x) calculated from the electrode mass and applied current for a Na//P2-Na x CoO 2 cell cycled at C/20 from 1.5 V to 4.3 V (Figure 3a) and 1.5V to 4.6V (Figure 3b). Figure 3a highlights the complexity of the P2-Na x CoO 2 system with the presence of many voltage steps corresponding to the formation of phases with an ordering between filled and vacant Na + sites for specific sodium contents, in agreement with previous studies [START_REF] Lei | Synthesis and Stoichiometry of Different Layered Sodium Cobalt Oxides[END_REF][START_REF] Berthelot | Electrochemical Investigation of the P2-Na x CoO 2 Phase Diagram[END_REF] . In the 1.5V-4.3V voltage range, all structural transitions are reversible and the cell exhibits a rather good capacity retention (Figure SI-4).

When the cell is cycled up to 4.6 V, an additional voltage jump is clearly observed for x = 1/3 probably due to the presence of a peculiar ordered phase for this composition that will be discussed below. To our knowledge this phase has not been reported before, although its structure was predicted by DFT calculations [START_REF] Zhang | Theory of Sodium Ordering in Na x CoO 2[END_REF][START_REF] Wang | Effect of Electronic Correlation on the Na Ordering of Na x CoO 2[END_REF] . The first discharge from 4.6 V (Figure 3b Note that the (004) peak intensity slightly decreases which could be the sign of the P2 structure instability and the beginning of sheets glidings as often observed in layered materials [START_REF] Yabuuchi | P2-Type Na x [Fe 1/2 Mn 1/2 ]O 2 Made from Earth-Abundant Elements for Rechargeable Na Batteries[END_REF][START_REF] Billaud | Na 0.67 Mn 1-x Mg x O 2 (0 < X < 0.2): A High Capacity Cathode for Sodium-Ion Batteries[END_REF] . In order to further investigate the phases structural modifications at high voltage, another cell was studied by in-situ SXRD, charged up to 4.1V at C/10 followed by a succession of potentiostatic steps between 4.1 and 4.8 V. SXRD patterns were collected at the end of the relaxation period when the material reached an equilibrium state (Figure 5 and Figure SI-7). In agreement with the operando study, from 4.1 V to 4.55 V, the average structure remains P2. At 4.6 V, the P2 structure is still maintained but the (002) line starts to broaden and appears with some asymmetry toward higher 2Ɵ angle that is usually due to the formation of stacking faults (Figure 5a). One can expect some "O"-type stacking faults in this structure (Figure SI-8). From 4.65 to 4.8 V, the (00l) diffraction lines strongly broaden and shift to larger angle values characteristic of a decrease of the c hex lattice parameter. In addition, the (h0l) diffraction lines broaden and become asymmetric. The peaks asymmetry and the decrease of the c hex lattice parameter confirm the presence of sheets glidings leading to a strong disorder in the material. The sheets glidings were already reported by Komaba and coworkers [START_REF] Platova | Nuclear Quadrupole Resonance and X-Ray Investigation of the Structure of Na 2/3 CoO 2[END_REF] and more recently by Billaud et al. [START_REF] Yabuuchi | P2-Type Na x [Fe 1/2 Mn 1/2 ]O 2 Made from Earth-Abundant Elements for Rechargeable Na Batteries[END_REF] on other P2systems by ex-situ XRD. Note that in these papers, the phases were determined in an OP4 structure considering that the Na + ions are intercalated in prismatic and octahedral sites after the P2-OP4 phase transition. In our study, the phase obtained at high voltage is called "OP" phase and not OP4 due to the absence of a long range ordering along the c axis contrary to the OP4-(Na,Li)CoO 2 phase studied by some of us [START_REF] Biecher | Original Layered OP4-(Li,Na) x CoO 2 Phase: Insights on Its Structure, Electronic Structure, and Dynamics from Solid State NMR[END_REF][START_REF] Balsys | The Structure of Li 0.43 Na 0.36 CoO 1.96 Using Neutron Powder Diffraction[END_REF][START_REF] Berthelot | Reinvestigation of the OP4-(Li/Na)CoO 2 -Layered System and First Evidence of the (Li/Na/Na)CoO 2 Phase with OPP9 Oxygen Stacking[END_REF] . Note also that in a previous publication of our group, this disordered "OP" phase was called "Z" phase in the P2-Na x Mn 1/2 Fe 1/2 O 2 system [START_REF] Mortemard De Boisse | P2-Na x Mn 1/2 Fe 1/2 O 2 Phase Used as Positive Electrode in Na Batteries: Structural Changes Induced by the Electrochemical (De)intercalation Process[END_REF] . The formation of this disordered phase starts to appear at 4.6V and be, at least, partially the origin of the large polarization observed upon discharge. If the "O" defects appears first at the surface of the particles, Li + intercalation kinetics may be slow down versus intercalation in the "P" type interslab.

Evolution of the local structure and electronic structure upon cycling

Figure 8 represents the [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR spectra recorded for the P2-Na x CoO 2 compounds. In the 0.33 x 0.82 composition range, a single signal with a typical 2 nd order interaction lineshape is observed. Due to strong Na + ion mobility in the interslab space versus the NMR timescale, an average signal between the two Na e and Na f sites is observed [START_REF] Carlier | Sodium Ion Mobility in Na X CoO 2 (0.6 < X < 0.75) Cobaltites Studied by 23 Na MAS NMR[END_REF] . The spectra were fitted using the DMFit software [START_REF] Massiot | Modelling One-and Two-Dimensional Solid-State NMR Spectra[END_REF] and the parameters are reported in Table SI- Considering the large values obtained for the isotropic shifts, we consider that they result mainly from two contributions: the Fermi contact and the Knight interaction due to the presence of, respectively, localized and delocalized electrons. The Fermi contact shift depends on the magnitude of the electronic spin transferred from the transition metal ions to the probed nucleus through the chemical bonds (eq (1)). This interaction, therefore, strongly depends on the number of unpaired localized electrons of Co n+ ions, on their 3d-type orbital occupancy and on the local hybridization between the Co 3d, O 2p and Na 3s orbitals. It is therefore very sensitive to both the local structure and electronic structure. The Knight shift depends on the contribution of the Na 3s orbital at the Fermi level where the delocalized electrons are involved.

Thus, here we can expressed the total isotropic shift by: In order to interpret the NMR shift evolution and to make the link with the mechanisms involved during the Na + deintercalation, we need to consider several main factors: the local geometric evolution, the local electronic structure evolution and the transition from a "Curie-Weiss metal" regime to a "Pauli metal" regime occurring at x = 0.5 [START_REF] Landron | Ab Initio Evaluation of the Local Effective Interactions in the Superconducting Compound Na 0[END_REF] . For the first aspect, the mean Na-O distances increases as x decreases mainly due to the c-axis expansion up to x = 0.40 as already mentioned (Figure 7a Those factors are sometimes antagonist regarding their effect on the [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR isotropic signal position evolution depending of Na + content (x). For more clarity, we will discuss below the evolution observed versus the Na content in three composition regions as indicated in Figure 8 and9.

In Region I (0.67 x 0.82), the NMR shift increases as x decreases. The main contribution of shift here is clearly the Fermi contact interaction that becomes stronger as the number of paramagnetic Co 4+ ions increases, despite the weaker Co n+ -O-Na orbital overlap. Therefore, a majority of the electrons have a localized behavior in agreement with the increase of the molar magnetic susceptibility χ(T) observed for 0.64 ≤ x ≤ 0.7 by Igarashi et al [START_REF] Igarashi | Precise Control of Na Content in the Layered Cobaltate γ-Na x CoO 2[END_REF] . It is reasonable to believe then that new holes are mainly formed in the a 1g band of the electronic structure (Figure 2). In Region II (0.5 x 0.67), the NMR shift significantly decreases when x decreases despite the further oxidation of the Co 3+ (diamagnetic) ions to Co 4+ ions (paramagnetic).

Although, those samples are still in the "Curie-Weiss metal" as reported by Foo et al. [START_REF] Foo | Commensurability, and Metallicity in the Phase Diagram of the Layered Na x CoO 2[END_REF] , a moderate increase of χ(T) was observed for 0.52 ≤ x ≤ 0.58 [START_REF] Carlier | Sodium Ion Mobility in Na X CoO 2 (0.6 < X < 0.75) Cobaltites Studied by 23 Na MAS NMR[END_REF] with absolute value rather close to Pauli paramagnetic state. Since the observed shift is not constant, but decreases for 0.5 x 0.67, we believe that the evolution of the local geometry is the predominating effect here. The increase of the interslab space results in a weaker 3d Co n+ -2p O-3s Na orbital overlap and thus to a weaker electronic spin transfer to Na nuclei that is also associated to a progressive change in the localization of new holes from the a 1g to the e g ' band. In Region III (0.33 x 0.5), an important decrease of the NMR shifts, compared to the values in the Region II is observed. This can be interpreted by the change of physical properties of the P2-Na x CoO 2 phase from "Weiss metal" to "Pauli metal" that exhibits only delocalized electrons. The shift here is then mainly governed by the Knight interaction (the Na 3s orbital participation to the Knight shift seems to be almost constant during the Na + deintercalation). It is reasonable to believe then that new holes are mainly formed in the e g ' band.

Finally, the [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR signal of the phase prepared at high voltage (V = 4.6 V, x = 0.25) is largely broadened and three contributions can be observed. The presence of these new Na + environments is assigned to the formation of the "OP" disordered phase due to slabs glidings and agrees with the observation reported by Clément et al. [START_REF] Clément | Structurally Stable Mg-Doped P2-Na 2/3 Mn 1-y Mg y O 2 Sodium-Ion Battery Cathodes with High Rate Performance: Insights from Electrochemical, NMR and Diffraction Studies[END_REF] for Na x MnO 2 with x 0.2. Na + ions might be located in O-type or P-type interslab space.

In order to discuss the reversibility of the sodium deintercalation process, when the battery is cycled up to 4.6 V, XRD patterns and NMR spectra have been also recorded on materials recovered at 2.4 V vs Na + /Na after a charge at 4.3 V or at 4.6 V (Figure SI-13). No significant changes at the mean and local structure is observed despite a real difference in the electrochemical discharge curves (Figure 3), showing that the material did not undergo severe structure damage while the battery reached 4.6 V. Nevertheless, a much larger signal intensity is observed in the 0 ppm region indicating the formation of a diamagnetic phase, probably amorphous as not seen by XRD, that would result from a thick cathode electrolyte interphase (CEI) formation. This indicates that the high polarization observed during the first discharge while charging up to 4.6 V is mainly due to kinetics effects as discussed earlier since the electrochemical study shows a good reversibility at the second discharge. This kinetics limitation can either be due to the thick CEI that might be unstable and involved in a dynamic evolution of dissolution-deposition over charge-discharge cycles as reported for some layered oxides [START_REF] Zhang | Dynamic Evolution of Cathode Electrolyte Interphase (CEI) on High Voltage LiCoO 2 Cathode and Its Interaction with Li Anode[END_REF] and/or to the formation of the "OP" phase predominantly at the surface of the particles that would lead to a lower kinetics of Na + reintercalation or "asymmetric" (more Na+ reintercalated in the "P"type layers before the sheets are gliding back to P2).

Conclusion

Using Synchrotron XRD, we studied the P2-Na x CoO 2 phase diagram upon charge with a specific attention to the structural modifications occurring at high voltage: the presence of an ordered phase for x = 1/3 is observed above 4.3 V vs Na + /Na, the P2 stacking becomes unstable and slab glidings occur to form a disordered structure with a stacking of O-type and P-type interslab space. [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR has been used to investigate the local structure and electronic property evolution upon charge. This study reveals a complex behavior due to the presence of localized and delocalized electrons in the material resulting in a Fermi contact interaction and Knight interaction, respectively, that co-exist here with relative amount depending on the Na content.

The [START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na NMR shift evolution versus Na content was interpreted by considering the predominant influence of structural factors (linked to the strength of Co-O-Na orbital overlap and thus to the efficiency of spin transfer mechanism) or electronic (such as the formal quantity of paramagnetic Co 4+ ions, and the relative amount of localized and delocalized electrons).

Unfortunately Density Functional Theory (DFT) calculations to support the discussion of the shift evolution as we have done previously [START_REF] Carlier | Understanding the NMR Shifts in Paramagnetic Transition Metal Oxides Using Density Functional Theory Calculations[END_REF][START_REF] Castets | Multinuclear NMR and DFT Calculations on the LiFePO 4 •OH and FePO 4 •H 2 O Homeotypic Phases[END_REF][START_REF] Bamine | Understanding Local Defects in Li-Ion Battery Electrodes through Combined DFT/NMR Studies: Application to LiVPO 4 F[END_REF][START_REF] Nguyen | Density Functional Theory-Assisted 31 P and 23 Na Magic-Angle Spinning Nuclear Magnetic Resonance Study of the Na 3 V 2 (PO 4 ) 2 F 3 -Na 3 V 2 (PO 4 ) 2 FO 2 Solid Solution: Unraveling Its Local and Electronic Structures[END_REF] could not be used here. Indeed, modelling "Curie-Weiss metal" can be satisfactory done using DFT. For example, neither Generalized Gradient Approximation (GGA) nor GGA+U allow a correct description of the electronic structure of such layered cobalt oxides that exhibits localized and delocalized electrons. In GGA, the inherent self-interaction error has a detrimental effect on localized transition-metal d orbitals and the addition of a Hubbard U correction has been shown to yield some unphysical results for layered cobalt oxide systems as incorrect phase diagram for the P2-Na x CoO 2 system [START_REF] Hinuma | Temperature-Concentration Phase Diagram of P2 -Na x CoO 2 from First-Principles Calculations[END_REF] .

Finally, we could interpret the large polarization observed upon discharge while cycling up to 4.6V by a kinetic limitation that can either be due to the formation of a thick CEI at high voltage, probably unstable upon discharge and/or to the formation of the "OP" phase predominantly at the surface.

ASSOCIATED CONTENT Supporting Information

Experimental

Figure 2 :

 2 Figure 2: Schematic representation of the band structure of the Co ions in layered alkali cobaltates (a) without and (b) with considering the distortion of the CoO 6 octahedra perpendicularly to the CoO 2 layers. The angle represents the angle between the z axis parallel to the c hex parameter and a Co-O bond direction. For a regular octahedral = 54.7°, whereas ~ 61.2° for P2-.

  ) looks much smoother without clear voltage drops compared to the first discharge curve from 4.3V and is located at lower voltage. Nevertheless, the voltage jumps reappear clearly during the following second charge indicating that the P2 structure is maintained. Then upon further cycling, all discharges do not exhibit any voltage steps whereas they are clearly observed upon charge. The origin of this behavior could be due to a strong difference in kinetics of the mechanisms occurring upon discharge compared to the ones involved upon charge. Although still unclear, this observation will be discussed a bit more below. When cycling between 1.5 and 4.6 V, a rapid capacity fading is also observed with only 78 % retention of the capacity after 40 cycles (FigureSI-4). Note that we indicated on Figure3by dash lines, the selected voltages used to prepare the different P2-Na x CoO 2 phases used for the ex-situ[START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR and XRD studies performed on the same samples as described in the experimental section.

Figure 3 :Figure 4 .

 34 Figure 3: a) Two first galvanostatic cycles at C/20 of Na(s)//P2-Na x CoO 2 batteries in the 1.5-4.3 V range The colored dashed lines correspond to the voltage applied to synthetize the different Na x CoO 2 phases studied by ex-situ XRD and NMR. b) Four first galvanostatic cycles at C/20 of Na(s)//P2-Na x CoO 2 batteries in the 1.5 V-4.6 V range.Evolution of the average structure upon cyclingIn addition to our previous work carried out by in-situ XRD[START_REF] Berthelot | Electrochemical Investigation of the P2-Na x CoO 2 Phase Diagram[END_REF] , we could investigate the evolution

Figure 4 (

 4 Figure 4 (two columns): a) XRD patterns obtained by operando SXRD on the P2-Na x CoO 2 system on the first charge from 1.5 V to 4.6 V at C/10 in the 5 -30° range. The colored lines represent the XRD patterns of the peculiar phases Na 2/3 CoO 2 , Na 4/7 CoO 2 , Na 1/2 CoO 2 and Na 1/3 CoO 2 . b) Zoom in the 16.3 -18.5° range. c) Corresponding galvanostatic cycling at C/10 of the Na(s)//P2-Na x CoO 2 battery.

Figure 5 ( 3 .

 53 Figure 5 (two columns): a) XRD patterns of stabilized P2-Na x CoO 2 phases by potentiostatic steps at different voltages from 4.1 V to 4.8 V for 8.3 < 2 < 10.3° b) for 16 < 2 < 27°.

Figure 7a gives

  Figure 7a gives the evolution of the O-Na-O interslab and CoO 2 slab thicknesses. The O-Na-O

Figure 6 :

 6 Figure 6: Evolution of c hex lattice parameter as a function of x. The black crosses represent the lattice parameters extracted from the SXRD patterns and the red squares are the values calculated with Rietveld refinements from the ex-situ laboratory XRD patterns

Figure 7 :

 7 Figure 7: a) Evolution of the O-Na-O layers and CoO 2 slab thickness for P2-Na x CoO 2 (insert: schematic representation of the P2-stacking). Comparison of the different distances in b) P2-Na 0.67 CoO 2 and c) P2-Na 0.33 CoO 2

12 .

 12 It appears that the quadrupolar interaction decreases as the sodium content decreases and the isotropic shifts varies in different manner depending of the Na content as discussed in the following. The evolution of the mean quadrupolar constant might be due to several factors occurring upon Na + deintercalation associated to Co ions oxidation: evolution of the local geometry around Na + ions, evolution of the hopping rate between Na e and Na f and relative site occupancy and evolution of the Co-O covalency and partial charges.

Figure 8 :

 8 Figure 8: Evolution of 23 Na MAS NMR signals of the P2-Na x CoO 2 phases using a 7.05 T magnetic field ( = 30 kHz) and single pulse sequences. The signals intensities are arbitrary chosen to better visualize the shifts and the shape evolution.

  ), simultaneously the CoO 6 octahedra are compressed and exhibit shorter Co-O distances due to the increase of the Co-O bond covalency. Overall, we expect from this factor a lower efficiency of the Co n+ -O-Na spin transfer mechanism from Co to Na nucleus as x decreases if the Co n+ mean charge is maintained constant and thus a decrease of the Fermi contact shift. Additionally, the local electronic structure evolves as x decreases considering the oxidation of diamagnetic LS-Co 3+ (t 2g 6 e g 0 ) into paramagnetic LS-Co 4+ (t 2g 5 e g 0 ): the more paramagnetic is the mean Co n+ ion, the larger will be the Co n+ -O-Na spin transfer mechanism given local geometry, i.e, if the Co-O-Na distances and angles are maintained constant and thus an increase of the Fermi contact shift. Finally, the "Curie-Weiss metal" regime to a "Pauli metal" regime transition occurs at x = 0.5 15 due to the different relative a 1g and e' g orbitals occupancy and/or less localization effect of the charge on Co ions induced by adjacent Na + layers. An evolution from a Fermi contact shift to a Knight shift is therefore expected.

Figure 9 .

 9 Figure 9. Shift evolution and description of the phenomena responsible 23 Na MAS NMR isotropic as a function of the Na + content

  and refined XRD patterns and SEM images of the pristine material; Principle of the electrochemical preparation of the Na x CoO 2 compounds studied; Evolution of the specific capacity in the 1.5 -4.3 V range and in the 1.5 -4.6 V range at C/20; XRD refinement results on P2-Na x CoO 2 peculiar phases with Na+/vacancy ordering; In situ Potentiostatic measurements on P2-Na x CoO 2 as a function of time and corresponding XRD patterns; Representation of the CoO 2stacking in an ideal P2 structure and in a P2 structure with an O-type stacking fault after sheets glidings ; Ex-situ capillary laboratory XRD patterns of the Na x CoO 2 phases studied by[START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR and corresponding structural parameters after Rietveld refinements; fitted[START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na NMR parameters; XRD patterns[START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] Na MAS NMR spectra of the pristine P2-phase and two P2-Na x CoO 2 phases recovered at 2.4 V after a charge at 4.3 and 4.6 V (vs Na + /Na).

Na MAS NMR (Magic Angle Spinning Nuclear Magnetic Resonance) is a key tool to investigate both the local structure and electronic structure in such paramagnetic or metallic materials thank the hyperfine interactions (electron spin-nucleus spin coupling)[START_REF] Gotoh | 23 Na Solid-State NMR Analyses for Na-Ion Batteries and Materials[END_REF] . In

Acknowledgments

This work beneficiates from financial support from the Fond Franco-Canadien pour la Recherche (FFCR). ALBA synchrotron is acknowledged for granting beamtime through proposal no 2017092426. We are also grateful to Mathieu Duttine for its technical support.

AUTHOR INFORMATION

Corresponding Author

Dany Carlier -Univ. Bordeaux, CNRS, Bordeaux INP, ICMCB, UMR 5026, F-33600 Pessac, France; orcid.org/0000-0002-5086-4363

Authors